Datasets:
PrincetonPLI
/
LongProc

Dataset card Data Studio Files
xet
Community
1
id
stringlengths
16
18
input_prompt
stringlengths
16.4k
16.4k
reference_output
stringlengths
881
1.4k
metadata
stringlengths
94
104
countdown_0.5k_0
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [21, 16, 17, 26] Target: 46
# Search Procedure Initial number set: [21, 16, 17, 26], target: 46. Options for choosing two numbers: [(21, 16), (21, 17), (21, 26), (16, 17), (16, 26), (17, 26)]. |- Pick two numbers (21, 16) (numbers left: [17, 26]). Try possible operations. |- Try 21 + 16 = 37. Add 37 to the number set. Current number set: [37, 17, 26], target: 46. Options for choosing two numbers: [(37, 17), (37, 26), (17, 26)]. |- Pick two numbers (37, 17) (numbers left: [26]). Try possible operations. |- Try 37 + 17 = 54. Add 54 to the number set. Current number set: [54, 26], target: 46, just two numbers left. |- Try 54 + 26 = 80. Evaluate 80 != 46, drop this branch. |- Try 54 - 26 = 28. Evaluate 28 != 46, drop this branch. |- Try 54 * 26 = 1404. Evaluate 1404 != 46, drop this branch. |- Try 54 / 26 = 2.1. 2.1 is a decimal, drop this branch. |- Try 37 - 17 = 20. Add 20 to the number set. Current number set: [20, 26], target: 46, just two numbers left. |- Try 26 + 20 = 46. Evaluate 46 == 46, target found! Now we have found the target, let's trace back the solution: Final step: 26 + 20 = 46 The step before: 37 - 17 = 20 The first step: 21 + 16 = 37 Output the solution in the required format: <Solution> 21 + 16 = 37 37 - 17 = 20 26 + 20 = 46 </Solution>
{"nums": [21, 16, 17, 26], "target": 46, "solution": ["21 + 16 = 37", "37 - 17 = 20", "26 + 20 = 46"]}
countdown_0.5k_1
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [12, 48, 22, 27] Target: 11
# Search Procedure Initial number set: [12, 48, 22, 27], target: 11. Options for choosing two numbers: [(12, 48), (12, 22), (12, 27), (48, 22), (48, 27), (22, 27)]. |- Pick two numbers (12, 48) (numbers left: [22, 27]). Try possible operations. |- Try 48 + 12 = 60. Add 60 to the number set. Current number set: [60, 22, 27], target: 11. Options for choosing two numbers: [(60, 22), (60, 27), (22, 27)]. |- Pick two numbers (60, 22) (numbers left: [27]). Try possible operations. |- Try 60 + 22 = 82. Add 82 to the number set. Current number set: [82, 27], target: 11, just two numbers left. |- Try 82 + 27 = 109. Evaluate 109 != 11, drop this branch. |- Try 82 - 27 = 55. Evaluate 55 != 11, drop this branch. |- Try 82 * 27 = 2214. 2214 exceeds the maximum intermediate result, drop this branch. |- Try 82 / 27 = 3.0. 3.0 is a decimal, drop this branch. |- Try 60 - 22 = 38. Add 38 to the number set. Current number set: [38, 27], target: 11, just two numbers left. |- Try 38 + 27 = 65. Evaluate 65 != 11, drop this branch. |- Try 38 - 27 = 11. Evaluate 11 == 11, target found! Now we have found the target, let's trace back the solution: Final step: 38 - 27 = 11 The step before: 60 - 22 = 38 The first step: 48 + 12 = 60 Output the solution in the required format: <Solution> 48 + 12 = 60 60 - 22 = 38 38 - 27 = 11 </Solution>
{"nums": [12, 48, 22, 27], "target": 11, "solution": ["48 + 12 = 60", "60 - 22 = 38", "38 - 27 = 11"]}
countdown_0.5k_2
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [33, 20, 43, 27] Target: 37
# Search Procedure Initial number set: [33, 20, 43, 27], target: 37. Options for choosing two numbers: [(33, 20), (33, 43), (33, 27), (20, 43), (20, 27), (43, 27)]. |- Pick two numbers (33, 20) (numbers left: [43, 27]). Try possible operations. |- Try 33 + 20 = 53. Add 53 to the number set. Current number set: [53, 43, 27], target: 37. Options for choosing two numbers: [(53, 43), (53, 27), (43, 27)]. |- Pick two numbers (53, 43) (numbers left: [27]). Try possible operations. |- Try 53 + 43 = 96. Add 96 to the number set. Current number set: [96, 27], target: 37, just two numbers left. |- Try 96 + 27 = 123. Evaluate 123 != 37, drop this branch. |- Try 96 - 27 = 69. Evaluate 69 != 37, drop this branch. |- Try 96 * 27 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 96 / 27 = 3.6. 3.6 is a decimal, drop this branch. |- Try 53 - 43 = 10. Add 10 to the number set. Current number set: [10, 27], target: 37, just two numbers left. |- Try 27 + 10 = 37. Evaluate 37 == 37, target found! Now we have found the target, let's trace back the solution: Final step: 27 + 10 = 37 The step before: 53 - 43 = 10 The first step: 33 + 20 = 53 Output the solution in the required format: <Solution> 33 + 20 = 53 53 - 43 = 10 27 + 10 = 37 </Solution>
{"nums": [33, 20, 43, 27], "target": 37, "solution": ["33 + 20 = 53", "53 - 43 = 10", "27 + 10 = 37"]}
countdown_0.5k_3
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [39, 21, 49, 29] Target: 18
# Search Procedure Initial number set: [39, 21, 49, 29], target: 18. Options for choosing two numbers: [(39, 21), (39, 49), (39, 29), (21, 49), (21, 29), (49, 29)]. |- Pick two numbers (39, 21) (numbers left: [49, 29]). Try possible operations. |- Try 39 + 21 = 60. Add 60 to the number set. Current number set: [60, 49, 29], target: 18. Options for choosing two numbers: [(60, 49), (60, 29), (49, 29)]. |- Pick two numbers (60, 49) (numbers left: [29]). Try possible operations. |- Try 60 + 49 = 109. Add 109 to the number set. Current number set: [109, 29], target: 18, just two numbers left. |- Try 109 + 29 = 138. Evaluate 138 != 18, drop this branch. |- Try 109 - 29 = 80. Evaluate 80 != 18, drop this branch. |- Try 109 * 29 = 3161. 3161 exceeds the maximum intermediate result, drop this branch. |- Try 109 / 29 = 3.8. 3.8 is a decimal, drop this branch. |- Try 60 - 49 = 11. Add 11 to the number set. Current number set: [11, 29], target: 18, just two numbers left. |- Try 29 + 11 = 40. Evaluate 40 != 18, drop this branch. |- Try 29 - 11 = 18. Evaluate 18 == 18, target found! Now we have found the target, let's trace back the solution: Final step: 29 - 11 = 18 The step before: 60 - 49 = 11 The first step: 39 + 21 = 60 Output the solution in the required format: <Solution> 39 + 21 = 60 60 - 49 = 11 29 - 11 = 18 </Solution>
{"nums": [39, 21, 49, 29], "target": 18, "solution": ["39 + 21 = 60", "60 - 49 = 11", "29 - 11 = 18"]}
countdown_0.5k_4
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [5, 4, 11, 20] Target: 22
# Search Procedure Initial number set: [5, 4, 11, 20], target: 22. Options for choosing two numbers: [(5, 4), (5, 11), (5, 20), (4, 11), (4, 20), (11, 20)]. |- Pick two numbers (5, 4) (numbers left: [11, 20]). Try possible operations. |- Try 5 + 4 = 9. Add 9 to the number set. Current number set: [9, 11, 20], target: 22. Options for choosing two numbers: [(9, 11), (9, 20), (11, 20)]. |- Pick two numbers (9, 11) (numbers left: [20]). Try possible operations. |- Try 11 + 9 = 20. Add 20 to the number set. Current number set: [20, 20], target: 22, just two numbers left. |- Try 20 + 20 = 40. Evaluate 40 != 22, drop this branch. |- Try 20 - 20 = 0. Evaluate 0 != 22, drop this branch. |- Try 20 * 20 = 400. Evaluate 400 != 22, drop this branch. |- Try 20 / 20 = 1. Evaluate 1 != 22, drop this branch. |- Try 11 - 9 = 2. Add 2 to the number set. Current number set: [2, 20], target: 22, just two numbers left. |- Try 20 + 2 = 22. Evaluate 22 == 22, target found! Now we have found the target, let's trace back the solution: Final step: 20 + 2 = 22 The step before: 11 - 9 = 2 The first step: 5 + 4 = 9 Output the solution in the required format: <Solution> 5 + 4 = 9 11 - 9 = 2 20 + 2 = 22 </Solution>
{"nums": [5, 4, 11, 20], "target": 22, "solution": ["5 + 4 = 9", "11 - 9 = 2", "20 + 2 = 22"]}
countdown_0.5k_5
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [31, 23, 22, 36] Target: 40
# Search Procedure Initial number set: [31, 23, 22, 36], target: 40. Options for choosing two numbers: [(31, 23), (31, 22), (31, 36), (23, 22), (23, 36), (22, 36)]. |- Pick two numbers (31, 23) (numbers left: [22, 36]). Try possible operations. |- Try 31 + 23 = 54. Add 54 to the number set. Current number set: [54, 22, 36], target: 40. Options for choosing two numbers: [(54, 22), (54, 36), (22, 36)]. |- Pick two numbers (54, 22) (numbers left: [36]). Try possible operations. |- Try 54 + 22 = 76. Add 76 to the number set. Current number set: [76, 36], target: 40, just two numbers left. |- Try 76 + 36 = 112. Evaluate 112 != 40, drop this branch. |- Try 76 - 36 = 40. Evaluate 40 == 40, target found! Now we have found the target, let's trace back the solution: Final step: 76 - 36 = 40 The step before: 54 + 22 = 76 The first step: 31 + 23 = 54 Output the solution in the required format: <Solution> 31 + 23 = 54 54 + 22 = 76 76 - 36 = 40 </Solution>
{"nums": [31, 23, 22, 36], "target": 40, "solution": ["31 + 23 = 54", "54 + 22 = 76", "76 - 36 = 40"]}
countdown_0.5k_6
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [26, 42, 6, 22] Target: 40
# Search Procedure Initial number set: [26, 42, 6, 22], target: 40. Options for choosing two numbers: [(26, 42), (26, 6), (26, 22), (42, 6), (42, 22), (6, 22)]. |- Pick two numbers (26, 42) (numbers left: [6, 22]). Try possible operations. |- Try 42 + 26 = 68. Add 68 to the number set. Current number set: [68, 6, 22], target: 40. Options for choosing two numbers: [(68, 6), (68, 22), (6, 22)]. |- Pick two numbers (68, 6) (numbers left: [22]). Try possible operations. |- Try 68 + 6 = 74. Add 74 to the number set. Current number set: [74, 22], target: 40, just two numbers left. |- Try 74 + 22 = 96. Evaluate 96 != 40, drop this branch. |- Try 74 - 22 = 52. Evaluate 52 != 40, drop this branch. |- Try 74 * 22 = 1628. Evaluate 1628 != 40, drop this branch. |- Try 74 / 22 = 3.4. 3.4 is a decimal, drop this branch. |- Try 68 - 6 = 62. Add 62 to the number set. Current number set: [62, 22], target: 40, just two numbers left. |- Try 62 + 22 = 84. Evaluate 84 != 40, drop this branch. |- Try 62 - 22 = 40. Evaluate 40 == 40, target found! Now we have found the target, let's trace back the solution: Final step: 62 - 22 = 40 The step before: 68 - 6 = 62 The first step: 42 + 26 = 68 Output the solution in the required format: <Solution> 42 + 26 = 68 68 - 6 = 62 62 - 22 = 40 </Solution>
{"nums": [26, 42, 6, 22], "target": 40, "solution": ["42 + 26 = 68", "68 - 6 = 62", "62 - 22 = 40"]}
countdown_0.5k_7
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [19, 2, 43, 26] Target: 38
# Search Procedure Initial number set: [19, 2, 43, 26], target: 38. Options for choosing two numbers: [(19, 2), (19, 43), (19, 26), (2, 43), (2, 26), (43, 26)]. |- Pick two numbers (19, 2) (numbers left: [43, 26]). Try possible operations. |- Try 19 + 2 = 21. Add 21 to the number set. Current number set: [21, 43, 26], target: 38. Options for choosing two numbers: [(21, 43), (21, 26), (43, 26)]. |- Pick two numbers (21, 43) (numbers left: [26]). Try possible operations. |- Try 43 + 21 = 64. Add 64 to the number set. Current number set: [64, 26], target: 38, just two numbers left. |- Try 64 + 26 = 90. Evaluate 90 != 38, drop this branch. |- Try 64 - 26 = 38. Evaluate 38 == 38, target found! Now we have found the target, let's trace back the solution: Final step: 64 - 26 = 38 The step before: 43 + 21 = 64 The first step: 19 + 2 = 21 Output the solution in the required format: <Solution> 19 + 2 = 21 43 + 21 = 64 64 - 26 = 38 </Solution>
{"nums": [19, 2, 43, 26], "target": 38, "solution": ["19 + 2 = 21", "43 + 21 = 64", "64 - 26 = 38"]}
countdown_0.5k_8
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [4, 39, 48, 32] Target: 27
# Search Procedure Initial number set: [4, 39, 48, 32], target: 27. Options for choosing two numbers: [(4, 39), (4, 48), (4, 32), (39, 48), (39, 32), (48, 32)]. |- Pick two numbers (4, 39) (numbers left: [48, 32]). Try possible operations. |- Try 39 + 4 = 43. Add 43 to the number set. Current number set: [43, 48, 32], target: 27. Options for choosing two numbers: [(43, 48), (43, 32), (48, 32)]. |- Pick two numbers (43, 48) (numbers left: [32]). Try possible operations. |- Try 48 + 43 = 91. Add 91 to the number set. Current number set: [91, 32], target: 27, just two numbers left. |- Try 91 + 32 = 123. Evaluate 123 != 27, drop this branch. |- Try 91 - 32 = 59. Evaluate 59 != 27, drop this branch. |- Try 91 * 32 = 2912. 2912 exceeds the maximum intermediate result, drop this branch. |- Try 91 / 32 = 2.8. 2.8 is a decimal, drop this branch. |- Try 48 - 43 = 5. Add 5 to the number set. Current number set: [5, 32], target: 27, just two numbers left. |- Try 32 + 5 = 37. Evaluate 37 != 27, drop this branch. |- Try 32 - 5 = 27. Evaluate 27 == 27, target found! Now we have found the target, let's trace back the solution: Final step: 32 - 5 = 27 The step before: 48 - 43 = 5 The first step: 39 + 4 = 43 Output the solution in the required format: <Solution> 39 + 4 = 43 48 - 43 = 5 32 - 5 = 27 </Solution>
{"nums": [4, 39, 48, 32], "target": 27, "solution": ["39 + 4 = 43", "48 - 43 = 5", "32 - 5 = 27"]}
countdown_0.5k_9
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [39, 2, 4, 22] Target: 15
# Search Procedure Initial number set: [39, 2, 4, 22], target: 15. Options for choosing two numbers: [(39, 2), (39, 4), (39, 22), (2, 4), (2, 22), (4, 22)]. |- Pick two numbers (39, 2) (numbers left: [4, 22]). Try possible operations. |- Try 39 + 2 = 41. Add 41 to the number set. Current number set: [41, 4, 22], target: 15. Options for choosing two numbers: [(41, 4), (41, 22), (4, 22)]. |- Pick two numbers (41, 4) (numbers left: [22]). Try possible operations. |- Try 41 + 4 = 45. Add 45 to the number set. Current number set: [45, 22], target: 15, just two numbers left. |- Try 45 + 22 = 67. Evaluate 67 != 15, drop this branch. |- Try 45 - 22 = 23. Evaluate 23 != 15, drop this branch. |- Try 45 * 22 = 990. Evaluate 990 != 15, drop this branch. |- Try 45 / 22 = 2.0. 2.0 is a decimal, drop this branch. |- Try 41 - 4 = 37. Add 37 to the number set. Current number set: [37, 22], target: 15, just two numbers left. |- Try 37 + 22 = 59. Evaluate 59 != 15, drop this branch. |- Try 37 - 22 = 15. Evaluate 15 == 15, target found! Now we have found the target, let's trace back the solution: Final step: 37 - 22 = 15 The step before: 41 - 4 = 37 The first step: 39 + 2 = 41 Output the solution in the required format: <Solution> 39 + 2 = 41 41 - 4 = 37 37 - 22 = 15 </Solution>
{"nums": [39, 2, 4, 22], "target": 15, "solution": ["39 + 2 = 41", "41 - 4 = 37", "37 - 22 = 15"]}
countdown_0.5k_10
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [34, 3, 43, 22] Target: 28
# Search Procedure Initial number set: [34, 3, 43, 22], target: 28. Options for choosing two numbers: [(34, 3), (34, 43), (34, 22), (3, 43), (3, 22), (43, 22)]. |- Pick two numbers (34, 3) (numbers left: [43, 22]). Try possible operations. |- Try 34 + 3 = 37. Add 37 to the number set. Current number set: [37, 43, 22], target: 28. Options for choosing two numbers: [(37, 43), (37, 22), (43, 22)]. |- Pick two numbers (37, 43) (numbers left: [22]). Try possible operations. |- Try 43 + 37 = 80. Add 80 to the number set. Current number set: [80, 22], target: 28, just two numbers left. |- Try 80 + 22 = 102. Evaluate 102 != 28, drop this branch. |- Try 80 - 22 = 58. Evaluate 58 != 28, drop this branch. |- Try 80 * 22 = 1760. Evaluate 1760 != 28, drop this branch. |- Try 80 / 22 = 3.6. 3.6 is a decimal, drop this branch. |- Try 43 - 37 = 6. Add 6 to the number set. Current number set: [6, 22], target: 28, just two numbers left. |- Try 22 + 6 = 28. Evaluate 28 == 28, target found! Now we have found the target, let's trace back the solution: Final step: 22 + 6 = 28 The step before: 43 - 37 = 6 The first step: 34 + 3 = 37 Output the solution in the required format: <Solution> 34 + 3 = 37 43 - 37 = 6 22 + 6 = 28 </Solution>
{"nums": [34, 3, 43, 22], "target": 28, "solution": ["34 + 3 = 37", "43 - 37 = 6", "22 + 6 = 28"]}
countdown_0.5k_11
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [2, 24, 22, 44] Target: 40
# Search Procedure Initial number set: [2, 24, 22, 44], target: 40. Options for choosing two numbers: [(2, 24), (2, 22), (2, 44), (24, 22), (24, 44), (22, 44)]. |- Pick two numbers (2, 24) (numbers left: [22, 44]). Try possible operations. |- Try 24 + 2 = 26. Add 26 to the number set. Current number set: [26, 22, 44], target: 40. Options for choosing two numbers: [(26, 22), (26, 44), (22, 44)]. |- Pick two numbers (26, 22) (numbers left: [44]). Try possible operations. |- Try 26 + 22 = 48. Add 48 to the number set. Current number set: [48, 44], target: 40, just two numbers left. |- Try 48 + 44 = 92. Evaluate 92 != 40, drop this branch. |- Try 48 - 44 = 4. Evaluate 4 != 40, drop this branch. |- Try 48 * 44 = 2112. 2112 exceeds the maximum intermediate result, drop this branch. |- Try 48 / 44 = 1.1. 1.1 is a decimal, drop this branch. |- Try 26 - 22 = 4. Add 4 to the number set. Current number set: [4, 44], target: 40, just two numbers left. |- Try 44 + 4 = 48. Evaluate 48 != 40, drop this branch. |- Try 44 - 4 = 40. Evaluate 40 == 40, target found! Now we have found the target, let's trace back the solution: Final step: 44 - 4 = 40 The step before: 26 - 22 = 4 The first step: 24 + 2 = 26 Output the solution in the required format: <Solution> 24 + 2 = 26 26 - 22 = 4 44 - 4 = 40 </Solution>
{"nums": [2, 24, 22, 44], "target": 40, "solution": ["24 + 2 = 26", "26 - 22 = 4", "44 - 4 = 40"]}
countdown_0.5k_12
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [35, 34, 32, 9] Target: 46
# Search Procedure Initial number set: [35, 34, 32, 9], target: 46. Options for choosing two numbers: [(35, 34), (35, 32), (35, 9), (34, 32), (34, 9), (32, 9)]. |- Pick two numbers (35, 34) (numbers left: [32, 9]). Try possible operations. |- Try 35 + 34 = 69. Add 69 to the number set. Current number set: [69, 32, 9], target: 46. Options for choosing two numbers: [(69, 32), (69, 9), (32, 9)]. |- Pick two numbers (69, 32) (numbers left: [9]). Try possible operations. |- Try 69 + 32 = 101. Add 101 to the number set. Current number set: [101, 9], target: 46, just two numbers left. |- Try 101 + 9 = 110. Evaluate 110 != 46, drop this branch. |- Try 101 - 9 = 92. Evaluate 92 != 46, drop this branch. |- Try 101 * 9 = 909. Evaluate 909 != 46, drop this branch. |- Try 101 / 9 = 11.2. 11.2 is a decimal, drop this branch. |- Try 69 - 32 = 37. Add 37 to the number set. Current number set: [37, 9], target: 46, just two numbers left. |- Try 37 + 9 = 46. Evaluate 46 == 46, target found! Now we have found the target, let's trace back the solution: Final step: 37 + 9 = 46 The step before: 69 - 32 = 37 The first step: 35 + 34 = 69 Output the solution in the required format: <Solution> 35 + 34 = 69 69 - 32 = 37 37 + 9 = 46 </Solution>
{"nums": [35, 34, 32, 9], "target": 46, "solution": ["35 + 34 = 69", "69 - 32 = 37", "37 + 9 = 46"]}
countdown_0.5k_13
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [21, 6, 26, 43] Target: 42
# Search Procedure Initial number set: [21, 6, 26, 43], target: 42. Options for choosing two numbers: [(21, 6), (21, 26), (21, 43), (6, 26), (6, 43), (26, 43)]. |- Pick two numbers (21, 6) (numbers left: [26, 43]). Try possible operations. |- Try 21 + 6 = 27. Add 27 to the number set. Current number set: [27, 26, 43], target: 42. Options for choosing two numbers: [(27, 26), (27, 43), (26, 43)]. |- Pick two numbers (27, 26) (numbers left: [43]). Try possible operations. |- Try 27 + 26 = 53. Add 53 to the number set. Current number set: [53, 43], target: 42, just two numbers left. |- Try 53 + 43 = 96. Evaluate 96 != 42, drop this branch. |- Try 53 - 43 = 10. Evaluate 10 != 42, drop this branch. |- Try 53 * 43 = 2279. 2279 exceeds the maximum intermediate result, drop this branch. |- Try 53 / 43 = 1.2. 1.2 is a decimal, drop this branch. |- Try 27 - 26 = 1. Add 1 to the number set. Current number set: [1, 43], target: 42, just two numbers left. |- Try 43 + 1 = 44. Evaluate 44 != 42, drop this branch. |- Try 43 - 1 = 42. Evaluate 42 == 42, target found! Now we have found the target, let's trace back the solution: Final step: 43 - 1 = 42 The step before: 27 - 26 = 1 The first step: 21 + 6 = 27 Output the solution in the required format: <Solution> 21 + 6 = 27 27 - 26 = 1 43 - 1 = 42 </Solution>
{"nums": [21, 6, 26, 43], "target": 42, "solution": ["21 + 6 = 27", "27 - 26 = 1", "43 - 1 = 42"]}
countdown_0.5k_14
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [30, 9, 33, 27] Target: 33
# Search Procedure Initial number set: [30, 9, 33, 27], target: 33. Options for choosing two numbers: [(30, 9), (30, 33), (30, 27), (9, 33), (9, 27), (33, 27)]. |- Pick two numbers (30, 9) (numbers left: [33, 27]). Try possible operations. |- Try 30 + 9 = 39. Add 39 to the number set. Current number set: [39, 33, 27], target: 33. Options for choosing two numbers: [(39, 33), (39, 27), (33, 27)]. |- Pick two numbers (39, 33) (numbers left: [27]). Try possible operations. |- Try 39 + 33 = 72. Add 72 to the number set. Current number set: [72, 27], target: 33, just two numbers left. |- Try 72 + 27 = 99. Evaluate 99 != 33, drop this branch. |- Try 72 - 27 = 45. Evaluate 45 != 33, drop this branch. |- Try 72 * 27 = 1944. Evaluate 1944 != 33, drop this branch. |- Try 72 / 27 = 2.7. 2.7 is a decimal, drop this branch. |- Try 39 - 33 = 6. Add 6 to the number set. Current number set: [6, 27], target: 33, just two numbers left. |- Try 27 + 6 = 33. Evaluate 33 == 33, target found! Now we have found the target, let's trace back the solution: Final step: 27 + 6 = 33 The step before: 39 - 33 = 6 The first step: 30 + 9 = 39 Output the solution in the required format: <Solution> 30 + 9 = 39 39 - 33 = 6 27 + 6 = 33 </Solution>
{"nums": [30, 9, 33, 27], "target": 33, "solution": ["30 + 9 = 39", "39 - 33 = 6", "27 + 6 = 33"]}
countdown_0.5k_15
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [31, 14, 22, 20] Target: 47
# Search Procedure Initial number set: [31, 14, 22, 20], target: 47. Options for choosing two numbers: [(31, 14), (31, 22), (31, 20), (14, 22), (14, 20), (22, 20)]. |- Pick two numbers (31, 14) (numbers left: [22, 20]). Try possible operations. |- Try 31 + 14 = 45. Add 45 to the number set. Current number set: [45, 22, 20], target: 47. Options for choosing two numbers: [(45, 22), (45, 20), (22, 20)]. |- Pick two numbers (45, 22) (numbers left: [20]). Try possible operations. |- Try 45 + 22 = 67. Add 67 to the number set. Current number set: [67, 20], target: 47, just two numbers left. |- Try 67 + 20 = 87. Evaluate 87 != 47, drop this branch. |- Try 67 - 20 = 47. Evaluate 47 == 47, target found! Now we have found the target, let's trace back the solution: Final step: 67 - 20 = 47 The step before: 45 + 22 = 67 The first step: 31 + 14 = 45 Output the solution in the required format: <Solution> 31 + 14 = 45 45 + 22 = 67 67 - 20 = 47 </Solution>
{"nums": [31, 14, 22, 20], "target": 47, "solution": ["31 + 14 = 45", "45 + 22 = 67", "67 - 20 = 47"]}
countdown_0.5k_16
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [15, 5, 22, 39] Target: 37
# Search Procedure Initial number set: [15, 5, 22, 39], target: 37. Options for choosing two numbers: [(15, 5), (15, 22), (15, 39), (5, 22), (5, 39), (22, 39)]. |- Pick two numbers (15, 5) (numbers left: [22, 39]). Try possible operations. |- Try 15 + 5 = 20. Add 20 to the number set. Current number set: [20, 22, 39], target: 37. Options for choosing two numbers: [(20, 22), (20, 39), (22, 39)]. |- Pick two numbers (20, 22) (numbers left: [39]). Try possible operations. |- Try 22 + 20 = 42. Add 42 to the number set. Current number set: [42, 39], target: 37, just two numbers left. |- Try 42 + 39 = 81. Evaluate 81 != 37, drop this branch. |- Try 42 - 39 = 3. Evaluate 3 != 37, drop this branch. |- Try 42 * 39 = 1638. Evaluate 1638 != 37, drop this branch. |- Try 42 / 39 = 1.1. 1.1 is a decimal, drop this branch. |- Try 22 - 20 = 2. Add 2 to the number set. Current number set: [2, 39], target: 37, just two numbers left. |- Try 39 + 2 = 41. Evaluate 41 != 37, drop this branch. |- Try 39 - 2 = 37. Evaluate 37 == 37, target found! Now we have found the target, let's trace back the solution: Final step: 39 - 2 = 37 The step before: 22 - 20 = 2 The first step: 15 + 5 = 20 Output the solution in the required format: <Solution> 15 + 5 = 20 22 - 20 = 2 39 - 2 = 37 </Solution>
{"nums": [15, 5, 22, 39], "target": 37, "solution": ["15 + 5 = 20", "22 - 20 = 2", "39 - 2 = 37"]}
countdown_0.5k_17
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [21, 19, 48, 45] Target: 37
# Search Procedure Initial number set: [21, 19, 48, 45], target: 37. Options for choosing two numbers: [(21, 19), (21, 48), (21, 45), (19, 48), (19, 45), (48, 45)]. |- Pick two numbers (21, 19) (numbers left: [48, 45]). Try possible operations. |- Try 21 + 19 = 40. Add 40 to the number set. Current number set: [40, 48, 45], target: 37. Options for choosing two numbers: [(40, 48), (40, 45), (48, 45)]. |- Pick two numbers (40, 48) (numbers left: [45]). Try possible operations. |- Try 48 + 40 = 88. Add 88 to the number set. Current number set: [88, 45], target: 37, just two numbers left. |- Try 88 + 45 = 133. Evaluate 133 != 37, drop this branch. |- Try 88 - 45 = 43. Evaluate 43 != 37, drop this branch. |- Try 88 * 45 = 3960. 3960 exceeds the maximum intermediate result, drop this branch. |- Try 88 / 45 = 2.0. 2.0 is a decimal, drop this branch. |- Try 48 - 40 = 8. Add 8 to the number set. Current number set: [8, 45], target: 37, just two numbers left. |- Try 45 + 8 = 53. Evaluate 53 != 37, drop this branch. |- Try 45 - 8 = 37. Evaluate 37 == 37, target found! Now we have found the target, let's trace back the solution: Final step: 45 - 8 = 37 The step before: 48 - 40 = 8 The first step: 21 + 19 = 40 Output the solution in the required format: <Solution> 21 + 19 = 40 48 - 40 = 8 45 - 8 = 37 </Solution>
{"nums": [21, 19, 48, 45], "target": 37, "solution": ["21 + 19 = 40", "48 - 40 = 8", "45 - 8 = 37"]}
countdown_0.5k_18
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [35, 33, 45, 27] Target: 50
# Search Procedure Initial number set: [35, 33, 45, 27], target: 50. Options for choosing two numbers: [(35, 33), (35, 45), (35, 27), (33, 45), (33, 27), (45, 27)]. |- Pick two numbers (35, 33) (numbers left: [45, 27]). Try possible operations. |- Try 35 + 33 = 68. Add 68 to the number set. Current number set: [68, 45, 27], target: 50. Options for choosing two numbers: [(68, 45), (68, 27), (45, 27)]. |- Pick two numbers (68, 45) (numbers left: [27]). Try possible operations. |- Try 68 + 45 = 113. Add 113 to the number set. Current number set: [113, 27], target: 50, just two numbers left. |- Try 113 + 27 = 140. Evaluate 140 != 50, drop this branch. |- Try 113 - 27 = 86. Evaluate 86 != 50, drop this branch. |- Try 113 * 27 = 3051. 3051 exceeds the maximum intermediate result, drop this branch. |- Try 113 / 27 = 4.2. 4.2 is a decimal, drop this branch. |- Try 68 - 45 = 23. Add 23 to the number set. Current number set: [23, 27], target: 50, just two numbers left. |- Try 27 + 23 = 50. Evaluate 50 == 50, target found! Now we have found the target, let's trace back the solution: Final step: 27 + 23 = 50 The step before: 68 - 45 = 23 The first step: 35 + 33 = 68 Output the solution in the required format: <Solution> 35 + 33 = 68 68 - 45 = 23 27 + 23 = 50 </Solution>
{"nums": [35, 33, 45, 27], "target": 50, "solution": ["35 + 33 = 68", "68 - 45 = 23", "27 + 23 = 50"]}
countdown_0.5k_19
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [26, 38, 8, 23] Target: 49
# Search Procedure Initial number set: [26, 38, 8, 23], target: 49. Options for choosing two numbers: [(26, 38), (26, 8), (26, 23), (38, 8), (38, 23), (8, 23)]. |- Pick two numbers (26, 38) (numbers left: [8, 23]). Try possible operations. |- Try 38 + 26 = 64. Add 64 to the number set. Current number set: [64, 8, 23], target: 49. Options for choosing two numbers: [(64, 8), (64, 23), (8, 23)]. |- Pick two numbers (64, 8) (numbers left: [23]). Try possible operations. |- Try 64 + 8 = 72. Add 72 to the number set. Current number set: [72, 23], target: 49, just two numbers left. |- Try 72 + 23 = 95. Evaluate 95 != 49, drop this branch. |- Try 72 - 23 = 49. Evaluate 49 == 49, target found! Now we have found the target, let's trace back the solution: Final step: 72 - 23 = 49 The step before: 64 + 8 = 72 The first step: 38 + 26 = 64 Output the solution in the required format: <Solution> 38 + 26 = 64 64 + 8 = 72 72 - 23 = 49 </Solution>
{"nums": [26, 38, 8, 23], "target": 49, "solution": ["38 + 26 = 64", "64 + 8 = 72", "72 - 23 = 49"]}
countdown_0.5k_20
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [6, 46, 7, 1] Target: 46
# Search Procedure Initial number set: [6, 46, 7, 1], target: 46. Options for choosing two numbers: [(6, 46), (6, 7), (6, 1), (46, 7), (46, 1), (7, 1)]. |- Pick two numbers (6, 46) (numbers left: [7, 1]). Try possible operations. |- Try 46 + 6 = 52. Add 52 to the number set. Current number set: [52, 7, 1], target: 46. Options for choosing two numbers: [(52, 7), (52, 1), (7, 1)]. |- Pick two numbers (52, 7) (numbers left: [1]). Try possible operations. |- Try 52 + 7 = 59. Add 59 to the number set. Current number set: [59, 1], target: 46, just two numbers left. |- Try 59 + 1 = 60. Evaluate 60 != 46, drop this branch. |- Try 59 - 1 = 58. Evaluate 58 != 46, drop this branch. |- Try 59 * 1 = 59. Evaluate 59 != 46, drop this branch. |- Try 59 / 1 = 59. Evaluate 59 != 46, drop this branch. |- Try 52 - 7 = 45. Add 45 to the number set. Current number set: [45, 1], target: 46, just two numbers left. |- Try 45 + 1 = 46. Evaluate 46 == 46, target found! Now we have found the target, let's trace back the solution: Final step: 45 + 1 = 46 The step before: 52 - 7 = 45 The first step: 46 + 6 = 52 Output the solution in the required format: <Solution> 46 + 6 = 52 52 - 7 = 45 45 + 1 = 46 </Solution>
{"nums": [6, 46, 7, 1], "target": 46, "solution": ["46 + 6 = 52", "52 - 7 = 45", "45 + 1 = 46"]}
countdown_0.5k_21
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [8, 30, 40, 39] Target: 37
# Search Procedure Initial number set: [8, 30, 40, 39], target: 37. Options for choosing two numbers: [(8, 30), (8, 40), (8, 39), (30, 40), (30, 39), (40, 39)]. |- Pick two numbers (8, 30) (numbers left: [40, 39]). Try possible operations. |- Try 30 + 8 = 38. Add 38 to the number set. Current number set: [38, 40, 39], target: 37. Options for choosing two numbers: [(38, 40), (38, 39), (40, 39)]. |- Pick two numbers (38, 40) (numbers left: [39]). Try possible operations. |- Try 40 + 38 = 78. Add 78 to the number set. Current number set: [78, 39], target: 37, just two numbers left. |- Try 78 + 39 = 117. Evaluate 117 != 37, drop this branch. |- Try 78 - 39 = 39. Evaluate 39 != 37, drop this branch. |- Try 78 * 39 = 3042. 3042 exceeds the maximum intermediate result, drop this branch. |- Try 78 / 39 = 2. Evaluate 2 != 37, drop this branch. |- Try 40 - 38 = 2. Add 2 to the number set. Current number set: [2, 39], target: 37, just two numbers left. |- Try 39 + 2 = 41. Evaluate 41 != 37, drop this branch. |- Try 39 - 2 = 37. Evaluate 37 == 37, target found! Now we have found the target, let's trace back the solution: Final step: 39 - 2 = 37 The step before: 40 - 38 = 2 The first step: 30 + 8 = 38 Output the solution in the required format: <Solution> 30 + 8 = 38 40 - 38 = 2 39 - 2 = 37 </Solution>
{"nums": [8, 30, 40, 39], "target": 37, "solution": ["30 + 8 = 38", "40 - 38 = 2", "39 - 2 = 37"]}
countdown_0.5k_22
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [18, 6, 3, 15] Target: 36
# Search Procedure Initial number set: [18, 6, 3, 15], target: 36. Options for choosing two numbers: [(18, 6), (18, 3), (18, 15), (6, 3), (6, 15), (3, 15)]. |- Pick two numbers (18, 6) (numbers left: [3, 15]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 3, 15], target: 36. Options for choosing two numbers: [(24, 3), (24, 15), (3, 15)]. |- Pick two numbers (24, 3) (numbers left: [15]). Try possible operations. |- Try 24 + 3 = 27. Add 27 to the number set. Current number set: [27, 15], target: 36, just two numbers left. |- Try 27 + 15 = 42. Evaluate 42 != 36, drop this branch. |- Try 27 - 15 = 12. Evaluate 12 != 36, drop this branch. |- Try 27 * 15 = 405. Evaluate 405 != 36, drop this branch. |- Try 27 / 15 = 1.8. 1.8 is a decimal, drop this branch. |- Try 24 - 3 = 21. Add 21 to the number set. Current number set: [21, 15], target: 36, just two numbers left. |- Try 21 + 15 = 36. Evaluate 36 == 36, target found! Now we have found the target, let's trace back the solution: Final step: 21 + 15 = 36 The step before: 24 - 3 = 21 The first step: 18 + 6 = 24 Output the solution in the required format: <Solution> 18 + 6 = 24 24 - 3 = 21 21 + 15 = 36 </Solution>
{"nums": [18, 6, 3, 15], "target": 36, "solution": ["18 + 6 = 24", "24 - 3 = 21", "21 + 15 = 36"]}
countdown_0.5k_23
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [8, 3, 30, 4] Target: 15
# Search Procedure Initial number set: [8, 3, 30, 4], target: 15. Options for choosing two numbers: [(8, 3), (8, 30), (8, 4), (3, 30), (3, 4), (30, 4)]. |- Pick two numbers (8, 3) (numbers left: [30, 4]). Try possible operations. |- Try 8 + 3 = 11. Add 11 to the number set. Current number set: [11, 30, 4], target: 15. Options for choosing two numbers: [(11, 30), (11, 4), (30, 4)]. |- Pick two numbers (11, 30) (numbers left: [4]). Try possible operations. |- Try 30 + 11 = 41. Add 41 to the number set. Current number set: [41, 4], target: 15, just two numbers left. |- Try 41 + 4 = 45. Evaluate 45 != 15, drop this branch. |- Try 41 - 4 = 37. Evaluate 37 != 15, drop this branch. |- Try 41 * 4 = 164. Evaluate 164 != 15, drop this branch. |- Try 41 / 4 = 10.2. 10.2 is a decimal, drop this branch. |- Try 30 - 11 = 19. Add 19 to the number set. Current number set: [19, 4], target: 15, just two numbers left. |- Try 19 + 4 = 23. Evaluate 23 != 15, drop this branch. |- Try 19 - 4 = 15. Evaluate 15 == 15, target found! Now we have found the target, let's trace back the solution: Final step: 19 - 4 = 15 The step before: 30 - 11 = 19 The first step: 8 + 3 = 11 Output the solution in the required format: <Solution> 8 + 3 = 11 30 - 11 = 19 19 - 4 = 15 </Solution>
{"nums": [8, 3, 30, 4], "target": 15, "solution": ["8 + 3 = 11", "30 - 11 = 19", "19 - 4 = 15"]}
countdown_0.5k_24
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [38, 7, 22, 20] Target: 43
# Search Procedure Initial number set: [38, 7, 22, 20], target: 43. Options for choosing two numbers: [(38, 7), (38, 22), (38, 20), (7, 22), (7, 20), (22, 20)]. |- Pick two numbers (38, 7) (numbers left: [22, 20]). Try possible operations. |- Try 38 + 7 = 45. Add 45 to the number set. Current number set: [45, 22, 20], target: 43. Options for choosing two numbers: [(45, 22), (45, 20), (22, 20)]. |- Pick two numbers (45, 22) (numbers left: [20]). Try possible operations. |- Try 45 + 22 = 67. Add 67 to the number set. Current number set: [67, 20], target: 43, just two numbers left. |- Try 67 + 20 = 87. Evaluate 87 != 43, drop this branch. |- Try 67 - 20 = 47. Evaluate 47 != 43, drop this branch. |- Try 67 * 20 = 1340. Evaluate 1340 != 43, drop this branch. |- Try 67 / 20 = 3.4. 3.4 is a decimal, drop this branch. |- Try 45 - 22 = 23. Add 23 to the number set. Current number set: [23, 20], target: 43, just two numbers left. |- Try 23 + 20 = 43. Evaluate 43 == 43, target found! Now we have found the target, let's trace back the solution: Final step: 23 + 20 = 43 The step before: 45 - 22 = 23 The first step: 38 + 7 = 45 Output the solution in the required format: <Solution> 38 + 7 = 45 45 - 22 = 23 23 + 20 = 43 </Solution>
{"nums": [38, 7, 22, 20], "target": 43, "solution": ["38 + 7 = 45", "45 - 22 = 23", "23 + 20 = 43"]}
countdown_0.5k_25
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [24, 4, 33, 11] Target: 16
# Search Procedure Initial number set: [24, 4, 33, 11], target: 16. Options for choosing two numbers: [(24, 4), (24, 33), (24, 11), (4, 33), (4, 11), (33, 11)]. |- Pick two numbers (24, 4) (numbers left: [33, 11]). Try possible operations. |- Try 24 + 4 = 28. Add 28 to the number set. Current number set: [28, 33, 11], target: 16. Options for choosing two numbers: [(28, 33), (28, 11), (33, 11)]. |- Pick two numbers (28, 33) (numbers left: [11]). Try possible operations. |- Try 33 + 28 = 61. Add 61 to the number set. Current number set: [61, 11], target: 16, just two numbers left. |- Try 61 + 11 = 72. Evaluate 72 != 16, drop this branch. |- Try 61 - 11 = 50. Evaluate 50 != 16, drop this branch. |- Try 61 * 11 = 671. Evaluate 671 != 16, drop this branch. |- Try 61 / 11 = 5.5. 5.5 is a decimal, drop this branch. |- Try 33 - 28 = 5. Add 5 to the number set. Current number set: [5, 11], target: 16, just two numbers left. |- Try 11 + 5 = 16. Evaluate 16 == 16, target found! Now we have found the target, let's trace back the solution: Final step: 11 + 5 = 16 The step before: 33 - 28 = 5 The first step: 24 + 4 = 28 Output the solution in the required format: <Solution> 24 + 4 = 28 33 - 28 = 5 11 + 5 = 16 </Solution>
{"nums": [24, 4, 33, 11], "target": 16, "solution": ["24 + 4 = 28", "33 - 28 = 5", "11 + 5 = 16"]}
countdown_0.5k_26
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [30, 4, 24, 46] Target: 12
# Search Procedure Initial number set: [30, 4, 24, 46], target: 12. Options for choosing two numbers: [(30, 4), (30, 24), (30, 46), (4, 24), (4, 46), (24, 46)]. |- Pick two numbers (30, 4) (numbers left: [24, 46]). Try possible operations. |- Try 30 + 4 = 34. Add 34 to the number set. Current number set: [34, 24, 46], target: 12. Options for choosing two numbers: [(34, 24), (34, 46), (24, 46)]. |- Pick two numbers (34, 24) (numbers left: [46]). Try possible operations. |- Try 34 + 24 = 58. Add 58 to the number set. Current number set: [58, 46], target: 12, just two numbers left. |- Try 58 + 46 = 104. Evaluate 104 != 12, drop this branch. |- Try 58 - 46 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 58 - 46 = 12 The step before: 34 + 24 = 58 The first step: 30 + 4 = 34 Output the solution in the required format: <Solution> 30 + 4 = 34 34 + 24 = 58 58 - 46 = 12 </Solution>
{"nums": [30, 4, 24, 46], "target": 12, "solution": ["30 + 4 = 34", "34 + 24 = 58", "58 - 46 = 12"]}
countdown_0.5k_27
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [1, 15, 17, 39] Target: 40
# Search Procedure Initial number set: [1, 15, 17, 39], target: 40. Options for choosing two numbers: [(1, 15), (1, 17), (1, 39), (15, 17), (15, 39), (17, 39)]. |- Pick two numbers (1, 15) (numbers left: [17, 39]). Try possible operations. |- Try 15 + 1 = 16. Add 16 to the number set. Current number set: [16, 17, 39], target: 40. Options for choosing two numbers: [(16, 17), (16, 39), (17, 39)]. |- Pick two numbers (16, 17) (numbers left: [39]). Try possible operations. |- Try 17 + 16 = 33. Add 33 to the number set. Current number set: [33, 39], target: 40, just two numbers left. |- Try 39 + 33 = 72. Evaluate 72 != 40, drop this branch. |- Try 39 - 33 = 6. Evaluate 6 != 40, drop this branch. |- Try 39 * 33 = 1287. Evaluate 1287 != 40, drop this branch. |- Try 39 / 33 = 1.2. 1.2 is a decimal, drop this branch. |- Try 17 - 16 = 1. Add 1 to the number set. Current number set: [1, 39], target: 40, just two numbers left. |- Try 39 + 1 = 40. Evaluate 40 == 40, target found! Now we have found the target, let's trace back the solution: Final step: 39 + 1 = 40 The step before: 17 - 16 = 1 The first step: 15 + 1 = 16 Output the solution in the required format: <Solution> 15 + 1 = 16 17 - 16 = 1 39 + 1 = 40 </Solution>
{"nums": [1, 15, 17, 39], "target": 40, "solution": ["15 + 1 = 16", "17 - 16 = 1", "39 + 1 = 40"]}
countdown_0.5k_28
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [27, 26, 25, 48] Target: 30
# Search Procedure Initial number set: [27, 26, 25, 48], target: 30. Options for choosing two numbers: [(27, 26), (27, 25), (27, 48), (26, 25), (26, 48), (25, 48)]. |- Pick two numbers (27, 26) (numbers left: [25, 48]). Try possible operations. |- Try 27 + 26 = 53. Add 53 to the number set. Current number set: [53, 25, 48], target: 30. Options for choosing two numbers: [(53, 25), (53, 48), (25, 48)]. |- Pick two numbers (53, 25) (numbers left: [48]). Try possible operations. |- Try 53 + 25 = 78. Add 78 to the number set. Current number set: [78, 48], target: 30, just two numbers left. |- Try 78 + 48 = 126. Evaluate 126 != 30, drop this branch. |- Try 78 - 48 = 30. Evaluate 30 == 30, target found! Now we have found the target, let's trace back the solution: Final step: 78 - 48 = 30 The step before: 53 + 25 = 78 The first step: 27 + 26 = 53 Output the solution in the required format: <Solution> 27 + 26 = 53 53 + 25 = 78 78 - 48 = 30 </Solution>
{"nums": [27, 26, 25, 48], "target": 30, "solution": ["27 + 26 = 53", "53 + 25 = 78", "78 - 48 = 30"]}
countdown_0.5k_29
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [4, 10, 9, 13] Target: 36
# Search Procedure Initial number set: [4, 10, 9, 13], target: 36. Options for choosing two numbers: [(4, 10), (4, 9), (4, 13), (10, 9), (10, 13), (9, 13)]. |- Pick two numbers (4, 10) (numbers left: [9, 13]). Try possible operations. |- Try 10 + 4 = 14. Add 14 to the number set. Current number set: [14, 9, 13], target: 36. Options for choosing two numbers: [(14, 9), (14, 13), (9, 13)]. |- Pick two numbers (14, 9) (numbers left: [13]). Try possible operations. |- Try 14 + 9 = 23. Add 23 to the number set. Current number set: [23, 13], target: 36, just two numbers left. |- Try 23 + 13 = 36. Evaluate 36 == 36, target found! Now we have found the target, let's trace back the solution: Final step: 23 + 13 = 36 The step before: 14 + 9 = 23 The first step: 10 + 4 = 14 Output the solution in the required format: <Solution> 10 + 4 = 14 14 + 9 = 23 23 + 13 = 36 </Solution>
{"nums": [4, 10, 9, 13], "target": 36, "solution": ["10 + 4 = 14", "14 + 9 = 23", "23 + 13 = 36"]}
countdown_0.5k_30
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [11, 8, 9, 45] Target: 35
# Search Procedure Initial number set: [11, 8, 9, 45], target: 35. Options for choosing two numbers: [(11, 8), (11, 9), (11, 45), (8, 9), (8, 45), (9, 45)]. |- Pick two numbers (11, 8) (numbers left: [9, 45]). Try possible operations. |- Try 11 + 8 = 19. Add 19 to the number set. Current number set: [19, 9, 45], target: 35. Options for choosing two numbers: [(19, 9), (19, 45), (9, 45)]. |- Pick two numbers (19, 9) (numbers left: [45]). Try possible operations. |- Try 19 + 9 = 28. Add 28 to the number set. Current number set: [28, 45], target: 35, just two numbers left. |- Try 45 + 28 = 73. Evaluate 73 != 35, drop this branch. |- Try 45 - 28 = 17. Evaluate 17 != 35, drop this branch. |- Try 45 * 28 = 1260. Evaluate 1260 != 35, drop this branch. |- Try 45 / 28 = 1.6. 1.6 is a decimal, drop this branch. |- Try 19 - 9 = 10. Add 10 to the number set. Current number set: [10, 45], target: 35, just two numbers left. |- Try 45 + 10 = 55. Evaluate 55 != 35, drop this branch. |- Try 45 - 10 = 35. Evaluate 35 == 35, target found! Now we have found the target, let's trace back the solution: Final step: 45 - 10 = 35 The step before: 19 - 9 = 10 The first step: 11 + 8 = 19 Output the solution in the required format: <Solution> 11 + 8 = 19 19 - 9 = 10 45 - 10 = 35 </Solution>
{"nums": [11, 8, 9, 45], "target": 35, "solution": ["11 + 8 = 19", "19 - 9 = 10", "45 - 10 = 35"]}
countdown_0.5k_31
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [21, 16, 22, 18] Target: 41
# Search Procedure Initial number set: [21, 16, 22, 18], target: 41. Options for choosing two numbers: [(21, 16), (21, 22), (21, 18), (16, 22), (16, 18), (22, 18)]. |- Pick two numbers (21, 16) (numbers left: [22, 18]). Try possible operations. |- Try 21 + 16 = 37. Add 37 to the number set. Current number set: [37, 22, 18], target: 41. Options for choosing two numbers: [(37, 22), (37, 18), (22, 18)]. |- Pick two numbers (37, 22) (numbers left: [18]). Try possible operations. |- Try 37 + 22 = 59. Add 59 to the number set. Current number set: [59, 18], target: 41, just two numbers left. |- Try 59 + 18 = 77. Evaluate 77 != 41, drop this branch. |- Try 59 - 18 = 41. Evaluate 41 == 41, target found! Now we have found the target, let's trace back the solution: Final step: 59 - 18 = 41 The step before: 37 + 22 = 59 The first step: 21 + 16 = 37 Output the solution in the required format: <Solution> 21 + 16 = 37 37 + 22 = 59 59 - 18 = 41 </Solution>
{"nums": [21, 16, 22, 18], "target": 41, "solution": ["21 + 16 = 37", "37 + 22 = 59", "59 - 18 = 41"]}
countdown_0.5k_32
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [15, 30, 22, 36] Target: 13
# Search Procedure Initial number set: [15, 30, 22, 36], target: 13. Options for choosing two numbers: [(15, 30), (15, 22), (15, 36), (30, 22), (30, 36), (22, 36)]. |- Pick two numbers (15, 30) (numbers left: [22, 36]). Try possible operations. |- Try 30 + 15 = 45. Add 45 to the number set. Current number set: [45, 22, 36], target: 13. Options for choosing two numbers: [(45, 22), (45, 36), (22, 36)]. |- Pick two numbers (45, 22) (numbers left: [36]). Try possible operations. |- Try 45 + 22 = 67. Add 67 to the number set. Current number set: [67, 36], target: 13, just two numbers left. |- Try 67 + 36 = 103. Evaluate 103 != 13, drop this branch. |- Try 67 - 36 = 31. Evaluate 31 != 13, drop this branch. |- Try 67 * 36 = 2412. 2412 exceeds the maximum intermediate result, drop this branch. |- Try 67 / 36 = 1.9. 1.9 is a decimal, drop this branch. |- Try 45 - 22 = 23. Add 23 to the number set. Current number set: [23, 36], target: 13, just two numbers left. |- Try 36 + 23 = 59. Evaluate 59 != 13, drop this branch. |- Try 36 - 23 = 13. Evaluate 13 == 13, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 23 = 13 The step before: 45 - 22 = 23 The first step: 30 + 15 = 45 Output the solution in the required format: <Solution> 30 + 15 = 45 45 - 22 = 23 36 - 23 = 13 </Solution>
{"nums": [15, 30, 22, 36], "target": 13, "solution": ["30 + 15 = 45", "45 - 22 = 23", "36 - 23 = 13"]}
countdown_0.5k_33
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [24, 46, 29, 8] Target: 49
# Search Procedure Initial number set: [24, 46, 29, 8], target: 49. Options for choosing two numbers: [(24, 46), (24, 29), (24, 8), (46, 29), (46, 8), (29, 8)]. |- Pick two numbers (24, 46) (numbers left: [29, 8]). Try possible operations. |- Try 46 + 24 = 70. Add 70 to the number set. Current number set: [70, 29, 8], target: 49. Options for choosing two numbers: [(70, 29), (70, 8), (29, 8)]. |- Pick two numbers (70, 29) (numbers left: [8]). Try possible operations. |- Try 70 + 29 = 99. Add 99 to the number set. Current number set: [99, 8], target: 49, just two numbers left. |- Try 99 + 8 = 107. Evaluate 107 != 49, drop this branch. |- Try 99 - 8 = 91. Evaluate 91 != 49, drop this branch. |- Try 99 * 8 = 792. Evaluate 792 != 49, drop this branch. |- Try 99 / 8 = 12.4. 12.4 is a decimal, drop this branch. |- Try 70 - 29 = 41. Add 41 to the number set. Current number set: [41, 8], target: 49, just two numbers left. |- Try 41 + 8 = 49. Evaluate 49 == 49, target found! Now we have found the target, let's trace back the solution: Final step: 41 + 8 = 49 The step before: 70 - 29 = 41 The first step: 46 + 24 = 70 Output the solution in the required format: <Solution> 46 + 24 = 70 70 - 29 = 41 41 + 8 = 49 </Solution>
{"nums": [24, 46, 29, 8], "target": 49, "solution": ["46 + 24 = 70", "70 - 29 = 41", "41 + 8 = 49"]}
countdown_0.5k_34
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [48, 36, 41, 14] Target: 29
# Search Procedure Initial number set: [48, 36, 41, 14], target: 29. Options for choosing two numbers: [(48, 36), (48, 41), (48, 14), (36, 41), (36, 14), (41, 14)]. |- Pick two numbers (48, 36) (numbers left: [41, 14]). Try possible operations. |- Try 48 + 36 = 84. Add 84 to the number set. Current number set: [84, 41, 14], target: 29. Options for choosing two numbers: [(84, 41), (84, 14), (41, 14)]. |- Pick two numbers (84, 41) (numbers left: [14]). Try possible operations. |- Try 84 + 41 = 125. Add 125 to the number set. Current number set: [125, 14], target: 29, just two numbers left. |- Try 125 + 14 = 139. Evaluate 139 != 29, drop this branch. |- Try 125 - 14 = 111. Evaluate 111 != 29, drop this branch. |- Try 125 * 14 = 1750. Evaluate 1750 != 29, drop this branch. |- Try 125 / 14 = 8.9. 8.9 is a decimal, drop this branch. |- Try 84 - 41 = 43. Add 43 to the number set. Current number set: [43, 14], target: 29, just two numbers left. |- Try 43 + 14 = 57. Evaluate 57 != 29, drop this branch. |- Try 43 - 14 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 43 - 14 = 29 The step before: 84 - 41 = 43 The first step: 48 + 36 = 84 Output the solution in the required format: <Solution> 48 + 36 = 84 84 - 41 = 43 43 - 14 = 29 </Solution>
{"nums": [48, 36, 41, 14], "target": 29, "solution": ["48 + 36 = 84", "84 - 41 = 43", "43 - 14 = 29"]}
countdown_0.5k_35
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [12, 17, 36, 5] Target: 13
# Search Procedure Initial number set: [12, 17, 36, 5], target: 13. Options for choosing two numbers: [(12, 17), (12, 36), (12, 5), (17, 36), (17, 5), (36, 5)]. |- Pick two numbers (12, 17) (numbers left: [36, 5]). Try possible operations. |- Try 17 + 12 = 29. Add 29 to the number set. Current number set: [29, 36, 5], target: 13. Options for choosing two numbers: [(29, 36), (29, 5), (36, 5)]. |- Pick two numbers (29, 36) (numbers left: [5]). Try possible operations. |- Try 36 + 29 = 65. Add 65 to the number set. Current number set: [65, 5], target: 13, just two numbers left. |- Try 65 + 5 = 70. Evaluate 70 != 13, drop this branch. |- Try 65 - 5 = 60. Evaluate 60 != 13, drop this branch. |- Try 65 * 5 = 325. Evaluate 325 != 13, drop this branch. |- Try 65 / 5 = 13. Evaluate 13 == 13, target found! Now we have found the target, let's trace back the solution: Final step: 65 / 5 = 13 The step before: 36 + 29 = 65 The first step: 17 + 12 = 29 Output the solution in the required format: <Solution> 17 + 12 = 29 36 + 29 = 65 65 / 5 = 13 </Solution>
{"nums": [12, 17, 36, 5], "target": 13, "solution": ["17 + 12 = 29", "36 + 29 = 65", "65 / 5 = 13"]}
countdown_0.5k_36
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [21, 2, 25, 44] Target: 46
# Search Procedure Initial number set: [21, 2, 25, 44], target: 46. Options for choosing two numbers: [(21, 2), (21, 25), (21, 44), (2, 25), (2, 44), (25, 44)]. |- Pick two numbers (21, 2) (numbers left: [25, 44]). Try possible operations. |- Try 21 + 2 = 23. Add 23 to the number set. Current number set: [23, 25, 44], target: 46. Options for choosing two numbers: [(23, 25), (23, 44), (25, 44)]. |- Pick two numbers (23, 25) (numbers left: [44]). Try possible operations. |- Try 25 + 23 = 48. Add 48 to the number set. Current number set: [48, 44], target: 46, just two numbers left. |- Try 48 + 44 = 92. Evaluate 92 != 46, drop this branch. |- Try 48 - 44 = 4. Evaluate 4 != 46, drop this branch. |- Try 48 * 44 = 2112. 2112 exceeds the maximum intermediate result, drop this branch. |- Try 48 / 44 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 - 23 = 2. Add 2 to the number set. Current number set: [2, 44], target: 46, just two numbers left. |- Try 44 + 2 = 46. Evaluate 46 == 46, target found! Now we have found the target, let's trace back the solution: Final step: 44 + 2 = 46 The step before: 25 - 23 = 2 The first step: 21 + 2 = 23 Output the solution in the required format: <Solution> 21 + 2 = 23 25 - 23 = 2 44 + 2 = 46 </Solution>
{"nums": [21, 2, 25, 44], "target": 46, "solution": ["21 + 2 = 23", "25 - 23 = 2", "44 + 2 = 46"]}
countdown_0.5k_37
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [42, 16, 7, 19] Target: 46
# Search Procedure Initial number set: [42, 16, 7, 19], target: 46. Options for choosing two numbers: [(42, 16), (42, 7), (42, 19), (16, 7), (16, 19), (7, 19)]. |- Pick two numbers (42, 16) (numbers left: [7, 19]). Try possible operations. |- Try 42 + 16 = 58. Add 58 to the number set. Current number set: [58, 7, 19], target: 46. Options for choosing two numbers: [(58, 7), (58, 19), (7, 19)]. |- Pick two numbers (58, 7) (numbers left: [19]). Try possible operations. |- Try 58 + 7 = 65. Add 65 to the number set. Current number set: [65, 19], target: 46, just two numbers left. |- Try 65 + 19 = 84. Evaluate 84 != 46, drop this branch. |- Try 65 - 19 = 46. Evaluate 46 == 46, target found! Now we have found the target, let's trace back the solution: Final step: 65 - 19 = 46 The step before: 58 + 7 = 65 The first step: 42 + 16 = 58 Output the solution in the required format: <Solution> 42 + 16 = 58 58 + 7 = 65 65 - 19 = 46 </Solution>
{"nums": [42, 16, 7, 19], "target": 46, "solution": ["42 + 16 = 58", "58 + 7 = 65", "65 - 19 = 46"]}
countdown_0.5k_38
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [34, 26, 23, 23] Target: 14
# Search Procedure Initial number set: [34, 26, 23, 23], target: 14. Options for choosing two numbers: [(34, 26), (34, 23), (34, 23), (26, 23), (26, 23), (23, 23)]. |- Pick two numbers (34, 26) (numbers left: [23, 23]). Try possible operations. |- Try 34 + 26 = 60. Add 60 to the number set. Current number set: [60, 23, 23], target: 14. Options for choosing two numbers: [(60, 23), (60, 23), (23, 23)]. |- Pick two numbers (60, 23) (numbers left: [23]). Try possible operations. |- Try 60 + 23 = 83. Add 83 to the number set. Current number set: [83, 23], target: 14, just two numbers left. |- Try 83 + 23 = 106. Evaluate 106 != 14, drop this branch. |- Try 83 - 23 = 60. Evaluate 60 != 14, drop this branch. |- Try 83 * 23 = 1909. Evaluate 1909 != 14, drop this branch. |- Try 83 / 23 = 3.6. 3.6 is a decimal, drop this branch. |- Try 60 - 23 = 37. Add 37 to the number set. Current number set: [37, 23], target: 14, just two numbers left. |- Try 37 + 23 = 60. Evaluate 60 != 14, drop this branch. |- Try 37 - 23 = 14. Evaluate 14 == 14, target found! Now we have found the target, let's trace back the solution: Final step: 37 - 23 = 14 The step before: 60 - 23 = 37 The first step: 34 + 26 = 60 Output the solution in the required format: <Solution> 34 + 26 = 60 60 - 23 = 37 37 - 23 = 14 </Solution>
{"nums": [34, 26, 23, 23], "target": 14, "solution": ["34 + 26 = 60", "60 - 23 = 37", "37 - 23 = 14"]}
countdown_0.5k_39
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [17, 15, 24, 6] Target: 14
# Search Procedure Initial number set: [17, 15, 24, 6], target: 14. Options for choosing two numbers: [(17, 15), (17, 24), (17, 6), (15, 24), (15, 6), (24, 6)]. |- Pick two numbers (17, 15) (numbers left: [24, 6]). Try possible operations. |- Try 17 + 15 = 32. Add 32 to the number set. Current number set: [32, 24, 6], target: 14. Options for choosing two numbers: [(32, 24), (32, 6), (24, 6)]. |- Pick two numbers (32, 24) (numbers left: [6]). Try possible operations. |- Try 32 + 24 = 56. Add 56 to the number set. Current number set: [56, 6], target: 14, just two numbers left. |- Try 56 + 6 = 62. Evaluate 62 != 14, drop this branch. |- Try 56 - 6 = 50. Evaluate 50 != 14, drop this branch. |- Try 56 * 6 = 336. Evaluate 336 != 14, drop this branch. |- Try 56 / 6 = 9.3. 9.3 is a decimal, drop this branch. |- Try 32 - 24 = 8. Add 8 to the number set. Current number set: [8, 6], target: 14, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 == 14, target found! Now we have found the target, let's trace back the solution: Final step: 8 + 6 = 14 The step before: 32 - 24 = 8 The first step: 17 + 15 = 32 Output the solution in the required format: <Solution> 17 + 15 = 32 32 - 24 = 8 8 + 6 = 14 </Solution>
{"nums": [17, 15, 24, 6], "target": 14, "solution": ["17 + 15 = 32", "32 - 24 = 8", "8 + 6 = 14"]}
countdown_0.5k_40
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [16, 42, 1, 11] Target: 48
# Search Procedure Initial number set: [16, 42, 1, 11], target: 48. Options for choosing two numbers: [(16, 42), (16, 1), (16, 11), (42, 1), (42, 11), (1, 11)]. |- Pick two numbers (16, 42) (numbers left: [1, 11]). Try possible operations. |- Try 42 + 16 = 58. Add 58 to the number set. Current number set: [58, 1, 11], target: 48. Options for choosing two numbers: [(58, 1), (58, 11), (1, 11)]. |- Pick two numbers (58, 1) (numbers left: [11]). Try possible operations. |- Try 58 + 1 = 59. Add 59 to the number set. Current number set: [59, 11], target: 48, just two numbers left. |- Try 59 + 11 = 70. Evaluate 70 != 48, drop this branch. |- Try 59 - 11 = 48. Evaluate 48 == 48, target found! Now we have found the target, let's trace back the solution: Final step: 59 - 11 = 48 The step before: 58 + 1 = 59 The first step: 42 + 16 = 58 Output the solution in the required format: <Solution> 42 + 16 = 58 58 + 1 = 59 59 - 11 = 48 </Solution>
{"nums": [16, 42, 1, 11], "target": 48, "solution": ["42 + 16 = 58", "58 + 1 = 59", "59 - 11 = 48"]}
countdown_0.5k_41
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [5, 25, 41, 15] Target: 26
# Search Procedure Initial number set: [5, 25, 41, 15], target: 26. Options for choosing two numbers: [(5, 25), (5, 41), (5, 15), (25, 41), (25, 15), (41, 15)]. |- Pick two numbers (5, 25) (numbers left: [41, 15]). Try possible operations. |- Try 25 + 5 = 30. Add 30 to the number set. Current number set: [30, 41, 15], target: 26. Options for choosing two numbers: [(30, 41), (30, 15), (41, 15)]. |- Pick two numbers (30, 41) (numbers left: [15]). Try possible operations. |- Try 41 + 30 = 71. Add 71 to the number set. Current number set: [71, 15], target: 26, just two numbers left. |- Try 71 + 15 = 86. Evaluate 86 != 26, drop this branch. |- Try 71 - 15 = 56. Evaluate 56 != 26, drop this branch. |- Try 71 * 15 = 1065. Evaluate 1065 != 26, drop this branch. |- Try 71 / 15 = 4.7. 4.7 is a decimal, drop this branch. |- Try 41 - 30 = 11. Add 11 to the number set. Current number set: [11, 15], target: 26, just two numbers left. |- Try 15 + 11 = 26. Evaluate 26 == 26, target found! Now we have found the target, let's trace back the solution: Final step: 15 + 11 = 26 The step before: 41 - 30 = 11 The first step: 25 + 5 = 30 Output the solution in the required format: <Solution> 25 + 5 = 30 41 - 30 = 11 15 + 11 = 26 </Solution>
{"nums": [5, 25, 41, 15], "target": 26, "solution": ["25 + 5 = 30", "41 - 30 = 11", "15 + 11 = 26"]}
countdown_0.5k_42
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [16, 48, 16, 17] Target: 31
# Search Procedure Initial number set: [16, 48, 16, 17], target: 31. Options for choosing two numbers: [(16, 48), (16, 16), (16, 17), (48, 16), (48, 17), (16, 17)]. |- Pick two numbers (16, 48) (numbers left: [16, 17]). Try possible operations. |- Try 48 + 16 = 64. Add 64 to the number set. Current number set: [64, 16, 17], target: 31. Options for choosing two numbers: [(64, 16), (64, 17), (16, 17)]. |- Pick two numbers (64, 16) (numbers left: [17]). Try possible operations. |- Try 64 + 16 = 80. Add 80 to the number set. Current number set: [80, 17], target: 31, just two numbers left. |- Try 80 + 17 = 97. Evaluate 97 != 31, drop this branch. |- Try 80 - 17 = 63. Evaluate 63 != 31, drop this branch. |- Try 80 * 17 = 1360. Evaluate 1360 != 31, drop this branch. |- Try 80 / 17 = 4.7. 4.7 is a decimal, drop this branch. |- Try 64 - 16 = 48. Add 48 to the number set. Current number set: [48, 17], target: 31, just two numbers left. |- Try 48 + 17 = 65. Evaluate 65 != 31, drop this branch. |- Try 48 - 17 = 31. Evaluate 31 == 31, target found! Now we have found the target, let's trace back the solution: Final step: 48 - 17 = 31 The step before: 64 - 16 = 48 The first step: 48 + 16 = 64 Output the solution in the required format: <Solution> 48 + 16 = 64 64 - 16 = 48 48 - 17 = 31 </Solution>
{"nums": [16, 48, 16, 17], "target": 31, "solution": ["48 + 16 = 64", "64 - 16 = 48", "48 - 17 = 31"]}
countdown_0.5k_43
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [44, 43, 32, 40] Target: 15
# Search Procedure Initial number set: [44, 43, 32, 40], target: 15. Options for choosing two numbers: [(44, 43), (44, 32), (44, 40), (43, 32), (43, 40), (32, 40)]. |- Pick two numbers (44, 43) (numbers left: [32, 40]). Try possible operations. |- Try 44 + 43 = 87. Add 87 to the number set. Current number set: [87, 32, 40], target: 15. Options for choosing two numbers: [(87, 32), (87, 40), (32, 40)]. |- Pick two numbers (87, 32) (numbers left: [40]). Try possible operations. |- Try 87 + 32 = 119. Add 119 to the number set. Current number set: [119, 40], target: 15, just two numbers left. |- Try 119 + 40 = 159. Evaluate 159 != 15, drop this branch. |- Try 119 - 40 = 79. Evaluate 79 != 15, drop this branch. |- Try 119 * 40 = 4760. 4760 exceeds the maximum intermediate result, drop this branch. |- Try 119 / 40 = 3.0. 3.0 is a decimal, drop this branch. |- Try 87 - 32 = 55. Add 55 to the number set. Current number set: [55, 40], target: 15, just two numbers left. |- Try 55 + 40 = 95. Evaluate 95 != 15, drop this branch. |- Try 55 - 40 = 15. Evaluate 15 == 15, target found! Now we have found the target, let's trace back the solution: Final step: 55 - 40 = 15 The step before: 87 - 32 = 55 The first step: 44 + 43 = 87 Output the solution in the required format: <Solution> 44 + 43 = 87 87 - 32 = 55 55 - 40 = 15 </Solution>
{"nums": [44, 43, 32, 40], "target": 15, "solution": ["44 + 43 = 87", "87 - 32 = 55", "55 - 40 = 15"]}
countdown_0.5k_44
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [48, 10, 31, 1] Target: 27
# Search Procedure Initial number set: [48, 10, 31, 1], target: 27. Options for choosing two numbers: [(48, 10), (48, 31), (48, 1), (10, 31), (10, 1), (31, 1)]. |- Pick two numbers (48, 10) (numbers left: [31, 1]). Try possible operations. |- Try 48 + 10 = 58. Add 58 to the number set. Current number set: [58, 31, 1], target: 27. Options for choosing two numbers: [(58, 31), (58, 1), (31, 1)]. |- Pick two numbers (58, 31) (numbers left: [1]). Try possible operations. |- Try 58 + 31 = 89. Add 89 to the number set. Current number set: [89, 1], target: 27, just two numbers left. |- Try 89 + 1 = 90. Evaluate 90 != 27, drop this branch. |- Try 89 - 1 = 88. Evaluate 88 != 27, drop this branch. |- Try 89 * 1 = 89. Evaluate 89 != 27, drop this branch. |- Try 89 / 1 = 89. Evaluate 89 != 27, drop this branch. |- Try 58 - 31 = 27. Add 27 to the number set. Current number set: [27, 1], target: 27, just two numbers left. |- Try 27 + 1 = 28. Evaluate 28 != 27, drop this branch. |- Try 27 - 1 = 26. Evaluate 26 != 27, drop this branch. |- Try 27 * 1 = 27. Evaluate 27 == 27, target found! Now we have found the target, let's trace back the solution: Final step: 27 * 1 = 27 The step before: 58 - 31 = 27 The first step: 48 + 10 = 58 Output the solution in the required format: <Solution> 48 + 10 = 58 58 - 31 = 27 27 * 1 = 27 </Solution>
{"nums": [48, 10, 31, 1], "target": 27, "solution": ["48 + 10 = 58", "58 - 31 = 27", "27 * 1 = 27"]}
countdown_0.5k_45
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [22, 27, 41, 40] Target: 50
# Search Procedure Initial number set: [22, 27, 41, 40], target: 50. Options for choosing two numbers: [(22, 27), (22, 41), (22, 40), (27, 41), (27, 40), (41, 40)]. |- Pick two numbers (22, 27) (numbers left: [41, 40]). Try possible operations. |- Try 27 + 22 = 49. Add 49 to the number set. Current number set: [49, 41, 40], target: 50. Options for choosing two numbers: [(49, 41), (49, 40), (41, 40)]. |- Pick two numbers (49, 41) (numbers left: [40]). Try possible operations. |- Try 49 + 41 = 90. Add 90 to the number set. Current number set: [90, 40], target: 50, just two numbers left. |- Try 90 + 40 = 130. Evaluate 130 != 50, drop this branch. |- Try 90 - 40 = 50. Evaluate 50 == 50, target found! Now we have found the target, let's trace back the solution: Final step: 90 - 40 = 50 The step before: 49 + 41 = 90 The first step: 27 + 22 = 49 Output the solution in the required format: <Solution> 27 + 22 = 49 49 + 41 = 90 90 - 40 = 50 </Solution>
{"nums": [22, 27, 41, 40], "target": 50, "solution": ["27 + 22 = 49", "49 + 41 = 90", "90 - 40 = 50"]}
countdown_0.5k_46
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [2, 26, 34, 49] Target: 13
# Search Procedure Initial number set: [2, 26, 34, 49], target: 13. Options for choosing two numbers: [(2, 26), (2, 34), (2, 49), (26, 34), (26, 49), (34, 49)]. |- Pick two numbers (2, 26) (numbers left: [34, 49]). Try possible operations. |- Try 26 + 2 = 28. Add 28 to the number set. Current number set: [28, 34, 49], target: 13. Options for choosing two numbers: [(28, 34), (28, 49), (34, 49)]. |- Pick two numbers (28, 34) (numbers left: [49]). Try possible operations. |- Try 34 + 28 = 62. Add 62 to the number set. Current number set: [62, 49], target: 13, just two numbers left. |- Try 62 + 49 = 111. Evaluate 111 != 13, drop this branch. |- Try 62 - 49 = 13. Evaluate 13 == 13, target found! Now we have found the target, let's trace back the solution: Final step: 62 - 49 = 13 The step before: 34 + 28 = 62 The first step: 26 + 2 = 28 Output the solution in the required format: <Solution> 26 + 2 = 28 34 + 28 = 62 62 - 49 = 13 </Solution>
{"nums": [2, 26, 34, 49], "target": 13, "solution": ["26 + 2 = 28", "34 + 28 = 62", "62 - 49 = 13"]}
countdown_0.5k_47
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [26, 5, 35, 31] Target: 27
# Search Procedure Initial number set: [26, 5, 35, 31], target: 27. Options for choosing two numbers: [(26, 5), (26, 35), (26, 31), (5, 35), (5, 31), (35, 31)]. |- Pick two numbers (26, 5) (numbers left: [35, 31]). Try possible operations. |- Try 26 + 5 = 31. Add 31 to the number set. Current number set: [31, 35, 31], target: 27. Options for choosing two numbers: [(31, 35), (31, 31), (35, 31)]. |- Pick two numbers (31, 35) (numbers left: [31]). Try possible operations. |- Try 35 + 31 = 66. Add 66 to the number set. Current number set: [66, 31], target: 27, just two numbers left. |- Try 66 + 31 = 97. Evaluate 97 != 27, drop this branch. |- Try 66 - 31 = 35. Evaluate 35 != 27, drop this branch. |- Try 66 * 31 = 2046. 2046 exceeds the maximum intermediate result, drop this branch. |- Try 66 / 31 = 2.1. 2.1 is a decimal, drop this branch. |- Try 35 - 31 = 4. Add 4 to the number set. Current number set: [4, 31], target: 27, just two numbers left. |- Try 31 + 4 = 35. Evaluate 35 != 27, drop this branch. |- Try 31 - 4 = 27. Evaluate 27 == 27, target found! Now we have found the target, let's trace back the solution: Final step: 31 - 4 = 27 The step before: 35 - 31 = 4 The first step: 26 + 5 = 31 Output the solution in the required format: <Solution> 26 + 5 = 31 35 - 31 = 4 31 - 4 = 27 </Solution>
{"nums": [26, 5, 35, 31], "target": 27, "solution": ["26 + 5 = 31", "35 - 31 = 4", "31 - 4 = 27"]}
countdown_0.5k_48
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [24, 40, 16, 21] Target: 27
# Search Procedure Initial number set: [24, 40, 16, 21], target: 27. Options for choosing two numbers: [(24, 40), (24, 16), (24, 21), (40, 16), (40, 21), (16, 21)]. |- Pick two numbers (24, 40) (numbers left: [16, 21]). Try possible operations. |- Try 40 + 24 = 64. Add 64 to the number set. Current number set: [64, 16, 21], target: 27. Options for choosing two numbers: [(64, 16), (64, 21), (16, 21)]. |- Pick two numbers (64, 16) (numbers left: [21]). Try possible operations. |- Try 64 + 16 = 80. Add 80 to the number set. Current number set: [80, 21], target: 27, just two numbers left. |- Try 80 + 21 = 101. Evaluate 101 != 27, drop this branch. |- Try 80 - 21 = 59. Evaluate 59 != 27, drop this branch. |- Try 80 * 21 = 1680. Evaluate 1680 != 27, drop this branch. |- Try 80 / 21 = 3.8. 3.8 is a decimal, drop this branch. |- Try 64 - 16 = 48. Add 48 to the number set. Current number set: [48, 21], target: 27, just two numbers left. |- Try 48 + 21 = 69. Evaluate 69 != 27, drop this branch. |- Try 48 - 21 = 27. Evaluate 27 == 27, target found! Now we have found the target, let's trace back the solution: Final step: 48 - 21 = 27 The step before: 64 - 16 = 48 The first step: 40 + 24 = 64 Output the solution in the required format: <Solution> 40 + 24 = 64 64 - 16 = 48 48 - 21 = 27 </Solution>
{"nums": [24, 40, 16, 21], "target": 27, "solution": ["40 + 24 = 64", "64 - 16 = 48", "48 - 21 = 27"]}
countdown_0.5k_49
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [5, 1, 14, 42] Target: 34
# Search Procedure Initial number set: [5, 1, 14, 42], target: 34. Options for choosing two numbers: [(5, 1), (5, 14), (5, 42), (1, 14), (1, 42), (14, 42)]. |- Pick two numbers (5, 1) (numbers left: [14, 42]). Try possible operations. |- Try 5 + 1 = 6. Add 6 to the number set. Current number set: [6, 14, 42], target: 34. Options for choosing two numbers: [(6, 14), (6, 42), (14, 42)]. |- Pick two numbers (6, 14) (numbers left: [42]). Try possible operations. |- Try 14 + 6 = 20. Add 20 to the number set. Current number set: [20, 42], target: 34, just two numbers left. |- Try 42 + 20 = 62. Evaluate 62 != 34, drop this branch. |- Try 42 - 20 = 22. Evaluate 22 != 34, drop this branch. |- Try 42 * 20 = 840. Evaluate 840 != 34, drop this branch. |- Try 42 / 20 = 2.1. 2.1 is a decimal, drop this branch. |- Try 14 - 6 = 8. Add 8 to the number set. Current number set: [8, 42], target: 34, just two numbers left. |- Try 42 + 8 = 50. Evaluate 50 != 34, drop this branch. |- Try 42 - 8 = 34. Evaluate 34 == 34, target found! Now we have found the target, let's trace back the solution: Final step: 42 - 8 = 34 The step before: 14 - 6 = 8 The first step: 5 + 1 = 6 Output the solution in the required format: <Solution> 5 + 1 = 6 14 - 6 = 8 42 - 8 = 34 </Solution>
{"nums": [5, 1, 14, 42], "target": 34, "solution": ["5 + 1 = 6", "14 - 6 = 8", "42 - 8 = 34"]}
countdown_0.5k_50
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [36, 48, 27, 24] Target: 33
# Search Procedure Initial number set: [36, 48, 27, 24], target: 33. Options for choosing two numbers: [(36, 48), (36, 27), (36, 24), (48, 27), (48, 24), (27, 24)]. |- Pick two numbers (36, 48) (numbers left: [27, 24]). Try possible operations. |- Try 48 + 36 = 84. Add 84 to the number set. Current number set: [84, 27, 24], target: 33. Options for choosing two numbers: [(84, 27), (84, 24), (27, 24)]. |- Pick two numbers (84, 27) (numbers left: [24]). Try possible operations. |- Try 84 + 27 = 111. Add 111 to the number set. Current number set: [111, 24], target: 33, just two numbers left. |- Try 111 + 24 = 135. Evaluate 135 != 33, drop this branch. |- Try 111 - 24 = 87. Evaluate 87 != 33, drop this branch. |- Try 111 * 24 = 2664. 2664 exceeds the maximum intermediate result, drop this branch. |- Try 111 / 24 = 4.6. 4.6 is a decimal, drop this branch. |- Try 84 - 27 = 57. Add 57 to the number set. Current number set: [57, 24], target: 33, just two numbers left. |- Try 57 + 24 = 81. Evaluate 81 != 33, drop this branch. |- Try 57 - 24 = 33. Evaluate 33 == 33, target found! Now we have found the target, let's trace back the solution: Final step: 57 - 24 = 33 The step before: 84 - 27 = 57 The first step: 48 + 36 = 84 Output the solution in the required format: <Solution> 48 + 36 = 84 84 - 27 = 57 57 - 24 = 33 </Solution>
{"nums": [36, 48, 27, 24], "target": 33, "solution": ["48 + 36 = 84", "84 - 27 = 57", "57 - 24 = 33"]}
countdown_0.5k_51
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [20, 15, 14, 21] Target: 42
# Search Procedure Initial number set: [20, 15, 14, 21], target: 42. Options for choosing two numbers: [(20, 15), (20, 14), (20, 21), (15, 14), (15, 21), (14, 21)]. |- Pick two numbers (20, 15) (numbers left: [14, 21]). Try possible operations. |- Try 20 + 15 = 35. Add 35 to the number set. Current number set: [35, 14, 21], target: 42. Options for choosing two numbers: [(35, 14), (35, 21), (14, 21)]. |- Pick two numbers (35, 14) (numbers left: [21]). Try possible operations. |- Try 35 + 14 = 49. Add 49 to the number set. Current number set: [49, 21], target: 42, just two numbers left. |- Try 49 + 21 = 70. Evaluate 70 != 42, drop this branch. |- Try 49 - 21 = 28. Evaluate 28 != 42, drop this branch. |- Try 49 * 21 = 1029. Evaluate 1029 != 42, drop this branch. |- Try 49 / 21 = 2.3. 2.3 is a decimal, drop this branch. |- Try 35 - 14 = 21. Add 21 to the number set. Current number set: [21, 21], target: 42, just two numbers left. |- Try 21 + 21 = 42. Evaluate 42 == 42, target found! Now we have found the target, let's trace back the solution: Final step: 21 + 21 = 42 The step before: 35 - 14 = 21 The first step: 20 + 15 = 35 Output the solution in the required format: <Solution> 20 + 15 = 35 35 - 14 = 21 21 + 21 = 42 </Solution>
{"nums": [20, 15, 14, 21], "target": 42, "solution": ["20 + 15 = 35", "35 - 14 = 21", "21 + 21 = 42"]}
countdown_0.5k_52
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [36, 37, 37, 10] Target: 11
# Search Procedure Initial number set: [36, 37, 37, 10], target: 11. Options for choosing two numbers: [(36, 37), (36, 37), (36, 10), (37, 37), (37, 10), (37, 10)]. |- Pick two numbers (36, 37) (numbers left: [37, 10]). Try possible operations. |- Try 37 + 36 = 73. Add 73 to the number set. Current number set: [73, 37, 10], target: 11. Options for choosing two numbers: [(73, 37), (73, 10), (37, 10)]. |- Pick two numbers (73, 37) (numbers left: [10]). Try possible operations. |- Try 73 + 37 = 110. Add 110 to the number set. Current number set: [110, 10], target: 11, just two numbers left. |- Try 110 + 10 = 120. Evaluate 120 != 11, drop this branch. |- Try 110 - 10 = 100. Evaluate 100 != 11, drop this branch. |- Try 110 * 10 = 1100. Evaluate 1100 != 11, drop this branch. |- Try 110 / 10 = 11. Evaluate 11 == 11, target found! Now we have found the target, let's trace back the solution: Final step: 110 / 10 = 11 The step before: 73 + 37 = 110 The first step: 37 + 36 = 73 Output the solution in the required format: <Solution> 37 + 36 = 73 73 + 37 = 110 110 / 10 = 11 </Solution>
{"nums": [36, 37, 37, 10], "target": 11, "solution": ["37 + 36 = 73", "73 + 37 = 110", "110 / 10 = 11"]}
countdown_0.5k_53
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [31, 12, 32, 45] Target: 30
# Search Procedure Initial number set: [31, 12, 32, 45], target: 30. Options for choosing two numbers: [(31, 12), (31, 32), (31, 45), (12, 32), (12, 45), (32, 45)]. |- Pick two numbers (31, 12) (numbers left: [32, 45]). Try possible operations. |- Try 31 + 12 = 43. Add 43 to the number set. Current number set: [43, 32, 45], target: 30. Options for choosing two numbers: [(43, 32), (43, 45), (32, 45)]. |- Pick two numbers (43, 32) (numbers left: [45]). Try possible operations. |- Try 43 + 32 = 75. Add 75 to the number set. Current number set: [75, 45], target: 30, just two numbers left. |- Try 75 + 45 = 120. Evaluate 120 != 30, drop this branch. |- Try 75 - 45 = 30. Evaluate 30 == 30, target found! Now we have found the target, let's trace back the solution: Final step: 75 - 45 = 30 The step before: 43 + 32 = 75 The first step: 31 + 12 = 43 Output the solution in the required format: <Solution> 31 + 12 = 43 43 + 32 = 75 75 - 45 = 30 </Solution>
{"nums": [31, 12, 32, 45], "target": 30, "solution": ["31 + 12 = 43", "43 + 32 = 75", "75 - 45 = 30"]}
countdown_0.5k_54
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [10, 9, 22, 6] Target: 35
# Search Procedure Initial number set: [10, 9, 22, 6], target: 35. Options for choosing two numbers: [(10, 9), (10, 22), (10, 6), (9, 22), (9, 6), (22, 6)]. |- Pick two numbers (10, 9) (numbers left: [22, 6]). Try possible operations. |- Try 10 + 9 = 19. Add 19 to the number set. Current number set: [19, 22, 6], target: 35. Options for choosing two numbers: [(19, 22), (19, 6), (22, 6)]. |- Pick two numbers (19, 22) (numbers left: [6]). Try possible operations. |- Try 22 + 19 = 41. Add 41 to the number set. Current number set: [41, 6], target: 35, just two numbers left. |- Try 41 + 6 = 47. Evaluate 47 != 35, drop this branch. |- Try 41 - 6 = 35. Evaluate 35 == 35, target found! Now we have found the target, let's trace back the solution: Final step: 41 - 6 = 35 The step before: 22 + 19 = 41 The first step: 10 + 9 = 19 Output the solution in the required format: <Solution> 10 + 9 = 19 22 + 19 = 41 41 - 6 = 35 </Solution>
{"nums": [10, 9, 22, 6], "target": 35, "solution": ["10 + 9 = 19", "22 + 19 = 41", "41 - 6 = 35"]}
countdown_0.5k_55
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [16, 5, 18, 28] Target: 25
# Search Procedure Initial number set: [16, 5, 18, 28], target: 25. Options for choosing two numbers: [(16, 5), (16, 18), (16, 28), (5, 18), (5, 28), (18, 28)]. |- Pick two numbers (16, 5) (numbers left: [18, 28]). Try possible operations. |- Try 16 + 5 = 21. Add 21 to the number set. Current number set: [21, 18, 28], target: 25. Options for choosing two numbers: [(21, 18), (21, 28), (18, 28)]. |- Pick two numbers (21, 18) (numbers left: [28]). Try possible operations. |- Try 21 + 18 = 39. Add 39 to the number set. Current number set: [39, 28], target: 25, just two numbers left. |- Try 39 + 28 = 67. Evaluate 67 != 25, drop this branch. |- Try 39 - 28 = 11. Evaluate 11 != 25, drop this branch. |- Try 39 * 28 = 1092. Evaluate 1092 != 25, drop this branch. |- Try 39 / 28 = 1.4. 1.4 is a decimal, drop this branch. |- Try 21 - 18 = 3. Add 3 to the number set. Current number set: [3, 28], target: 25, just two numbers left. |- Try 28 + 3 = 31. Evaluate 31 != 25, drop this branch. |- Try 28 - 3 = 25. Evaluate 25 == 25, target found! Now we have found the target, let's trace back the solution: Final step: 28 - 3 = 25 The step before: 21 - 18 = 3 The first step: 16 + 5 = 21 Output the solution in the required format: <Solution> 16 + 5 = 21 21 - 18 = 3 28 - 3 = 25 </Solution>
{"nums": [16, 5, 18, 28], "target": 25, "solution": ["16 + 5 = 21", "21 - 18 = 3", "28 - 3 = 25"]}
countdown_0.5k_56
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [35, 6, 12, 13] Target: 16
# Search Procedure Initial number set: [35, 6, 12, 13], target: 16. Options for choosing two numbers: [(35, 6), (35, 12), (35, 13), (6, 12), (6, 13), (12, 13)]. |- Pick two numbers (35, 6) (numbers left: [12, 13]). Try possible operations. |- Try 35 + 6 = 41. Add 41 to the number set. Current number set: [41, 12, 13], target: 16. Options for choosing two numbers: [(41, 12), (41, 13), (12, 13)]. |- Pick two numbers (41, 12) (numbers left: [13]). Try possible operations. |- Try 41 + 12 = 53. Add 53 to the number set. Current number set: [53, 13], target: 16, just two numbers left. |- Try 53 + 13 = 66. Evaluate 66 != 16, drop this branch. |- Try 53 - 13 = 40. Evaluate 40 != 16, drop this branch. |- Try 53 * 13 = 689. Evaluate 689 != 16, drop this branch. |- Try 53 / 13 = 4.1. 4.1 is a decimal, drop this branch. |- Try 41 - 12 = 29. Add 29 to the number set. Current number set: [29, 13], target: 16, just two numbers left. |- Try 29 + 13 = 42. Evaluate 42 != 16, drop this branch. |- Try 29 - 13 = 16. Evaluate 16 == 16, target found! Now we have found the target, let's trace back the solution: Final step: 29 - 13 = 16 The step before: 41 - 12 = 29 The first step: 35 + 6 = 41 Output the solution in the required format: <Solution> 35 + 6 = 41 41 - 12 = 29 29 - 13 = 16 </Solution>
{"nums": [35, 6, 12, 13], "target": 16, "solution": ["35 + 6 = 41", "41 - 12 = 29", "29 - 13 = 16"]}
countdown_0.5k_57
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [21, 5, 36, 35] Target: 27
# Search Procedure Initial number set: [21, 5, 36, 35], target: 27. Options for choosing two numbers: [(21, 5), (21, 36), (21, 35), (5, 36), (5, 35), (36, 35)]. |- Pick two numbers (21, 5) (numbers left: [36, 35]). Try possible operations. |- Try 21 + 5 = 26. Add 26 to the number set. Current number set: [26, 36, 35], target: 27. Options for choosing two numbers: [(26, 36), (26, 35), (36, 35)]. |- Pick two numbers (26, 36) (numbers left: [35]). Try possible operations. |- Try 36 + 26 = 62. Add 62 to the number set. Current number set: [62, 35], target: 27, just two numbers left. |- Try 62 + 35 = 97. Evaluate 97 != 27, drop this branch. |- Try 62 - 35 = 27. Evaluate 27 == 27, target found! Now we have found the target, let's trace back the solution: Final step: 62 - 35 = 27 The step before: 36 + 26 = 62 The first step: 21 + 5 = 26 Output the solution in the required format: <Solution> 21 + 5 = 26 36 + 26 = 62 62 - 35 = 27 </Solution>
{"nums": [21, 5, 36, 35], "target": 27, "solution": ["21 + 5 = 26", "36 + 26 = 62", "62 - 35 = 27"]}
countdown_0.5k_58
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [12, 25, 32, 31] Target: 36
# Search Procedure Initial number set: [12, 25, 32, 31], target: 36. Options for choosing two numbers: [(12, 25), (12, 32), (12, 31), (25, 32), (25, 31), (32, 31)]. |- Pick two numbers (12, 25) (numbers left: [32, 31]). Try possible operations. |- Try 25 + 12 = 37. Add 37 to the number set. Current number set: [37, 32, 31], target: 36. Options for choosing two numbers: [(37, 32), (37, 31), (32, 31)]. |- Pick two numbers (37, 32) (numbers left: [31]). Try possible operations. |- Try 37 + 32 = 69. Add 69 to the number set. Current number set: [69, 31], target: 36, just two numbers left. |- Try 69 + 31 = 100. Evaluate 100 != 36, drop this branch. |- Try 69 - 31 = 38. Evaluate 38 != 36, drop this branch. |- Try 69 * 31 = 2139. 2139 exceeds the maximum intermediate result, drop this branch. |- Try 69 / 31 = 2.2. 2.2 is a decimal, drop this branch. |- Try 37 - 32 = 5. Add 5 to the number set. Current number set: [5, 31], target: 36, just two numbers left. |- Try 31 + 5 = 36. Evaluate 36 == 36, target found! Now we have found the target, let's trace back the solution: Final step: 31 + 5 = 36 The step before: 37 - 32 = 5 The first step: 25 + 12 = 37 Output the solution in the required format: <Solution> 25 + 12 = 37 37 - 32 = 5 31 + 5 = 36 </Solution>
{"nums": [12, 25, 32, 31], "target": 36, "solution": ["25 + 12 = 37", "37 - 32 = 5", "31 + 5 = 36"]}
countdown_0.5k_59
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [45, 42, 46, 11] Target: 30
# Search Procedure Initial number set: [45, 42, 46, 11], target: 30. Options for choosing two numbers: [(45, 42), (45, 46), (45, 11), (42, 46), (42, 11), (46, 11)]. |- Pick two numbers (45, 42) (numbers left: [46, 11]). Try possible operations. |- Try 45 + 42 = 87. Add 87 to the number set. Current number set: [87, 46, 11], target: 30. Options for choosing two numbers: [(87, 46), (87, 11), (46, 11)]. |- Pick two numbers (87, 46) (numbers left: [11]). Try possible operations. |- Try 87 + 46 = 133. Add 133 to the number set. Current number set: [133, 11], target: 30, just two numbers left. |- Try 133 + 11 = 144. Evaluate 144 != 30, drop this branch. |- Try 133 - 11 = 122. Evaluate 122 != 30, drop this branch. |- Try 133 * 11 = 1463. Evaluate 1463 != 30, drop this branch. |- Try 133 / 11 = 12.1. 12.1 is a decimal, drop this branch. |- Try 87 - 46 = 41. Add 41 to the number set. Current number set: [41, 11], target: 30, just two numbers left. |- Try 41 + 11 = 52. Evaluate 52 != 30, drop this branch. |- Try 41 - 11 = 30. Evaluate 30 == 30, target found! Now we have found the target, let's trace back the solution: Final step: 41 - 11 = 30 The step before: 87 - 46 = 41 The first step: 45 + 42 = 87 Output the solution in the required format: <Solution> 45 + 42 = 87 87 - 46 = 41 41 - 11 = 30 </Solution>
{"nums": [45, 42, 46, 11], "target": 30, "solution": ["45 + 42 = 87", "87 - 46 = 41", "41 - 11 = 30"]}
countdown_0.5k_60
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [25, 3, 11, 22] Target: 17
# Search Procedure Initial number set: [25, 3, 11, 22], target: 17. Options for choosing two numbers: [(25, 3), (25, 11), (25, 22), (3, 11), (3, 22), (11, 22)]. |- Pick two numbers (25, 3) (numbers left: [11, 22]). Try possible operations. |- Try 25 + 3 = 28. Add 28 to the number set. Current number set: [28, 11, 22], target: 17. Options for choosing two numbers: [(28, 11), (28, 22), (11, 22)]. |- Pick two numbers (28, 11) (numbers left: [22]). Try possible operations. |- Try 28 + 11 = 39. Add 39 to the number set. Current number set: [39, 22], target: 17, just two numbers left. |- Try 39 + 22 = 61. Evaluate 61 != 17, drop this branch. |- Try 39 - 22 = 17. Evaluate 17 == 17, target found! Now we have found the target, let's trace back the solution: Final step: 39 - 22 = 17 The step before: 28 + 11 = 39 The first step: 25 + 3 = 28 Output the solution in the required format: <Solution> 25 + 3 = 28 28 + 11 = 39 39 - 22 = 17 </Solution>
{"nums": [25, 3, 11, 22], "target": 17, "solution": ["25 + 3 = 28", "28 + 11 = 39", "39 - 22 = 17"]}
countdown_0.5k_61
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [27, 12, 11, 44] Target: 16
# Search Procedure Initial number set: [27, 12, 11, 44], target: 16. Options for choosing two numbers: [(27, 12), (27, 11), (27, 44), (12, 11), (12, 44), (11, 44)]. |- Pick two numbers (27, 12) (numbers left: [11, 44]). Try possible operations. |- Try 27 + 12 = 39. Add 39 to the number set. Current number set: [39, 11, 44], target: 16. Options for choosing two numbers: [(39, 11), (39, 44), (11, 44)]. |- Pick two numbers (39, 11) (numbers left: [44]). Try possible operations. |- Try 39 + 11 = 50. Add 50 to the number set. Current number set: [50, 44], target: 16, just two numbers left. |- Try 50 + 44 = 94. Evaluate 94 != 16, drop this branch. |- Try 50 - 44 = 6. Evaluate 6 != 16, drop this branch. |- Try 50 * 44 = 2200. 2200 exceeds the maximum intermediate result, drop this branch. |- Try 50 / 44 = 1.1. 1.1 is a decimal, drop this branch. |- Try 39 - 11 = 28. Add 28 to the number set. Current number set: [28, 44], target: 16, just two numbers left. |- Try 44 + 28 = 72. Evaluate 72 != 16, drop this branch. |- Try 44 - 28 = 16. Evaluate 16 == 16, target found! Now we have found the target, let's trace back the solution: Final step: 44 - 28 = 16 The step before: 39 - 11 = 28 The first step: 27 + 12 = 39 Output the solution in the required format: <Solution> 27 + 12 = 39 39 - 11 = 28 44 - 28 = 16 </Solution>
{"nums": [27, 12, 11, 44], "target": 16, "solution": ["27 + 12 = 39", "39 - 11 = 28", "44 - 28 = 16"]}
countdown_0.5k_62
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [39, 37, 36, 12] Target: 28
# Search Procedure Initial number set: [39, 37, 36, 12], target: 28. Options for choosing two numbers: [(39, 37), (39, 36), (39, 12), (37, 36), (37, 12), (36, 12)]. |- Pick two numbers (39, 37) (numbers left: [36, 12]). Try possible operations. |- Try 39 + 37 = 76. Add 76 to the number set. Current number set: [76, 36, 12], target: 28. Options for choosing two numbers: [(76, 36), (76, 12), (36, 12)]. |- Pick two numbers (76, 36) (numbers left: [12]). Try possible operations. |- Try 76 + 36 = 112. Add 112 to the number set. Current number set: [112, 12], target: 28, just two numbers left. |- Try 112 + 12 = 124. Evaluate 124 != 28, drop this branch. |- Try 112 - 12 = 100. Evaluate 100 != 28, drop this branch. |- Try 112 * 12 = 1344. Evaluate 1344 != 28, drop this branch. |- Try 112 / 12 = 9.3. 9.3 is a decimal, drop this branch. |- Try 76 - 36 = 40. Add 40 to the number set. Current number set: [40, 12], target: 28, just two numbers left. |- Try 40 + 12 = 52. Evaluate 52 != 28, drop this branch. |- Try 40 - 12 = 28. Evaluate 28 == 28, target found! Now we have found the target, let's trace back the solution: Final step: 40 - 12 = 28 The step before: 76 - 36 = 40 The first step: 39 + 37 = 76 Output the solution in the required format: <Solution> 39 + 37 = 76 76 - 36 = 40 40 - 12 = 28 </Solution>
{"nums": [39, 37, 36, 12], "target": 28, "solution": ["39 + 37 = 76", "76 - 36 = 40", "40 - 12 = 28"]}
countdown_0.5k_63
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [45, 14, 40, 4] Target: 23
# Search Procedure Initial number set: [45, 14, 40, 4], target: 23. Options for choosing two numbers: [(45, 14), (45, 40), (45, 4), (14, 40), (14, 4), (40, 4)]. |- Pick two numbers (45, 14) (numbers left: [40, 4]). Try possible operations. |- Try 45 + 14 = 59. Add 59 to the number set. Current number set: [59, 40, 4], target: 23. Options for choosing two numbers: [(59, 40), (59, 4), (40, 4)]. |- Pick two numbers (59, 40) (numbers left: [4]). Try possible operations. |- Try 59 + 40 = 99. Add 99 to the number set. Current number set: [99, 4], target: 23, just two numbers left. |- Try 99 + 4 = 103. Evaluate 103 != 23, drop this branch. |- Try 99 - 4 = 95. Evaluate 95 != 23, drop this branch. |- Try 99 * 4 = 396. Evaluate 396 != 23, drop this branch. |- Try 99 / 4 = 24.8. 24.8 is a decimal, drop this branch. |- Try 59 - 40 = 19. Add 19 to the number set. Current number set: [19, 4], target: 23, just two numbers left. |- Try 19 + 4 = 23. Evaluate 23 == 23, target found! Now we have found the target, let's trace back the solution: Final step: 19 + 4 = 23 The step before: 59 - 40 = 19 The first step: 45 + 14 = 59 Output the solution in the required format: <Solution> 45 + 14 = 59 59 - 40 = 19 19 + 4 = 23 </Solution>
{"nums": [45, 14, 40, 4], "target": 23, "solution": ["45 + 14 = 59", "59 - 40 = 19", "19 + 4 = 23"]}
countdown_0.5k_64
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [7, 34, 44, 9] Target: 12
# Search Procedure Initial number set: [7, 34, 44, 9], target: 12. Options for choosing two numbers: [(7, 34), (7, 44), (7, 9), (34, 44), (34, 9), (44, 9)]. |- Pick two numbers (7, 34) (numbers left: [44, 9]). Try possible operations. |- Try 34 + 7 = 41. Add 41 to the number set. Current number set: [41, 44, 9], target: 12. Options for choosing two numbers: [(41, 44), (41, 9), (44, 9)]. |- Pick two numbers (41, 44) (numbers left: [9]). Try possible operations. |- Try 44 + 41 = 85. Add 85 to the number set. Current number set: [85, 9], target: 12, just two numbers left. |- Try 85 + 9 = 94. Evaluate 94 != 12, drop this branch. |- Try 85 - 9 = 76. Evaluate 76 != 12, drop this branch. |- Try 85 * 9 = 765. Evaluate 765 != 12, drop this branch. |- Try 85 / 9 = 9.4. 9.4 is a decimal, drop this branch. |- Try 44 - 41 = 3. Add 3 to the number set. Current number set: [3, 9], target: 12, just two numbers left. |- Try 9 + 3 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 9 + 3 = 12 The step before: 44 - 41 = 3 The first step: 34 + 7 = 41 Output the solution in the required format: <Solution> 34 + 7 = 41 44 - 41 = 3 9 + 3 = 12 </Solution>
{"nums": [7, 34, 44, 9], "target": 12, "solution": ["34 + 7 = 41", "44 - 41 = 3", "9 + 3 = 12"]}
countdown_0.5k_65
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [45, 30, 7, 30] Target: 38
# Search Procedure Initial number set: [45, 30, 7, 30], target: 38. Options for choosing two numbers: [(45, 30), (45, 7), (45, 30), (30, 7), (30, 30), (7, 30)]. |- Pick two numbers (45, 30) (numbers left: [7, 30]). Try possible operations. |- Try 45 + 30 = 75. Add 75 to the number set. Current number set: [75, 7, 30], target: 38. Options for choosing two numbers: [(75, 7), (75, 30), (7, 30)]. |- Pick two numbers (75, 7) (numbers left: [30]). Try possible operations. |- Try 75 + 7 = 82. Add 82 to the number set. Current number set: [82, 30], target: 38, just two numbers left. |- Try 82 + 30 = 112. Evaluate 112 != 38, drop this branch. |- Try 82 - 30 = 52. Evaluate 52 != 38, drop this branch. |- Try 82 * 30 = 2460. 2460 exceeds the maximum intermediate result, drop this branch. |- Try 82 / 30 = 2.7. 2.7 is a decimal, drop this branch. |- Try 75 - 7 = 68. Add 68 to the number set. Current number set: [68, 30], target: 38, just two numbers left. |- Try 68 + 30 = 98. Evaluate 98 != 38, drop this branch. |- Try 68 - 30 = 38. Evaluate 38 == 38, target found! Now we have found the target, let's trace back the solution: Final step: 68 - 30 = 38 The step before: 75 - 7 = 68 The first step: 45 + 30 = 75 Output the solution in the required format: <Solution> 45 + 30 = 75 75 - 7 = 68 68 - 30 = 38 </Solution>
{"nums": [45, 30, 7, 30], "target": 38, "solution": ["45 + 30 = 75", "75 - 7 = 68", "68 - 30 = 38"]}
countdown_0.5k_66
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [16, 40, 16, 36] Target: 36
# Search Procedure Initial number set: [16, 40, 16, 36], target: 36. Options for choosing two numbers: [(16, 40), (16, 16), (16, 36), (40, 16), (40, 36), (16, 36)]. |- Pick two numbers (16, 40) (numbers left: [16, 36]). Try possible operations. |- Try 40 + 16 = 56. Add 56 to the number set. Current number set: [56, 16, 36], target: 36. Options for choosing two numbers: [(56, 16), (56, 36), (16, 36)]. |- Pick two numbers (56, 16) (numbers left: [36]). Try possible operations. |- Try 56 + 16 = 72. Add 72 to the number set. Current number set: [72, 36], target: 36, just two numbers left. |- Try 72 + 36 = 108. Evaluate 108 != 36, drop this branch. |- Try 72 - 36 = 36. Evaluate 36 == 36, target found! Now we have found the target, let's trace back the solution: Final step: 72 - 36 = 36 The step before: 56 + 16 = 72 The first step: 40 + 16 = 56 Output the solution in the required format: <Solution> 40 + 16 = 56 56 + 16 = 72 72 - 36 = 36 </Solution>
{"nums": [16, 40, 16, 36], "target": 36, "solution": ["40 + 16 = 56", "56 + 16 = 72", "72 - 36 = 36"]}
countdown_0.5k_67
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [46, 24, 8, 38] Target: 40
# Search Procedure Initial number set: [46, 24, 8, 38], target: 40. Options for choosing two numbers: [(46, 24), (46, 8), (46, 38), (24, 8), (24, 38), (8, 38)]. |- Pick two numbers (46, 24) (numbers left: [8, 38]). Try possible operations. |- Try 46 + 24 = 70. Add 70 to the number set. Current number set: [70, 8, 38], target: 40. Options for choosing two numbers: [(70, 8), (70, 38), (8, 38)]. |- Pick two numbers (70, 8) (numbers left: [38]). Try possible operations. |- Try 70 + 8 = 78. Add 78 to the number set. Current number set: [78, 38], target: 40, just two numbers left. |- Try 78 + 38 = 116. Evaluate 116 != 40, drop this branch. |- Try 78 - 38 = 40. Evaluate 40 == 40, target found! Now we have found the target, let's trace back the solution: Final step: 78 - 38 = 40 The step before: 70 + 8 = 78 The first step: 46 + 24 = 70 Output the solution in the required format: <Solution> 46 + 24 = 70 70 + 8 = 78 78 - 38 = 40 </Solution>
{"nums": [46, 24, 8, 38], "target": 40, "solution": ["46 + 24 = 70", "70 + 8 = 78", "78 - 38 = 40"]}
countdown_0.5k_68
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [31, 37, 47, 22] Target: 43
# Search Procedure Initial number set: [31, 37, 47, 22], target: 43. Options for choosing two numbers: [(31, 37), (31, 47), (31, 22), (37, 47), (37, 22), (47, 22)]. |- Pick two numbers (31, 37) (numbers left: [47, 22]). Try possible operations. |- Try 37 + 31 = 68. Add 68 to the number set. Current number set: [68, 47, 22], target: 43. Options for choosing two numbers: [(68, 47), (68, 22), (47, 22)]. |- Pick two numbers (68, 47) (numbers left: [22]). Try possible operations. |- Try 68 + 47 = 115. Add 115 to the number set. Current number set: [115, 22], target: 43, just two numbers left. |- Try 115 + 22 = 137. Evaluate 137 != 43, drop this branch. |- Try 115 - 22 = 93. Evaluate 93 != 43, drop this branch. |- Try 115 * 22 = 2530. 2530 exceeds the maximum intermediate result, drop this branch. |- Try 115 / 22 = 5.2. 5.2 is a decimal, drop this branch. |- Try 68 - 47 = 21. Add 21 to the number set. Current number set: [21, 22], target: 43, just two numbers left. |- Try 22 + 21 = 43. Evaluate 43 == 43, target found! Now we have found the target, let's trace back the solution: Final step: 22 + 21 = 43 The step before: 68 - 47 = 21 The first step: 37 + 31 = 68 Output the solution in the required format: <Solution> 37 + 31 = 68 68 - 47 = 21 22 + 21 = 43 </Solution>
{"nums": [31, 37, 47, 22], "target": 43, "solution": ["37 + 31 = 68", "68 - 47 = 21", "22 + 21 = 43"]}
countdown_0.5k_69
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [18, 17, 9, 40] Target: 14
# Search Procedure Initial number set: [18, 17, 9, 40], target: 14. Options for choosing two numbers: [(18, 17), (18, 9), (18, 40), (17, 9), (17, 40), (9, 40)]. |- Pick two numbers (18, 17) (numbers left: [9, 40]). Try possible operations. |- Try 18 + 17 = 35. Add 35 to the number set. Current number set: [35, 9, 40], target: 14. Options for choosing two numbers: [(35, 9), (35, 40), (9, 40)]. |- Pick two numbers (35, 9) (numbers left: [40]). Try possible operations. |- Try 35 + 9 = 44. Add 44 to the number set. Current number set: [44, 40], target: 14, just two numbers left. |- Try 44 + 40 = 84. Evaluate 84 != 14, drop this branch. |- Try 44 - 40 = 4. Evaluate 4 != 14, drop this branch. |- Try 44 * 40 = 1760. Evaluate 1760 != 14, drop this branch. |- Try 44 / 40 = 1.1. 1.1 is a decimal, drop this branch. |- Try 35 - 9 = 26. Add 26 to the number set. Current number set: [26, 40], target: 14, just two numbers left. |- Try 40 + 26 = 66. Evaluate 66 != 14, drop this branch. |- Try 40 - 26 = 14. Evaluate 14 == 14, target found! Now we have found the target, let's trace back the solution: Final step: 40 - 26 = 14 The step before: 35 - 9 = 26 The first step: 18 + 17 = 35 Output the solution in the required format: <Solution> 18 + 17 = 35 35 - 9 = 26 40 - 26 = 14 </Solution>
{"nums": [18, 17, 9, 40], "target": 14, "solution": ["18 + 17 = 35", "35 - 9 = 26", "40 - 26 = 14"]}
countdown_0.5k_70
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [11, 14, 37, 47] Target: 15
# Search Procedure Initial number set: [11, 14, 37, 47], target: 15. Options for choosing two numbers: [(11, 14), (11, 37), (11, 47), (14, 37), (14, 47), (37, 47)]. |- Pick two numbers (11, 14) (numbers left: [37, 47]). Try possible operations. |- Try 14 + 11 = 25. Add 25 to the number set. Current number set: [25, 37, 47], target: 15. Options for choosing two numbers: [(25, 37), (25, 47), (37, 47)]. |- Pick two numbers (25, 37) (numbers left: [47]). Try possible operations. |- Try 37 + 25 = 62. Add 62 to the number set. Current number set: [62, 47], target: 15, just two numbers left. |- Try 62 + 47 = 109. Evaluate 109 != 15, drop this branch. |- Try 62 - 47 = 15. Evaluate 15 == 15, target found! Now we have found the target, let's trace back the solution: Final step: 62 - 47 = 15 The step before: 37 + 25 = 62 The first step: 14 + 11 = 25 Output the solution in the required format: <Solution> 14 + 11 = 25 37 + 25 = 62 62 - 47 = 15 </Solution>
{"nums": [11, 14, 37, 47], "target": 15, "solution": ["14 + 11 = 25", "37 + 25 = 62", "62 - 47 = 15"]}
countdown_0.5k_71
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [17, 10, 34, 28] Target: 35
# Search Procedure Initial number set: [17, 10, 34, 28], target: 35. Options for choosing two numbers: [(17, 10), (17, 34), (17, 28), (10, 34), (10, 28), (34, 28)]. |- Pick two numbers (17, 10) (numbers left: [34, 28]). Try possible operations. |- Try 17 + 10 = 27. Add 27 to the number set. Current number set: [27, 34, 28], target: 35. Options for choosing two numbers: [(27, 34), (27, 28), (34, 28)]. |- Pick two numbers (27, 34) (numbers left: [28]). Try possible operations. |- Try 34 + 27 = 61. Add 61 to the number set. Current number set: [61, 28], target: 35, just two numbers left. |- Try 61 + 28 = 89. Evaluate 89 != 35, drop this branch. |- Try 61 - 28 = 33. Evaluate 33 != 35, drop this branch. |- Try 61 * 28 = 1708. Evaluate 1708 != 35, drop this branch. |- Try 61 / 28 = 2.2. 2.2 is a decimal, drop this branch. |- Try 34 - 27 = 7. Add 7 to the number set. Current number set: [7, 28], target: 35, just two numbers left. |- Try 28 + 7 = 35. Evaluate 35 == 35, target found! Now we have found the target, let's trace back the solution: Final step: 28 + 7 = 35 The step before: 34 - 27 = 7 The first step: 17 + 10 = 27 Output the solution in the required format: <Solution> 17 + 10 = 27 34 - 27 = 7 28 + 7 = 35 </Solution>
{"nums": [17, 10, 34, 28], "target": 35, "solution": ["17 + 10 = 27", "34 - 27 = 7", "28 + 7 = 35"]}
countdown_0.5k_72
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [3, 25, 43, 13] Target: 28
# Search Procedure Initial number set: [3, 25, 43, 13], target: 28. Options for choosing two numbers: [(3, 25), (3, 43), (3, 13), (25, 43), (25, 13), (43, 13)]. |- Pick two numbers (3, 25) (numbers left: [43, 13]). Try possible operations. |- Try 25 + 3 = 28. Add 28 to the number set. Current number set: [28, 43, 13], target: 28. Options for choosing two numbers: [(28, 43), (28, 13), (43, 13)]. |- Pick two numbers (28, 43) (numbers left: [13]). Try possible operations. |- Try 43 + 28 = 71. Add 71 to the number set. Current number set: [71, 13], target: 28, just two numbers left. |- Try 71 + 13 = 84. Evaluate 84 != 28, drop this branch. |- Try 71 - 13 = 58. Evaluate 58 != 28, drop this branch. |- Try 71 * 13 = 923. Evaluate 923 != 28, drop this branch. |- Try 71 / 13 = 5.5. 5.5 is a decimal, drop this branch. |- Try 43 - 28 = 15. Add 15 to the number set. Current number set: [15, 13], target: 28, just two numbers left. |- Try 15 + 13 = 28. Evaluate 28 == 28, target found! Now we have found the target, let's trace back the solution: Final step: 15 + 13 = 28 The step before: 43 - 28 = 15 The first step: 25 + 3 = 28 Output the solution in the required format: <Solution> 25 + 3 = 28 43 - 28 = 15 15 + 13 = 28 </Solution>
{"nums": [3, 25, 43, 13], "target": 28, "solution": ["25 + 3 = 28", "43 - 28 = 15", "15 + 13 = 28"]}
countdown_0.5k_73
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [10, 48, 38, 15] Target: 35
# Search Procedure Initial number set: [10, 48, 38, 15], target: 35. Options for choosing two numbers: [(10, 48), (10, 38), (10, 15), (48, 38), (48, 15), (38, 15)]. |- Pick two numbers (10, 48) (numbers left: [38, 15]). Try possible operations. |- Try 48 + 10 = 58. Add 58 to the number set. Current number set: [58, 38, 15], target: 35. Options for choosing two numbers: [(58, 38), (58, 15), (38, 15)]. |- Pick two numbers (58, 38) (numbers left: [15]). Try possible operations. |- Try 58 + 38 = 96. Add 96 to the number set. Current number set: [96, 15], target: 35, just two numbers left. |- Try 96 + 15 = 111. Evaluate 111 != 35, drop this branch. |- Try 96 - 15 = 81. Evaluate 81 != 35, drop this branch. |- Try 96 * 15 = 1440. Evaluate 1440 != 35, drop this branch. |- Try 96 / 15 = 6.4. 6.4 is a decimal, drop this branch. |- Try 58 - 38 = 20. Add 20 to the number set. Current number set: [20, 15], target: 35, just two numbers left. |- Try 20 + 15 = 35. Evaluate 35 == 35, target found! Now we have found the target, let's trace back the solution: Final step: 20 + 15 = 35 The step before: 58 - 38 = 20 The first step: 48 + 10 = 58 Output the solution in the required format: <Solution> 48 + 10 = 58 58 - 38 = 20 20 + 15 = 35 </Solution>
{"nums": [10, 48, 38, 15], "target": 35, "solution": ["48 + 10 = 58", "58 - 38 = 20", "20 + 15 = 35"]}
countdown_0.5k_74
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [14, 18, 4, 13] Target: 49
# Search Procedure Initial number set: [14, 18, 4, 13], target: 49. Options for choosing two numbers: [(14, 18), (14, 4), (14, 13), (18, 4), (18, 13), (4, 13)]. |- Pick two numbers (14, 18) (numbers left: [4, 13]). Try possible operations. |- Try 18 + 14 = 32. Add 32 to the number set. Current number set: [32, 4, 13], target: 49. Options for choosing two numbers: [(32, 4), (32, 13), (4, 13)]. |- Pick two numbers (32, 4) (numbers left: [13]). Try possible operations. |- Try 32 + 4 = 36. Add 36 to the number set. Current number set: [36, 13], target: 49, just two numbers left. |- Try 36 + 13 = 49. Evaluate 49 == 49, target found! Now we have found the target, let's trace back the solution: Final step: 36 + 13 = 49 The step before: 32 + 4 = 36 The first step: 18 + 14 = 32 Output the solution in the required format: <Solution> 18 + 14 = 32 32 + 4 = 36 36 + 13 = 49 </Solution>
{"nums": [14, 18, 4, 13], "target": 49, "solution": ["18 + 14 = 32", "32 + 4 = 36", "36 + 13 = 49"]}
countdown_0.5k_75
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [12, 11, 43, 3] Target: 17
# Search Procedure Initial number set: [12, 11, 43, 3], target: 17. Options for choosing two numbers: [(12, 11), (12, 43), (12, 3), (11, 43), (11, 3), (43, 3)]. |- Pick two numbers (12, 11) (numbers left: [43, 3]). Try possible operations. |- Try 12 + 11 = 23. Add 23 to the number set. Current number set: [23, 43, 3], target: 17. Options for choosing two numbers: [(23, 43), (23, 3), (43, 3)]. |- Pick two numbers (23, 43) (numbers left: [3]). Try possible operations. |- Try 43 + 23 = 66. Add 66 to the number set. Current number set: [66, 3], target: 17, just two numbers left. |- Try 66 + 3 = 69. Evaluate 69 != 17, drop this branch. |- Try 66 - 3 = 63. Evaluate 63 != 17, drop this branch. |- Try 66 * 3 = 198. Evaluate 198 != 17, drop this branch. |- Try 66 / 3 = 22. Evaluate 22 != 17, drop this branch. |- Try 43 - 23 = 20. Add 20 to the number set. Current number set: [20, 3], target: 17, just two numbers left. |- Try 20 + 3 = 23. Evaluate 23 != 17, drop this branch. |- Try 20 - 3 = 17. Evaluate 17 == 17, target found! Now we have found the target, let's trace back the solution: Final step: 20 - 3 = 17 The step before: 43 - 23 = 20 The first step: 12 + 11 = 23 Output the solution in the required format: <Solution> 12 + 11 = 23 43 - 23 = 20 20 - 3 = 17 </Solution>
{"nums": [12, 11, 43, 3], "target": 17, "solution": ["12 + 11 = 23", "43 - 23 = 20", "20 - 3 = 17"]}
countdown_0.5k_76
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [45, 5, 13, 30] Target: 33
# Search Procedure Initial number set: [45, 5, 13, 30], target: 33. Options for choosing two numbers: [(45, 5), (45, 13), (45, 30), (5, 13), (5, 30), (13, 30)]. |- Pick two numbers (45, 5) (numbers left: [13, 30]). Try possible operations. |- Try 45 + 5 = 50. Add 50 to the number set. Current number set: [50, 13, 30], target: 33. Options for choosing two numbers: [(50, 13), (50, 30), (13, 30)]. |- Pick two numbers (50, 13) (numbers left: [30]). Try possible operations. |- Try 50 + 13 = 63. Add 63 to the number set. Current number set: [63, 30], target: 33, just two numbers left. |- Try 63 + 30 = 93. Evaluate 93 != 33, drop this branch. |- Try 63 - 30 = 33. Evaluate 33 == 33, target found! Now we have found the target, let's trace back the solution: Final step: 63 - 30 = 33 The step before: 50 + 13 = 63 The first step: 45 + 5 = 50 Output the solution in the required format: <Solution> 45 + 5 = 50 50 + 13 = 63 63 - 30 = 33 </Solution>
{"nums": [45, 5, 13, 30], "target": 33, "solution": ["45 + 5 = 50", "50 + 13 = 63", "63 - 30 = 33"]}
countdown_0.5k_77
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [8, 49, 18, 32] Target: 43
# Search Procedure Initial number set: [8, 49, 18, 32], target: 43. Options for choosing two numbers: [(8, 49), (8, 18), (8, 32), (49, 18), (49, 32), (18, 32)]. |- Pick two numbers (8, 49) (numbers left: [18, 32]). Try possible operations. |- Try 49 + 8 = 57. Add 57 to the number set. Current number set: [57, 18, 32], target: 43. Options for choosing two numbers: [(57, 18), (57, 32), (18, 32)]. |- Pick two numbers (57, 18) (numbers left: [32]). Try possible operations. |- Try 57 + 18 = 75. Add 75 to the number set. Current number set: [75, 32], target: 43, just two numbers left. |- Try 75 + 32 = 107. Evaluate 107 != 43, drop this branch. |- Try 75 - 32 = 43. Evaluate 43 == 43, target found! Now we have found the target, let's trace back the solution: Final step: 75 - 32 = 43 The step before: 57 + 18 = 75 The first step: 49 + 8 = 57 Output the solution in the required format: <Solution> 49 + 8 = 57 57 + 18 = 75 75 - 32 = 43 </Solution>
{"nums": [8, 49, 18, 32], "target": 43, "solution": ["49 + 8 = 57", "57 + 18 = 75", "75 - 32 = 43"]}
countdown_0.5k_78
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [24, 9, 29, 39] Target: 23
# Search Procedure Initial number set: [24, 9, 29, 39], target: 23. Options for choosing two numbers: [(24, 9), (24, 29), (24, 39), (9, 29), (9, 39), (29, 39)]. |- Pick two numbers (24, 9) (numbers left: [29, 39]). Try possible operations. |- Try 24 + 9 = 33. Add 33 to the number set. Current number set: [33, 29, 39], target: 23. Options for choosing two numbers: [(33, 29), (33, 39), (29, 39)]. |- Pick two numbers (33, 29) (numbers left: [39]). Try possible operations. |- Try 33 + 29 = 62. Add 62 to the number set. Current number set: [62, 39], target: 23, just two numbers left. |- Try 62 + 39 = 101. Evaluate 101 != 23, drop this branch. |- Try 62 - 39 = 23. Evaluate 23 == 23, target found! Now we have found the target, let's trace back the solution: Final step: 62 - 39 = 23 The step before: 33 + 29 = 62 The first step: 24 + 9 = 33 Output the solution in the required format: <Solution> 24 + 9 = 33 33 + 29 = 62 62 - 39 = 23 </Solution>
{"nums": [24, 9, 29, 39], "target": 23, "solution": ["24 + 9 = 33", "33 + 29 = 62", "62 - 39 = 23"]}
countdown_0.5k_79
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [38, 23, 23, 11] Target: 27
# Search Procedure Initial number set: [38, 23, 23, 11], target: 27. Options for choosing two numbers: [(38, 23), (38, 23), (38, 11), (23, 23), (23, 11), (23, 11)]. |- Pick two numbers (38, 23) (numbers left: [23, 11]). Try possible operations. |- Try 38 + 23 = 61. Add 61 to the number set. Current number set: [61, 23, 11], target: 27. Options for choosing two numbers: [(61, 23), (61, 11), (23, 11)]. |- Pick two numbers (61, 23) (numbers left: [11]). Try possible operations. |- Try 61 + 23 = 84. Add 84 to the number set. Current number set: [84, 11], target: 27, just two numbers left. |- Try 84 + 11 = 95. Evaluate 95 != 27, drop this branch. |- Try 84 - 11 = 73. Evaluate 73 != 27, drop this branch. |- Try 84 * 11 = 924. Evaluate 924 != 27, drop this branch. |- Try 84 / 11 = 7.6. 7.6 is a decimal, drop this branch. |- Try 61 - 23 = 38. Add 38 to the number set. Current number set: [38, 11], target: 27, just two numbers left. |- Try 38 + 11 = 49. Evaluate 49 != 27, drop this branch. |- Try 38 - 11 = 27. Evaluate 27 == 27, target found! Now we have found the target, let's trace back the solution: Final step: 38 - 11 = 27 The step before: 61 - 23 = 38 The first step: 38 + 23 = 61 Output the solution in the required format: <Solution> 38 + 23 = 61 61 - 23 = 38 38 - 11 = 27 </Solution>
{"nums": [38, 23, 23, 11], "target": 27, "solution": ["38 + 23 = 61", "61 - 23 = 38", "38 - 11 = 27"]}
countdown_0.5k_80
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [2, 12, 25, 39] Target: 50
# Search Procedure Initial number set: [2, 12, 25, 39], target: 50. Options for choosing two numbers: [(2, 12), (2, 25), (2, 39), (12, 25), (12, 39), (25, 39)]. |- Pick two numbers (2, 12) (numbers left: [25, 39]). Try possible operations. |- Try 12 + 2 = 14. Add 14 to the number set. Current number set: [14, 25, 39], target: 50. Options for choosing two numbers: [(14, 25), (14, 39), (25, 39)]. |- Pick two numbers (14, 25) (numbers left: [39]). Try possible operations. |- Try 25 + 14 = 39. Add 39 to the number set. Current number set: [39, 39], target: 50, just two numbers left. |- Try 39 + 39 = 78. Evaluate 78 != 50, drop this branch. |- Try 39 - 39 = 0. Evaluate 0 != 50, drop this branch. |- Try 39 * 39 = 1521. Evaluate 1521 != 50, drop this branch. |- Try 39 / 39 = 1. Evaluate 1 != 50, drop this branch. |- Try 25 - 14 = 11. Add 11 to the number set. Current number set: [11, 39], target: 50, just two numbers left. |- Try 39 + 11 = 50. Evaluate 50 == 50, target found! Now we have found the target, let's trace back the solution: Final step: 39 + 11 = 50 The step before: 25 - 14 = 11 The first step: 12 + 2 = 14 Output the solution in the required format: <Solution> 12 + 2 = 14 25 - 14 = 11 39 + 11 = 50 </Solution>
{"nums": [2, 12, 25, 39], "target": 50, "solution": ["12 + 2 = 14", "25 - 14 = 11", "39 + 11 = 50"]}
countdown_0.5k_81
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [10, 20, 40, 40] Target: 30
# Search Procedure Initial number set: [10, 20, 40, 40], target: 30. Options for choosing two numbers: [(10, 20), (10, 40), (10, 40), (20, 40), (20, 40), (40, 40)]. |- Pick two numbers (10, 20) (numbers left: [40, 40]). Try possible operations. |- Try 20 + 10 = 30. Add 30 to the number set. Current number set: [30, 40, 40], target: 30. Options for choosing two numbers: [(30, 40), (30, 40), (40, 40)]. |- Pick two numbers (30, 40) (numbers left: [40]). Try possible operations. |- Try 40 + 30 = 70. Add 70 to the number set. Current number set: [70, 40], target: 30, just two numbers left. |- Try 70 + 40 = 110. Evaluate 110 != 30, drop this branch. |- Try 70 - 40 = 30. Evaluate 30 == 30, target found! Now we have found the target, let's trace back the solution: Final step: 70 - 40 = 30 The step before: 40 + 30 = 70 The first step: 20 + 10 = 30 Output the solution in the required format: <Solution> 20 + 10 = 30 40 + 30 = 70 70 - 40 = 30 </Solution>
{"nums": [10, 20, 40, 40], "target": 30, "solution": ["20 + 10 = 30", "40 + 30 = 70", "70 - 40 = 30"]}
countdown_0.5k_82
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [7, 38, 22, 18] Target: 49
# Search Procedure Initial number set: [7, 38, 22, 18], target: 49. Options for choosing two numbers: [(7, 38), (7, 22), (7, 18), (38, 22), (38, 18), (22, 18)]. |- Pick two numbers (7, 38) (numbers left: [22, 18]). Try possible operations. |- Try 38 + 7 = 45. Add 45 to the number set. Current number set: [45, 22, 18], target: 49. Options for choosing two numbers: [(45, 22), (45, 18), (22, 18)]. |- Pick two numbers (45, 22) (numbers left: [18]). Try possible operations. |- Try 45 + 22 = 67. Add 67 to the number set. Current number set: [67, 18], target: 49, just two numbers left. |- Try 67 + 18 = 85. Evaluate 85 != 49, drop this branch. |- Try 67 - 18 = 49. Evaluate 49 == 49, target found! Now we have found the target, let's trace back the solution: Final step: 67 - 18 = 49 The step before: 45 + 22 = 67 The first step: 38 + 7 = 45 Output the solution in the required format: <Solution> 38 + 7 = 45 45 + 22 = 67 67 - 18 = 49 </Solution>
{"nums": [7, 38, 22, 18], "target": 49, "solution": ["38 + 7 = 45", "45 + 22 = 67", "67 - 18 = 49"]}
countdown_0.5k_83
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [28, 27, 21, 5] Target: 29
# Search Procedure Initial number set: [28, 27, 21, 5], target: 29. Options for choosing two numbers: [(28, 27), (28, 21), (28, 5), (27, 21), (27, 5), (21, 5)]. |- Pick two numbers (28, 27) (numbers left: [21, 5]). Try possible operations. |- Try 28 + 27 = 55. Add 55 to the number set. Current number set: [55, 21, 5], target: 29. Options for choosing two numbers: [(55, 21), (55, 5), (21, 5)]. |- Pick two numbers (55, 21) (numbers left: [5]). Try possible operations. |- Try 55 + 21 = 76. Add 76 to the number set. Current number set: [76, 5], target: 29, just two numbers left. |- Try 76 + 5 = 81. Evaluate 81 != 29, drop this branch. |- Try 76 - 5 = 71. Evaluate 71 != 29, drop this branch. |- Try 76 * 5 = 380. Evaluate 380 != 29, drop this branch. |- Try 76 / 5 = 15.2. 15.2 is a decimal, drop this branch. |- Try 55 - 21 = 34. Add 34 to the number set. Current number set: [34, 5], target: 29, just two numbers left. |- Try 34 + 5 = 39. Evaluate 39 != 29, drop this branch. |- Try 34 - 5 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 34 - 5 = 29 The step before: 55 - 21 = 34 The first step: 28 + 27 = 55 Output the solution in the required format: <Solution> 28 + 27 = 55 55 - 21 = 34 34 - 5 = 29 </Solution>
{"nums": [28, 27, 21, 5], "target": 29, "solution": ["28 + 27 = 55", "55 - 21 = 34", "34 - 5 = 29"]}
countdown_0.5k_84
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [34, 17, 32, 6] Target: 25
# Search Procedure Initial number set: [34, 17, 32, 6], target: 25. Options for choosing two numbers: [(34, 17), (34, 32), (34, 6), (17, 32), (17, 6), (32, 6)]. |- Pick two numbers (34, 17) (numbers left: [32, 6]). Try possible operations. |- Try 34 + 17 = 51. Add 51 to the number set. Current number set: [51, 32, 6], target: 25. Options for choosing two numbers: [(51, 32), (51, 6), (32, 6)]. |- Pick two numbers (51, 32) (numbers left: [6]). Try possible operations. |- Try 51 + 32 = 83. Add 83 to the number set. Current number set: [83, 6], target: 25, just two numbers left. |- Try 83 + 6 = 89. Evaluate 89 != 25, drop this branch. |- Try 83 - 6 = 77. Evaluate 77 != 25, drop this branch. |- Try 83 * 6 = 498. Evaluate 498 != 25, drop this branch. |- Try 83 / 6 = 13.8. 13.8 is a decimal, drop this branch. |- Try 51 - 32 = 19. Add 19 to the number set. Current number set: [19, 6], target: 25, just two numbers left. |- Try 19 + 6 = 25. Evaluate 25 == 25, target found! Now we have found the target, let's trace back the solution: Final step: 19 + 6 = 25 The step before: 51 - 32 = 19 The first step: 34 + 17 = 51 Output the solution in the required format: <Solution> 34 + 17 = 51 51 - 32 = 19 19 + 6 = 25 </Solution>
{"nums": [34, 17, 32, 6], "target": 25, "solution": ["34 + 17 = 51", "51 - 32 = 19", "19 + 6 = 25"]}
countdown_0.5k_85
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [4, 32, 35, 24] Target: 47
# Search Procedure Initial number set: [4, 32, 35, 24], target: 47. Options for choosing two numbers: [(4, 32), (4, 35), (4, 24), (32, 35), (32, 24), (35, 24)]. |- Pick two numbers (4, 32) (numbers left: [35, 24]). Try possible operations. |- Try 32 + 4 = 36. Add 36 to the number set. Current number set: [36, 35, 24], target: 47. Options for choosing two numbers: [(36, 35), (36, 24), (35, 24)]. |- Pick two numbers (36, 35) (numbers left: [24]). Try possible operations. |- Try 36 + 35 = 71. Add 71 to the number set. Current number set: [71, 24], target: 47, just two numbers left. |- Try 71 + 24 = 95. Evaluate 95 != 47, drop this branch. |- Try 71 - 24 = 47. Evaluate 47 == 47, target found! Now we have found the target, let's trace back the solution: Final step: 71 - 24 = 47 The step before: 36 + 35 = 71 The first step: 32 + 4 = 36 Output the solution in the required format: <Solution> 32 + 4 = 36 36 + 35 = 71 71 - 24 = 47 </Solution>
{"nums": [4, 32, 35, 24], "target": 47, "solution": ["32 + 4 = 36", "36 + 35 = 71", "71 - 24 = 47"]}
countdown_0.5k_86
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [17, 12, 16, 26] Target: 39
# Search Procedure Initial number set: [17, 12, 16, 26], target: 39. Options for choosing two numbers: [(17, 12), (17, 16), (17, 26), (12, 16), (12, 26), (16, 26)]. |- Pick two numbers (17, 12) (numbers left: [16, 26]). Try possible operations. |- Try 17 + 12 = 29. Add 29 to the number set. Current number set: [29, 16, 26], target: 39. Options for choosing two numbers: [(29, 16), (29, 26), (16, 26)]. |- Pick two numbers (29, 16) (numbers left: [26]). Try possible operations. |- Try 29 + 16 = 45. Add 45 to the number set. Current number set: [45, 26], target: 39, just two numbers left. |- Try 45 + 26 = 71. Evaluate 71 != 39, drop this branch. |- Try 45 - 26 = 19. Evaluate 19 != 39, drop this branch. |- Try 45 * 26 = 1170. Evaluate 1170 != 39, drop this branch. |- Try 45 / 26 = 1.7. 1.7 is a decimal, drop this branch. |- Try 29 - 16 = 13. Add 13 to the number set. Current number set: [13, 26], target: 39, just two numbers left. |- Try 26 + 13 = 39. Evaluate 39 == 39, target found! Now we have found the target, let's trace back the solution: Final step: 26 + 13 = 39 The step before: 29 - 16 = 13 The first step: 17 + 12 = 29 Output the solution in the required format: <Solution> 17 + 12 = 29 29 - 16 = 13 26 + 13 = 39 </Solution>
{"nums": [17, 12, 16, 26], "target": 39, "solution": ["17 + 12 = 29", "29 - 16 = 13", "26 + 13 = 39"]}
countdown_0.5k_87
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [12, 28, 41, 28] Target: 27
# Search Procedure Initial number set: [12, 28, 41, 28], target: 27. Options for choosing two numbers: [(12, 28), (12, 41), (12, 28), (28, 41), (28, 28), (41, 28)]. |- Pick two numbers (12, 28) (numbers left: [41, 28]). Try possible operations. |- Try 28 + 12 = 40. Add 40 to the number set. Current number set: [40, 41, 28], target: 27. Options for choosing two numbers: [(40, 41), (40, 28), (41, 28)]. |- Pick two numbers (40, 41) (numbers left: [28]). Try possible operations. |- Try 41 + 40 = 81. Add 81 to the number set. Current number set: [81, 28], target: 27, just two numbers left. |- Try 81 + 28 = 109. Evaluate 109 != 27, drop this branch. |- Try 81 - 28 = 53. Evaluate 53 != 27, drop this branch. |- Try 81 * 28 = 2268. 2268 exceeds the maximum intermediate result, drop this branch. |- Try 81 / 28 = 2.9. 2.9 is a decimal, drop this branch. |- Try 41 - 40 = 1. Add 1 to the number set. Current number set: [1, 28], target: 27, just two numbers left. |- Try 28 + 1 = 29. Evaluate 29 != 27, drop this branch. |- Try 28 - 1 = 27. Evaluate 27 == 27, target found! Now we have found the target, let's trace back the solution: Final step: 28 - 1 = 27 The step before: 41 - 40 = 1 The first step: 28 + 12 = 40 Output the solution in the required format: <Solution> 28 + 12 = 40 41 - 40 = 1 28 - 1 = 27 </Solution>
{"nums": [12, 28, 41, 28], "target": 27, "solution": ["28 + 12 = 40", "41 - 40 = 1", "28 - 1 = 27"]}
countdown_0.5k_88
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [2, 29, 10, 1] Target: 40
# Search Procedure Initial number set: [2, 29, 10, 1], target: 40. Options for choosing two numbers: [(2, 29), (2, 10), (2, 1), (29, 10), (29, 1), (10, 1)]. |- Pick two numbers (2, 29) (numbers left: [10, 1]). Try possible operations. |- Try 29 + 2 = 31. Add 31 to the number set. Current number set: [31, 10, 1], target: 40. Options for choosing two numbers: [(31, 10), (31, 1), (10, 1)]. |- Pick two numbers (31, 10) (numbers left: [1]). Try possible operations. |- Try 31 + 10 = 41. Add 41 to the number set. Current number set: [41, 1], target: 40, just two numbers left. |- Try 41 + 1 = 42. Evaluate 42 != 40, drop this branch. |- Try 41 - 1 = 40. Evaluate 40 == 40, target found! Now we have found the target, let's trace back the solution: Final step: 41 - 1 = 40 The step before: 31 + 10 = 41 The first step: 29 + 2 = 31 Output the solution in the required format: <Solution> 29 + 2 = 31 31 + 10 = 41 41 - 1 = 40 </Solution>
{"nums": [2, 29, 10, 1], "target": 40, "solution": ["29 + 2 = 31", "31 + 10 = 41", "41 - 1 = 40"]}
countdown_0.5k_89
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [36, 6, 13, 40] Target: 11
# Search Procedure Initial number set: [36, 6, 13, 40], target: 11. Options for choosing two numbers: [(36, 6), (36, 13), (36, 40), (6, 13), (6, 40), (13, 40)]. |- Pick two numbers (36, 6) (numbers left: [13, 40]). Try possible operations. |- Try 36 + 6 = 42. Add 42 to the number set. Current number set: [42, 13, 40], target: 11. Options for choosing two numbers: [(42, 13), (42, 40), (13, 40)]. |- Pick two numbers (42, 13) (numbers left: [40]). Try possible operations. |- Try 42 + 13 = 55. Add 55 to the number set. Current number set: [55, 40], target: 11, just two numbers left. |- Try 55 + 40 = 95. Evaluate 95 != 11, drop this branch. |- Try 55 - 40 = 15. Evaluate 15 != 11, drop this branch. |- Try 55 * 40 = 2200. 2200 exceeds the maximum intermediate result, drop this branch. |- Try 55 / 40 = 1.4. 1.4 is a decimal, drop this branch. |- Try 42 - 13 = 29. Add 29 to the number set. Current number set: [29, 40], target: 11, just two numbers left. |- Try 40 + 29 = 69. Evaluate 69 != 11, drop this branch. |- Try 40 - 29 = 11. Evaluate 11 == 11, target found! Now we have found the target, let's trace back the solution: Final step: 40 - 29 = 11 The step before: 42 - 13 = 29 The first step: 36 + 6 = 42 Output the solution in the required format: <Solution> 36 + 6 = 42 42 - 13 = 29 40 - 29 = 11 </Solution>
{"nums": [36, 6, 13, 40], "target": 11, "solution": ["36 + 6 = 42", "42 - 13 = 29", "40 - 29 = 11"]}
countdown_0.5k_90
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [7, 21, 10, 27] Target: 11
# Search Procedure Initial number set: [7, 21, 10, 27], target: 11. Options for choosing two numbers: [(7, 21), (7, 10), (7, 27), (21, 10), (21, 27), (10, 27)]. |- Pick two numbers (7, 21) (numbers left: [10, 27]). Try possible operations. |- Try 21 + 7 = 28. Add 28 to the number set. Current number set: [28, 10, 27], target: 11. Options for choosing two numbers: [(28, 10), (28, 27), (10, 27)]. |- Pick two numbers (28, 10) (numbers left: [27]). Try possible operations. |- Try 28 + 10 = 38. Add 38 to the number set. Current number set: [38, 27], target: 11, just two numbers left. |- Try 38 + 27 = 65. Evaluate 65 != 11, drop this branch. |- Try 38 - 27 = 11. Evaluate 11 == 11, target found! Now we have found the target, let's trace back the solution: Final step: 38 - 27 = 11 The step before: 28 + 10 = 38 The first step: 21 + 7 = 28 Output the solution in the required format: <Solution> 21 + 7 = 28 28 + 10 = 38 38 - 27 = 11 </Solution>
{"nums": [7, 21, 10, 27], "target": 11, "solution": ["21 + 7 = 28", "28 + 10 = 38", "38 - 27 = 11"]}
countdown_0.5k_91
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [17, 25, 8, 4] Target: 30
# Search Procedure Initial number set: [17, 25, 8, 4], target: 30. Options for choosing two numbers: [(17, 25), (17, 8), (17, 4), (25, 8), (25, 4), (8, 4)]. |- Pick two numbers (17, 25) (numbers left: [8, 4]). Try possible operations. |- Try 25 + 17 = 42. Add 42 to the number set. Current number set: [42, 8, 4], target: 30. Options for choosing two numbers: [(42, 8), (42, 4), (8, 4)]. |- Pick two numbers (42, 8) (numbers left: [4]). Try possible operations. |- Try 42 + 8 = 50. Add 50 to the number set. Current number set: [50, 4], target: 30, just two numbers left. |- Try 50 + 4 = 54. Evaluate 54 != 30, drop this branch. |- Try 50 - 4 = 46. Evaluate 46 != 30, drop this branch. |- Try 50 * 4 = 200. Evaluate 200 != 30, drop this branch. |- Try 50 / 4 = 12.5. 12.5 is a decimal, drop this branch. |- Try 42 - 8 = 34. Add 34 to the number set. Current number set: [34, 4], target: 30, just two numbers left. |- Try 34 + 4 = 38. Evaluate 38 != 30, drop this branch. |- Try 34 - 4 = 30. Evaluate 30 == 30, target found! Now we have found the target, let's trace back the solution: Final step: 34 - 4 = 30 The step before: 42 - 8 = 34 The first step: 25 + 17 = 42 Output the solution in the required format: <Solution> 25 + 17 = 42 42 - 8 = 34 34 - 4 = 30 </Solution>
{"nums": [17, 25, 8, 4], "target": 30, "solution": ["25 + 17 = 42", "42 - 8 = 34", "34 - 4 = 30"]}
countdown_0.5k_92
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [41, 15, 2, 18] Target: 36
# Search Procedure Initial number set: [41, 15, 2, 18], target: 36. Options for choosing two numbers: [(41, 15), (41, 2), (41, 18), (15, 2), (15, 18), (2, 18)]. |- Pick two numbers (41, 15) (numbers left: [2, 18]). Try possible operations. |- Try 41 + 15 = 56. Add 56 to the number set. Current number set: [56, 2, 18], target: 36. Options for choosing two numbers: [(56, 2), (56, 18), (2, 18)]. |- Pick two numbers (56, 2) (numbers left: [18]). Try possible operations. |- Try 56 + 2 = 58. Add 58 to the number set. Current number set: [58, 18], target: 36, just two numbers left. |- Try 58 + 18 = 76. Evaluate 76 != 36, drop this branch. |- Try 58 - 18 = 40. Evaluate 40 != 36, drop this branch. |- Try 58 * 18 = 1044. Evaluate 1044 != 36, drop this branch. |- Try 58 / 18 = 3.2. 3.2 is a decimal, drop this branch. |- Try 56 - 2 = 54. Add 54 to the number set. Current number set: [54, 18], target: 36, just two numbers left. |- Try 54 + 18 = 72. Evaluate 72 != 36, drop this branch. |- Try 54 - 18 = 36. Evaluate 36 == 36, target found! Now we have found the target, let's trace back the solution: Final step: 54 - 18 = 36 The step before: 56 - 2 = 54 The first step: 41 + 15 = 56 Output the solution in the required format: <Solution> 41 + 15 = 56 56 - 2 = 54 54 - 18 = 36 </Solution>
{"nums": [41, 15, 2, 18], "target": 36, "solution": ["41 + 15 = 56", "56 - 2 = 54", "54 - 18 = 36"]}
countdown_0.5k_93
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [6, 42, 42, 5] Target: 18
# Search Procedure Initial number set: [6, 42, 42, 5], target: 18. Options for choosing two numbers: [(6, 42), (6, 42), (6, 5), (42, 42), (42, 5), (42, 5)]. |- Pick two numbers (6, 42) (numbers left: [42, 5]). Try possible operations. |- Try 42 + 6 = 48. Add 48 to the number set. Current number set: [48, 42, 5], target: 18. Options for choosing two numbers: [(48, 42), (48, 5), (42, 5)]. |- Pick two numbers (48, 42) (numbers left: [5]). Try possible operations. |- Try 48 + 42 = 90. Add 90 to the number set. Current number set: [90, 5], target: 18, just two numbers left. |- Try 90 + 5 = 95. Evaluate 95 != 18, drop this branch. |- Try 90 - 5 = 85. Evaluate 85 != 18, drop this branch. |- Try 90 * 5 = 450. Evaluate 450 != 18, drop this branch. |- Try 90 / 5 = 18. Evaluate 18 == 18, target found! Now we have found the target, let's trace back the solution: Final step: 90 / 5 = 18 The step before: 48 + 42 = 90 The first step: 42 + 6 = 48 Output the solution in the required format: <Solution> 42 + 6 = 48 48 + 42 = 90 90 / 5 = 18 </Solution>
{"nums": [6, 42, 42, 5], "target": 18, "solution": ["42 + 6 = 48", "48 + 42 = 90", "90 / 5 = 18"]}
countdown_0.5k_94
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [32, 24, 34, 49] Target: 27
# Search Procedure Initial number set: [32, 24, 34, 49], target: 27. Options for choosing two numbers: [(32, 24), (32, 34), (32, 49), (24, 34), (24, 49), (34, 49)]. |- Pick two numbers (32, 24) (numbers left: [34, 49]). Try possible operations. |- Try 32 + 24 = 56. Add 56 to the number set. Current number set: [56, 34, 49], target: 27. Options for choosing two numbers: [(56, 34), (56, 49), (34, 49)]. |- Pick two numbers (56, 34) (numbers left: [49]). Try possible operations. |- Try 56 + 34 = 90. Add 90 to the number set. Current number set: [90, 49], target: 27, just two numbers left. |- Try 90 + 49 = 139. Evaluate 139 != 27, drop this branch. |- Try 90 - 49 = 41. Evaluate 41 != 27, drop this branch. |- Try 90 * 49 = 4410. 4410 exceeds the maximum intermediate result, drop this branch. |- Try 90 / 49 = 1.8. 1.8 is a decimal, drop this branch. |- Try 56 - 34 = 22. Add 22 to the number set. Current number set: [22, 49], target: 27, just two numbers left. |- Try 49 + 22 = 71. Evaluate 71 != 27, drop this branch. |- Try 49 - 22 = 27. Evaluate 27 == 27, target found! Now we have found the target, let's trace back the solution: Final step: 49 - 22 = 27 The step before: 56 - 34 = 22 The first step: 32 + 24 = 56 Output the solution in the required format: <Solution> 32 + 24 = 56 56 - 34 = 22 49 - 22 = 27 </Solution>
{"nums": [32, 24, 34, 49], "target": 27, "solution": ["32 + 24 = 56", "56 - 34 = 22", "49 - 22 = 27"]}
countdown_0.5k_95
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [27, 35, 29, 21] Target: 12
# Search Procedure Initial number set: [27, 35, 29, 21], target: 12. Options for choosing two numbers: [(27, 35), (27, 29), (27, 21), (35, 29), (35, 21), (29, 21)]. |- Pick two numbers (27, 35) (numbers left: [29, 21]). Try possible operations. |- Try 35 + 27 = 62. Add 62 to the number set. Current number set: [62, 29, 21], target: 12. Options for choosing two numbers: [(62, 29), (62, 21), (29, 21)]. |- Pick two numbers (62, 29) (numbers left: [21]). Try possible operations. |- Try 62 + 29 = 91. Add 91 to the number set. Current number set: [91, 21], target: 12, just two numbers left. |- Try 91 + 21 = 112. Evaluate 112 != 12, drop this branch. |- Try 91 - 21 = 70. Evaluate 70 != 12, drop this branch. |- Try 91 * 21 = 1911. Evaluate 1911 != 12, drop this branch. |- Try 91 / 21 = 4.3. 4.3 is a decimal, drop this branch. |- Try 62 - 29 = 33. Add 33 to the number set. Current number set: [33, 21], target: 12, just two numbers left. |- Try 33 + 21 = 54. Evaluate 54 != 12, drop this branch. |- Try 33 - 21 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 33 - 21 = 12 The step before: 62 - 29 = 33 The first step: 35 + 27 = 62 Output the solution in the required format: <Solution> 35 + 27 = 62 62 - 29 = 33 33 - 21 = 12 </Solution>
{"nums": [27, 35, 29, 21], "target": 12, "solution": ["35 + 27 = 62", "62 - 29 = 33", "33 - 21 = 12"]}
countdown_0.5k_96
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [20, 38, 35, 10] Target: 13
# Search Procedure Initial number set: [20, 38, 35, 10], target: 13. Options for choosing two numbers: [(20, 38), (20, 35), (20, 10), (38, 35), (38, 10), (35, 10)]. |- Pick two numbers (20, 38) (numbers left: [35, 10]). Try possible operations. |- Try 38 + 20 = 58. Add 58 to the number set. Current number set: [58, 35, 10], target: 13. Options for choosing two numbers: [(58, 35), (58, 10), (35, 10)]. |- Pick two numbers (58, 35) (numbers left: [10]). Try possible operations. |- Try 58 + 35 = 93. Add 93 to the number set. Current number set: [93, 10], target: 13, just two numbers left. |- Try 93 + 10 = 103. Evaluate 103 != 13, drop this branch. |- Try 93 - 10 = 83. Evaluate 83 != 13, drop this branch. |- Try 93 * 10 = 930. Evaluate 930 != 13, drop this branch. |- Try 93 / 10 = 9.3. 9.3 is a decimal, drop this branch. |- Try 58 - 35 = 23. Add 23 to the number set. Current number set: [23, 10], target: 13, just two numbers left. |- Try 23 + 10 = 33. Evaluate 33 != 13, drop this branch. |- Try 23 - 10 = 13. Evaluate 13 == 13, target found! Now we have found the target, let's trace back the solution: Final step: 23 - 10 = 13 The step before: 58 - 35 = 23 The first step: 38 + 20 = 58 Output the solution in the required format: <Solution> 38 + 20 = 58 58 - 35 = 23 23 - 10 = 13 </Solution>
{"nums": [20, 38, 35, 10], "target": 13, "solution": ["38 + 20 = 58", "58 - 35 = 23", "23 - 10 = 13"]}
countdown_0.5k_97
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [37, 22, 40, 20] Target: 39
# Search Procedure Initial number set: [37, 22, 40, 20], target: 39. Options for choosing two numbers: [(37, 22), (37, 40), (37, 20), (22, 40), (22, 20), (40, 20)]. |- Pick two numbers (37, 22) (numbers left: [40, 20]). Try possible operations. |- Try 37 + 22 = 59. Add 59 to the number set. Current number set: [59, 40, 20], target: 39. Options for choosing two numbers: [(59, 40), (59, 20), (40, 20)]. |- Pick two numbers (59, 40) (numbers left: [20]). Try possible operations. |- Try 59 + 40 = 99. Add 99 to the number set. Current number set: [99, 20], target: 39, just two numbers left. |- Try 99 + 20 = 119. Evaluate 119 != 39, drop this branch. |- Try 99 - 20 = 79. Evaluate 79 != 39, drop this branch. |- Try 99 * 20 = 1980. Evaluate 1980 != 39, drop this branch. |- Try 99 / 20 = 5.0. 5.0 is a decimal, drop this branch. |- Try 59 - 40 = 19. Add 19 to the number set. Current number set: [19, 20], target: 39, just two numbers left. |- Try 20 + 19 = 39. Evaluate 39 == 39, target found! Now we have found the target, let's trace back the solution: Final step: 20 + 19 = 39 The step before: 59 - 40 = 19 The first step: 37 + 22 = 59 Output the solution in the required format: <Solution> 37 + 22 = 59 59 - 40 = 19 20 + 19 = 39 </Solution>
{"nums": [37, 22, 40, 20], "target": 39, "solution": ["37 + 22 = 59", "59 - 40 = 19", "20 + 19 = 39"]}
countdown_0.5k_98
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [14, 42, 3, 16] Target: 37
# Search Procedure Initial number set: [14, 42, 3, 16], target: 37. Options for choosing two numbers: [(14, 42), (14, 3), (14, 16), (42, 3), (42, 16), (3, 16)]. |- Pick two numbers (14, 42) (numbers left: [3, 16]). Try possible operations. |- Try 42 + 14 = 56. Add 56 to the number set. Current number set: [56, 3, 16], target: 37. Options for choosing two numbers: [(56, 3), (56, 16), (3, 16)]. |- Pick two numbers (56, 3) (numbers left: [16]). Try possible operations. |- Try 56 + 3 = 59. Add 59 to the number set. Current number set: [59, 16], target: 37, just two numbers left. |- Try 59 + 16 = 75. Evaluate 75 != 37, drop this branch. |- Try 59 - 16 = 43. Evaluate 43 != 37, drop this branch. |- Try 59 * 16 = 944. Evaluate 944 != 37, drop this branch. |- Try 59 / 16 = 3.7. 3.7 is a decimal, drop this branch. |- Try 56 - 3 = 53. Add 53 to the number set. Current number set: [53, 16], target: 37, just two numbers left. |- Try 53 + 16 = 69. Evaluate 69 != 37, drop this branch. |- Try 53 - 16 = 37. Evaluate 37 == 37, target found! Now we have found the target, let's trace back the solution: Final step: 53 - 16 = 37 The step before: 56 - 3 = 53 The first step: 42 + 14 = 56 Output the solution in the required format: <Solution> 42 + 14 = 56 56 - 3 = 53 53 - 16 = 37 </Solution>
{"nums": [14, 42, 3, 16], "target": 37, "solution": ["42 + 14 = 56", "56 - 3 = 53", "53 - 16 = 37"]}
countdown_0.5k_99
[TASK] You will be given four numbers and a target number, your task is to find a way to use all four numbers exactly once, along with the basic operations (+, -, *, /), to reach the target number. [RULES] - You can use each number exactly once. - You can use the four basic operations (+, -, *, /). - The intermediate results must be integers (no decimals allowed). - The intermediate results must be positive. - The intermediate results will not exceed 2000. [APPROACH] We will solve the problem by searching. Starting from a given set of four numbers, we will follow this search process: - At each state, we will consider all possible number pairs (in order) from the current number set. Choose one pair and apply one of the four basic operations to them to obtain a new number. * If there are still numbers left, we will add the new number to the number set and continue the search. * If we have used all numbers, we will check if the new number is equal to the target number. If it is, we have found the solution. Otherwise, we will backtrack. - Suppose the two numbers we choose are a and b (where a >= b). We will try the four options (a + b), (a - b), (a * b), (a / b) to obtain the new number. Remember to always use the larger number as the first operand. - If the new number is a decimal, or exceeds the maximum intermediate result, we will discard this branch and backtrack. - We will continue this process until we reach the target number with four numbers used or exhaust all possible combinations. [EXAMPLES] # Example Numbers: [40, 19, 23, 7] Target: 29 # Search Procedure Initial number set: [40, 19, 23, 7], target: 29. Options for choosing two numbers: [(40, 19), (40, 23), (40, 7), (19, 23), (19, 7), (23, 7)]. |- Pick two numbers (40, 19) (numbers left: [23, 7]). Try possible operations. |- Try 40 + 19 = 59. Add 59 to the number set. Current number set: [59, 23, 7], target: 29. Options for choosing two numbers: [(59, 23), (59, 7), (23, 7)]. |- Pick two numbers (59, 23) (numbers left: [7]). Try possible operations. |- Try 59 + 23 = 82. Add 82 to the number set. Current number set: [82, 7], target: 29, just two numbers left. |- Try 82 + 7 = 89. Evaluate 89 != 29, drop this branch. |- Try 82 - 7 = 75. Evaluate 75 != 29, drop this branch. |- Try 82 * 7 = 574. Evaluate 574 != 29, drop this branch. |- Try 82 / 7 = 11.7. 11.7 is a decimal, drop this branch. |- Try 59 - 23 = 36. Add 36 to the number set. Current number set: [36, 7], target: 29, just two numbers left. |- Try 36 + 7 = 43. Evaluate 43 != 29, drop this branch. |- Try 36 - 7 = 29. Evaluate 29 == 29, target found! Now we have found the target, let's trace back the solution: Final step: 36 - 7 = 29 The step before: 59 - 23 = 36 The first step: 40 + 19 = 59 Output the solution in the required format: <Solution> 40 + 19 = 59 59 - 23 = 36 36 - 7 = 29 </Solution> # Example Numbers: [9, 16, 6, 18] Target: 12 # Search Procedure Initial number set: [9, 16, 6, 18], target: 12. Options for choosing two numbers: [(9, 16), (9, 6), (9, 18), (16, 6), (16, 18), (6, 18)]. |- Pick two numbers (9, 16) (numbers left: [6, 18]). Try possible operations. |- Try 16 + 9 = 25. Add 25 to the number set. Current number set: [25, 6, 18], target: 12. Options for choosing two numbers: [(25, 6), (25, 18), (6, 18)]. |- Pick two numbers (25, 6) (numbers left: [18]). Try possible operations. |- Try 25 + 6 = 31. Add 31 to the number set. Current number set: [31, 18], target: 12, just two numbers left. |- Try 31 + 18 = 49. Evaluate 49 != 12, drop this branch. |- Try 31 - 18 = 13. Evaluate 13 != 12, drop this branch. |- Try 31 * 18 = 558. Evaluate 558 != 12, drop this branch. |- Try 31 / 18 = 1.7. 1.7 is a decimal, drop this branch. |- Try 25 - 6 = 19. Add 19 to the number set. Current number set: [19, 18], target: 12, just two numbers left. |- Try 19 + 18 = 37. Evaluate 37 != 12, drop this branch. |- Try 19 - 18 = 1. Evaluate 1 != 12, drop this branch. |- Try 19 * 18 = 342. Evaluate 342 != 12, drop this branch. |- Try 19 / 18 = 1.1. 1.1 is a decimal, drop this branch. |- Try 25 * 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Pick two numbers (25, 18) (numbers left: [6]). Try possible operations. |- Try 25 + 18 = 43. Add 43 to the number set. Current number set: [43, 6], target: 12, just two numbers left. |- Try 43 + 6 = 49. Evaluate 49 != 12, drop this branch. |- Try 43 - 6 = 37. Evaluate 37 != 12, drop this branch. |- Try 43 * 6 = 258. Evaluate 258 != 12, drop this branch. |- Try 43 / 6 = 7.2. 7.2 is a decimal, drop this branch. |- Try 25 - 18 = 7. Add 7 to the number set. Current number set: [7, 6], target: 12, just two numbers left. |- Try 7 + 6 = 13. Evaluate 13 != 12, drop this branch. |- Try 7 - 6 = 1. Evaluate 1 != 12, drop this branch. |- Try 7 * 6 = 42. Evaluate 42 != 12, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Try 25 * 18 = 450. Add 450 to the number set. Current number set: [450, 6], target: 12, just two numbers left. |- Try 450 + 6 = 456. Evaluate 456 != 12, drop this branch. |- Try 450 - 6 = 444. Evaluate 444 != 12, drop this branch. |- Try 450 * 6 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 450 / 6 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 18 = 1.4. 1.4 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [25]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 25], target: 12, just two numbers left. |- Try 25 + 24 = 49. Evaluate 49 != 12, drop this branch. |- Try 25 - 24 = 1. Evaluate 1 != 12, drop this branch. |- Try 25 * 24 = 600. Evaluate 600 != 12, drop this branch. |- Try 25 / 24 = 1.0. 1.0 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 25], target: 12, just two numbers left. |- Try 25 + 12 = 37. Evaluate 37 != 12, drop this branch. |- Try 25 - 12 = 13. Evaluate 13 != 12, drop this branch. |- Try 25 * 12 = 300. Evaluate 300 != 12, drop this branch. |- Try 25 / 12 = 2.1. 2.1 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 25], target: 12, just two numbers left. |- Try 108 + 25 = 133. Evaluate 133 != 12, drop this branch. |- Try 108 - 25 = 83. Evaluate 83 != 12, drop this branch. |- Try 108 * 25 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 108 / 25 = 4.3. 4.3 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 25], target: 12, just two numbers left. |- Try 25 + 3 = 28. Evaluate 28 != 12, drop this branch. |- Try 25 - 3 = 22. Evaluate 22 != 12, drop this branch. |- Try 25 * 3 = 75. Evaluate 75 != 12, drop this branch. |- Try 25 / 3 = 8.3. 8.3 is a decimal, drop this branch. |- Try 16 - 9 = 7. Add 7 to the number set. Current number set: [7, 6, 18], target: 12. Options for choosing two numbers: [(7, 6), (7, 18), (6, 18)]. |- Pick two numbers (7, 6) (numbers left: [18]). Try possible operations. |- Try 7 + 6 = 13. Add 13 to the number set. Current number set: [13, 18], target: 12, just two numbers left. |- Try 18 + 13 = 31. Evaluate 31 != 12, drop this branch. |- Try 18 - 13 = 5. Evaluate 5 != 12, drop this branch. |- Try 18 * 13 = 234. Evaluate 234 != 12, drop this branch. |- Try 18 / 13 = 1.4. 1.4 is a decimal, drop this branch. |- Try 7 - 6 = 1. Add 1 to the number set. Current number set: [1, 18], target: 12, just two numbers left. |- Try 18 + 1 = 19. Evaluate 19 != 12, drop this branch. |- Try 18 - 1 = 17. Evaluate 17 != 12, drop this branch. |- Try 18 * 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 18 / 1 = 18. Evaluate 18 != 12, drop this branch. |- Try 7 * 6 = 42. Add 42 to the number set. Current number set: [42, 18], target: 12, just two numbers left. |- Try 42 + 18 = 60. Evaluate 60 != 12, drop this branch. |- Try 42 - 18 = 24. Evaluate 24 != 12, drop this branch. |- Try 42 * 18 = 756. Evaluate 756 != 12, drop this branch. |- Try 42 / 18 = 2.3. 2.3 is a decimal, drop this branch. |- Try 7 / 6 = 1.2. 1.2 is a decimal, drop this branch. |- Pick two numbers (7, 18) (numbers left: [6]). Try possible operations. |- Try 18 + 7 = 25. Add 25 to the number set. Current number set: [25, 6], target: 12, just two numbers left. |- Try 25 + 6 = 31. Evaluate 31 != 12, drop this branch. |- Try 25 - 6 = 19. Evaluate 19 != 12, drop this branch. |- Try 25 * 6 = 150. Evaluate 150 != 12, drop this branch. |- Try 25 / 6 = 4.2. 4.2 is a decimal, drop this branch. |- Try 18 - 7 = 11. Add 11 to the number set. Current number set: [11, 6], target: 12, just two numbers left. |- Try 11 + 6 = 17. Evaluate 17 != 12, drop this branch. |- Try 11 - 6 = 5. Evaluate 5 != 12, drop this branch. |- Try 11 * 6 = 66. Evaluate 66 != 12, drop this branch. |- Try 11 / 6 = 1.8. 1.8 is a decimal, drop this branch. |- Try 18 * 7 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 18 / 7 = 2.6. 2.6 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [7]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 7], target: 12, just two numbers left. |- Try 24 + 7 = 31. Evaluate 31 != 12, drop this branch. |- Try 24 - 7 = 17. Evaluate 17 != 12, drop this branch. |- Try 24 * 7 = 168. Evaluate 168 != 12, drop this branch. |- Try 24 / 7 = 3.4. 3.4 is a decimal, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 7], target: 12, just two numbers left. |- Try 12 + 7 = 19. Evaluate 19 != 12, drop this branch. |- Try 12 - 7 = 5. Evaluate 5 != 12, drop this branch. |- Try 12 * 7 = 84. Evaluate 84 != 12, drop this branch. |- Try 12 / 7 = 1.7. 1.7 is a decimal, drop this branch. |- Try 18 * 6 = 108. Add 108 to the number set. Current number set: [108, 7], target: 12, just two numbers left. |- Try 108 + 7 = 115. Evaluate 115 != 12, drop this branch. |- Try 108 - 7 = 101. Evaluate 101 != 12, drop this branch. |- Try 108 * 7 = 756. Evaluate 756 != 12, drop this branch. |- Try 108 / 7 = 15.4. 15.4 is a decimal, drop this branch. |- Try 18 / 6 = 3. Add 3 to the number set. Current number set: [3, 7], target: 12, just two numbers left. |- Try 7 + 3 = 10. Evaluate 10 != 12, drop this branch. |- Try 7 - 3 = 4. Evaluate 4 != 12, drop this branch. |- Try 7 * 3 = 21. Evaluate 21 != 12, drop this branch. |- Try 7 / 3 = 2.3. 2.3 is a decimal, drop this branch. |- Try 16 * 9 = 144. Add 144 to the number set. Current number set: [144, 6, 18], target: 12. Options for choosing two numbers: [(144, 6), (144, 18), (6, 18)]. |- Pick two numbers (144, 6) (numbers left: [18]). Try possible operations. |- Try 144 + 6 = 150. Add 150 to the number set. Current number set: [150, 18], target: 12, just two numbers left. |- Try 150 + 18 = 168. Evaluate 168 != 12, drop this branch. |- Try 150 - 18 = 132. Evaluate 132 != 12, drop this branch. |- Try 150 * 18 = 2700. 2700 exceeds the maximum intermediate result, drop this branch. |- Try 150 / 18 = 8.3. 8.3 is a decimal, drop this branch. |- Try 144 - 6 = 138. Add 138 to the number set. Current number set: [138, 18], target: 12, just two numbers left. |- Try 138 + 18 = 156. Evaluate 156 != 12, drop this branch. |- Try 138 - 18 = 120. Evaluate 120 != 12, drop this branch. |- Try 138 * 18 = 2484. 2484 exceeds the maximum intermediate result, drop this branch. |- Try 138 / 18 = 7.7. 7.7 is a decimal, drop this branch. |- Try 144 * 6 = 864. Add 864 to the number set. Current number set: [864, 18], target: 12, just two numbers left. |- Try 864 + 18 = 882. Evaluate 882 != 12, drop this branch. |- Try 864 - 18 = 846. Evaluate 846 != 12, drop this branch. |- Try 864 * 18 = 15552. 15552 exceeds the maximum intermediate result, drop this branch. |- Try 864 / 18 = 48. Evaluate 48 != 12, drop this branch. |- Try 144 / 6 = 24. Add 24 to the number set. Current number set: [24, 18], target: 12, just two numbers left. |- Try 24 + 18 = 42. Evaluate 42 != 12, drop this branch. |- Try 24 - 18 = 6. Evaluate 6 != 12, drop this branch. |- Try 24 * 18 = 432. Evaluate 432 != 12, drop this branch. |- Try 24 / 18 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (144, 18) (numbers left: [6]). Try possible operations. |- Try 144 + 18 = 162. Add 162 to the number set. Current number set: [162, 6], target: 12, just two numbers left. |- Try 162 + 6 = 168. Evaluate 168 != 12, drop this branch. |- Try 162 - 6 = 156. Evaluate 156 != 12, drop this branch. |- Try 162 * 6 = 972. Evaluate 972 != 12, drop this branch. |- Try 162 / 6 = 27. Evaluate 27 != 12, drop this branch. |- Try 144 - 18 = 126. Add 126 to the number set. Current number set: [126, 6], target: 12, just two numbers left. |- Try 126 + 6 = 132. Evaluate 132 != 12, drop this branch. |- Try 126 - 6 = 120. Evaluate 120 != 12, drop this branch. |- Try 126 * 6 = 756. Evaluate 756 != 12, drop this branch. |- Try 126 / 6 = 21. Evaluate 21 != 12, drop this branch. |- Try 144 * 18 = 2592. 2592 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 18 = 8. Add 8 to the number set. Current number set: [8, 6], target: 12, just two numbers left. |- Try 8 + 6 = 14. Evaluate 14 != 12, drop this branch. |- Try 8 - 6 = 2. Evaluate 2 != 12, drop this branch. |- Try 8 * 6 = 48. Evaluate 48 != 12, drop this branch. |- Try 8 / 6 = 1.3. 1.3 is a decimal, drop this branch. |- Pick two numbers (6, 18) (numbers left: [144]). Try possible operations. |- Try 18 + 6 = 24. Add 24 to the number set. Current number set: [24, 144], target: 12, just two numbers left. |- Try 144 + 24 = 168. Evaluate 168 != 12, drop this branch. |- Try 144 - 24 = 120. Evaluate 120 != 12, drop this branch. |- Try 144 * 24 = 3456. 3456 exceeds the maximum intermediate result, drop this branch. |- Try 144 / 24 = 6. Evaluate 6 != 12, drop this branch. |- Try 18 - 6 = 12. Add 12 to the number set. Current number set: [12, 144], target: 12, just two numbers left. |- Try 144 + 12 = 156. Evaluate 156 != 12, drop this branch. |- Try 144 - 12 = 132. Evaluate 132 != 12, drop this branch. |- Try 144 * 12 = 1728. Evaluate 1728 != 12, drop this branch. |- Try 144 / 12 = 12. Evaluate 12 == 12, target found! Now we have found the target, let's trace back the solution: Final step: 144 / 12 = 12 The step before: 18 - 6 = 12 The first step: 16 * 9 = 144 Output the solution in the required format: <Solution> 16 * 9 = 144 18 - 6 = 12 144 / 12 = 12 </Solution> [Problem] Now, solve the following problem. Note that: - Please carefully read the approach and examples provided above, and follow them to solve the problem. - Please ALWAYS include your search procedure (after # Search Procedure). The search procedure should follow the format of the examples provided above. - Please mark your answer with <Solution> and </Solution> tags. The solution should be a sequence of three equations exactly following the format of the examples above, with no additional text in between. Numbers: [39, 35, 10, 36] Target: 28
# Search Procedure Initial number set: [39, 35, 10, 36], target: 28. Options for choosing two numbers: [(39, 35), (39, 10), (39, 36), (35, 10), (35, 36), (10, 36)]. |- Pick two numbers (39, 35) (numbers left: [10, 36]). Try possible operations. |- Try 39 + 35 = 74. Add 74 to the number set. Current number set: [74, 10, 36], target: 28. Options for choosing two numbers: [(74, 10), (74, 36), (10, 36)]. |- Pick two numbers (74, 10) (numbers left: [36]). Try possible operations. |- Try 74 + 10 = 84. Add 84 to the number set. Current number set: [84, 36], target: 28, just two numbers left. |- Try 84 + 36 = 120. Evaluate 120 != 28, drop this branch. |- Try 84 - 36 = 48. Evaluate 48 != 28, drop this branch. |- Try 84 * 36 = 3024. 3024 exceeds the maximum intermediate result, drop this branch. |- Try 84 / 36 = 2.3. 2.3 is a decimal, drop this branch. |- Try 74 - 10 = 64. Add 64 to the number set. Current number set: [64, 36], target: 28, just two numbers left. |- Try 64 + 36 = 100. Evaluate 100 != 28, drop this branch. |- Try 64 - 36 = 28. Evaluate 28 == 28, target found! Now we have found the target, let's trace back the solution: Final step: 64 - 36 = 28 The step before: 74 - 10 = 64 The first step: 39 + 35 = 74 Output the solution in the required format: <Solution> 39 + 35 = 74 74 - 10 = 64 64 - 36 = 28 </Solution>
{"nums": [39, 35, 10, 36], "target": 28, "solution": ["39 + 35 = 74", "74 - 10 = 64", "64 - 36 = 28"]}