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Running
on
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Running
on
Zero
1.2 No 39
#81
by
mikoo2ww22
- opened
๐: Freedonia has fifty senators. Each senator is either honest
or corrupt. Suppose you know that at least one of the Freedonian senators is honest and that, given any two Freedonian senators, at least one is corrupt. Based on these
facts, can you determine how many Freedonian senators
are honest and how many are corrupt? If so, what is the
answer?
๐ค: Yes, we can use this information to determine how many Freedonian senators are honest and how many are corrupt. Here's a step-by-step solution:
- Since there are 50 senators in total, and at least one is honest, there must be at least 49 corrupt senators (since we cannot have more than half of them being honest).
- Now, suppose for the sake of contradiction that there are exactly 49 corrupt senators. This means that the remaining senator (the "at least one" mentioned in the problem statement) must be honest.
- But then, consider two random Freedonian senators, say Senator A and Senator B. Since at least one of them is corrupt, it must be the case that either Senator A or Senator B is corrupt.
- However, since we assumed that there were exactly 49 corrupt senators, it must be the case that Senator A is honest. Therefore, the number of honest senators is 1 + 1 = 2, and the number of corrupt senators is 49.
So, out of the 50 Freedonian senators, 2 are honest and 48 are corrupt.