Politician Puzzle Solution


The Politician

A fairly common answer is “50 honest and 50 crooked.” Another rather frequent one is “51 honest and 49 crooked.” Both answers are wrong. Now let us see how to find the correct solution.

We are given the information that at least one person is honest. Let us pick out any one honest person, whose name, say, is Frank. Now pick any of the remaining 99; call him John. By the second given condition, at least one of the two menFrank, John-is crooked. Since Frank is not crooked, it must be John. Since John arbitrarily represents any of the remaining 99 men, then each of those 99 men must be crooked. So the answer is that one is honest and 99 are crooked.

Another way of proving it is this: the statement that given any two, at least one is crooked, says nothing more or less than that given any two, they are not both honest; in other words, no two are honest. This means that at most one is honest. Also (by the first condition), at least one is honest. Hence exactly one is honest.

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