Logical Liars, Paradoxical Politicians


One is a truth-teller, one a liar, and one a normal person who sometimes lies and sometimes tells the truth. (They all know each other's status.) The reporter doesn't know who is who, but this time he can ask two Yes or No questions, each directed to a single informant, to determine Senator S's involvement. What questions should he ask and of whom should he ask them?

Answer: Since the previous puzzle showed that we can handle situations with truth-tellers and liars, our goal here is to use one of our questions to find an informant who isn't normal. Once we've located him, we've reduced the problem to the previous one.

Thus, the first question should be directed toward A and it should be: Are the following two statements -- you are a truth-teller, and B is normal -- either both true or both false?

Assume A answers Yes. If A is a truth-teller or a liar, then we know we can trust the answer, and B must be normal and hence C is not normal. If A is not a truth-teller or a liar, then he must be normal and again we conclude that C is not normal. Either way a Yes answer means C is not normal.

On the other hand, if A answers No and he is a truth-teller or a liar, then we can trust his answer and conclude that B is not normal. If A is not a truth-teller or a liar, then again we know that B is not normal since A is.

Either way a No means that B is not normal. If we get a Yes, we ask C the second question; if we get a No, we ask B the second question.

And what is the second question? It's the one posed in the first scenario involving only a truth-teller and a liar. Are the two statements -- 1) you are a truth-teller, and 2) the Senator is involved in this scandal -- either both true or both false?

Would There Were a Completely Dishonest Person

The moral of the story is that complete liars can be as informative as truth-tellers. Diogenes, who, as Greek legend had it, spent his life looking for a totally honest man, should have expanded the object of his search. A totally dishonest man would have done just as well. The problem is with those pesky critters who sometimes lie and sometimes tell the truth.

John Allen Paulos, a professor of mathematics at Temple University in Philadelphia, is the author of several best-selling books, including "Innumeracy," "A Mathematician Reads the Newspaper," and "Irreligion." He's on Twitter and his "Who's Counting?" column on ABCNews.com usually appears the first weekend of every month.

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