In Tripleland, natives always go in trios: a knight, a knave, and a liar.

Once I met such a trio, and I asked one of the natives a simple question ("simple" meaning, "of six words or less"); he answered, and I knew what type he was. Then, I asked another of the natives a different simple question; he answered, and I knew what type he was, and therefore, the type of the third one too.

"Logical" thinking: This cannot be. The natives could be in six possible orders. Two yes-no questions allow four combinations. Thus, you cannot pick one out of six with only two questions; you need one more!

How could this be? What's wrong with the reasoning above?

(In reply to re: Two ways by e.g.)

The question, or puzzle, was not to come up with the questions that would identify the three people, but rather to determine what was wrong with the reasoning of the '"logical" thinking'. The answer is that the "logical" thinker had discounted the possibility of truth tables like the one I presented.  Mine was an information theory answer to an information theory question.

To put it another way, the question was to show that the italicized statement did not constitute a proof that you couldn't identify one out of the six possible orders with only two yes-no questions.  The truth table I gave showed how, in general, it is possible, given luck, to determine one out of six possibilities with only two yes-no questions, even though, in the general case you'd need three.

If you want an example: the first question could have been "Are you a liar." If the answer was "yes", you know immediately the answerer is a knave. If your next question, to another of the three, is "was he right?" that will differentiate between the knight and the liar, and you're done.  If on the other hand the first respondent had said, "no, I'm not a liar", he could be either the knight or the liar, and the next question could not be framed to determine the whole sequence, and you'd need a third yes-no question.

 

Edited on September 2, 2006, 4:52 pm

Posted by Charlie on 2006-09-02 16:38:05