Leiter tells me that you can win a book. If you answer this question correctly, your name goes into a drawing.
A logician vacationing in the South Seas finds himself on an island inhabited by the two proverbial tribes of liars and truth-tellers. Members of one tribe always tell the truth; members of the other always lie.
He comes to a fork in a road and has to ask a native bystander which branch he should take to reach a village. He has no way of telling whether the native is a truth-teller or liar. The logician thinks a moment and then asks one question only. From the reply, he knows which road to take.
What question does he ask?
…For our purposes here, we’ll assume that the answer is confined to a single “yes†or “no.â€
I had heard this one years ago, and cannot now remember if I figured it out on my own or not. (Probably not, knowing my limitations.) The only snag is that either I’m wrong, or else the last instruction—”the answer is confined to a single “yes†or “no.——is slightly misleading. Here’s a hint: the instruction is playing fast and loose with the use/mention distinction. Oh, and if my solution works, then there are actually two questions he could ask.
Anyway, I submitted an answer. We’ll see what happens.
ben wolfson | 01-Oct-08 at 1:44 pm | Permalink
Assume WLOG that the left road goes to the village. Ask the guy, “if I asked you to tell me which road goes to the village, would you tell me to take the left road?”.
The guy is a truth-teller: he would, so he says “yes”.
The guy is a liar: he wouldn’t, so he says “yes”.
Suppose you ask instead “if I asked you to tell me which road goes to the village, would you tell me to take the right road?”.
The guy is a truth-teller: he wouldn’t, so he says “no”.
The guy is a liar: he would, so he says “no”.
Thus you can always ask “would you tell me to take road X, if I asked”, and if you get the answer “yes”, that’s the right road, and if you get the answer “no”, it’s the wrong road.
ben wolfson | 01-Oct-08 at 1:45 pm | Permalink
It’s not a use-mention thing, it’s secretly iterating the question and exploiting the fact that the honest man’s answers are stable (you can depend on him!) whereas the liar’s answers are unstable under that sort of condition.
Paul | 01-Oct-08 at 1:54 pm | Permalink
Cool. That’s much better than the question I had devised, and you’re right that it doesn’t involve use/mention. But mine did:
“If I were to ask the a member of the *other* tribe if this is the right road, what would he say?”
The answer in this case would be a mentioning of either “yes” or “no,” not a use of it. But it would work. So it’s only a bad answer if the condition is “it must be a yes/no” question.
I didn’t win the book. :(
ben wolfson | 01-Oct-08 at 2:07 pm | Permalink
Actually, I think that’s the sort of answer that Gardner himself gives in one of the books in which this puzzle appears—I couldn’t remember the right formulation of it, though.
Paul | 01-Oct-08 at 2:15 pm | Permalink
Yeah, the first two of three possible answers to the contest (revealed here) involve “what would he say” questions. And there is definitely something fishy about such an answer when one is told that all answers must be one word, “yes” or “no.”