4. The Fork in the Road

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HERE’S A RECENT twist on an old type of logic puzzle. 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 a liar. The logician thinks a moment,
then asks one question only. From the reply he knows which road to take. What
question does he ask?

The Answer

If we require that the question be answerable by “yes” or “no,” there are
several solutions, all exploiting the same basic gimmick. For example, the
logician points to one of the roads and says to the native, “If I were to ask you if
this road leads to the village, would you say ‘yes’?” The native is forced to give
the right answer, even if he is a liar! If the road does lead to the village, the liar
would say “no” to the direct question, but as the question is put, he lies and says
he would respond “yes.” Thus the logician can be certain that the road does lead
to the village, whether the respondent is a truth-teller or a liar. On the other hand,
if the road actually does not go to the village, the liar is forced in the same way
to reply “no” to the inquirer’s question.
A similar question would be, “If I asked a member of the other tribe whether
this road leads to the village, would he say ‘yes’?” To avoid the cloudiness that
results from a question within a question, perhaps this phrasing (suggested by
Warren C. Haggstrom, of Ann Arbor, Michigan) is best: “Of the two statements,
‘You are a liar’ and ‘This road leads to the village,’ is one and only one of them
true?” Here again, a “yes” answer indicates it is the road, a “no” answer that it
isn’t, regardless of whether the native lies or tells the truth.
Dennis Sciama, Cambridge University cosmologist, and John McCarthy of
Hanover, New Hampshire, called my attention to a delightful additional twist on
the problem. “Suppose,” Mr. McCarthy wrote (in a letter published in Scientific
American, April 1957), “the logician knows that ‘pish’ and ‘tush’ are the native
words for ‘yes’ and ‘no’ but has forgotten which is which, though otherwise he
can speak the native language. He can still determine which road leads to the
village.
“He points to one of the roads and asks, ‘If I asked you whether the road I am
pointing to is the road to the village would you say pish?’ If the native replies,
‘Pish,’ the logician can conclude that the road pointed to is the road to the village
even though he will still be in the dark as to whether the native is a liar or a
truth-teller and as to whether ‘pish’ means yes or no. If the native says, ‘Tush,’
he may draw the opposite conclusion.”
For hundreds of ingenious problems involving truth-tellers and liars, see the
puzzle books by mathematician-logician Raymond M. Smullyan.

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