Logic - Liars & Truthtellers (What Question Does She Ask?)

Date: 12 Mar 1995 00:21:59 -0500
From: Anonymous
Subject: Another Math Question

A logician vacationing in the South Seas finds herself on an 
island inhabited by the two proverbial tribes of liars and 
truth-tellers. The members of one tribe always tell the truth, 
and the members of the other always lie.  She comes to a fork 
in the road and has to ask a native bystander which path she 
should take to reach a village.  She has no way of telling 
whether the native is a liar or a truth-teller.  To further 
complicate matters, the natives of the island will only answer 
yes-or-no questions. The logician thinks a moment, then asks 
ONE question only. From the reply, she knows which road to 
take.  What question does she ask?

Thanks a whole lot!

Date: 12 Mar 1995 12:44:31 -0500
From: Dr. Ethan
Subject: Re: Another Math Question

Hello!

The key here is to ensure that you phrase the question so that
you will get the same answer from both parties.  To do this, you 
need to ask a question that will involve both of their opinions.

Think about this question:  Is water wet?
        The liar will say no, the truth teller yes.

But what if I ask this question:  "If I were to ask a member of the 
tribe that you don't belong to 'Is water wet?' what would he say?"

Well, the truth teller would answer truthfully that the liars would 
lie and say no.  The liar would lie and say that the truth tellers 
would say no.

So you see with this new way that we have phrased the question, 
we have guaranteed a falsehood, so the opposite must be true.  
Think how you might phrase a question related to your problem in 
a similar way.

Good Luck.

[See also the Classic Problem "Liars and Truthtellers" in the
Dr. Math FAQ: 
 
   http://mathforum.org/dr.math/faq/faq.liar.html   .]

Ethan, Doctor On Call