A Statement to Save a Life

Date: 05/28/97 at 12:17:43
From: Ray
Subject: How can this man save his life?

A man is lost in a forest and is trying to find his way out when he is 
found by a group of cannibals. The cannibals take him to their village 
and tell him that he can make one statement that will determine how he 
will die.  They tell him that if he tells a lie, he will die by poison 
and that if he tells the truth he will die by the bullet.  What 
statement will save his life?

Date: 05/29/97 at 19:14:41
From: Doctor Keith
Subject: Re: How can this man save his life?

Dear Ray,

Fun problem!

We can write down our information in table form:

     Statement is    |   Death by
    =================|============
       True          |   bullet
    -----------------|------------
       False         |   poison

We need to find a statement that will make their statement wrong if 
they carry it out.  Then presuming they are honorable, they cannot act 
(i.e., kill the traveller).  

It sounds good, but what does it mean? Well, we need to make our 
statement false if the man dies by a bullet and true if the man dies 
by poison. So the man tries the following:

   "You will kill me by poison."

Then let's see what happens. We have two cases to consider:

1) They shoot him. Then his statement was false, so if they were 
   honorable, they would have poisoned him, so they cannot shoot 
   him and be true to their word.

2) They poison him, thus he said a true statement and should have 
   been shot. Thus to poison him goes against their word, and they 
   cannot do it.

Thus they must let him live!  So we have a statement that works.

Of course we are assuming that the cannibals are all-knowing,
otherwise a statement like "My mother's name is Jen" would be a 
problem. In theory, your typical cannibal does not know the man's 
mother's name, so the validity of the statement is unknown. Thus to 
act could violate the cannibals word, and they supposedly won't do 
that.  

This brings up the whole area of unanswerable questions and statements 
that are not true or false. The command, "Eat your lunch" is neither 
true nor false in any circumstance, so what do they do?  This 
situation is not defined in the problem. Of course they could get 
tricky and poison the bullet to handle this. Why do I mention all 
this?  Well there is an obvious best answer:

   "You will kill me by poison."

But even given what seems a completely stated problem with a best 
answer, there are special cases where things fall apart. This is true 
in many areas of math and life in general and you just want to be on 
the lookout for how things can fail. In other words, when you first 
see a problem, read carefully and try to see whether something has not 
been stated clearly enough or where the problem lies.  Also note there 
is often ambiguity going from English (or any language) to math, and 
many problems like this come up from these issues.  

If these type of problems interest you, I suggest you look into what 
is called logic. I used an arguement in mathematical logic to get the 
best answer and to test it out. There are many great problems like 
this one out there so you can have lots of fun and develop very 
practical skills for later life.

If any of this is confusing let me know which part and I will clarify 
it.  Hope all of this helps.  

-Doctor Keith,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/