Adam, Bob, and Chuck, three perfectly intelligent logicians, are sitting facing each other with a hat on each of their heads so that each can see the others' hats but they cannot see their own. Each hat, they are told, has a (non-zero) positive integer on it, and the number on one hat is the sum of the numbers on the other two hats. The following conversation ensues:

Adam: I do not know the number on my hat.
Bob: I do not know the number on my hat.
Chuck: I do not know the number on my hat.
Adam: I do not know the number on my hat.
Bob: I do not know the number on my hat.
Chuck: I do not know the number on my hat.
Adam: I do not know the number on my hat.
Bob: I do not know the number on my hat.
Chuck: I do not know the number on my hat.
Adam: The number on my hat is 1691.

Adam was correct. What are the numbers on the other two hats?

(In reply to Another shot by JLo)

Gamer got the solution I was going for, which I thought was unique, but JLo your argument is convincing except for the fact that after a single loop around, A's reckoning fails because he knows he clearly does not see the numbers he is supposing himself to see according to your argument. The difference with Gamer's approach is that each "I don't know" can be seen to be equivalent to a mathematical statement about the numbers that person sees: for instance, A's initial statement is equivalent to the expression "B != C" and B's initial statement is equivalent to the expression "A != C and 2A != C", and so on. Can your solution be transformed in this way from the bottom up? If so I will change my solution before it is posted to admit that multiple solutions are possible.

Posted by Avin on 2006-08-16 09:30:37