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 OK, I'll have a shot at it... by JLo)

There is only one solution and it is unique. It can be arrived at purely deductively, and not by playing with interesting number properties as you did, JLo, so I will refrain from saying whether a guess was correct or not unless some reasoning is provided - of course it's possible for someone to guess the two numbers, but that would spoil the point of the puzzle. Think in the lines of other puzzles of this sort where a lack of information leads to some information.

Posted by Avin on 2006-08-15 09:30:12