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 Guys, seriously by Ben)

Heh! Psychic powers - I like that. Would have certainly made the problem easier!

Alas, the "I don't know" response doesn't endow psychic powers - just a hint about what ratios numbers there cannot be on the hats.

So, to start with, if the first person, A, says "I don't know" then we know for sure that B and C's values are not the same. (If they were the same, then A would know his was the sum of those two and would not have passed.)

At each stage of "I don't know", you are gaining knowledge about the possible ratios. At the end of the puzzle you have a number of possibilities for the ratios, where only 1 ratio allows a whole number solution.

HTH,
bumble

Posted by bumble on 2006-09-13 17:51:45