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?

This is not the solution, but it may be an approach to it. I think there's only a set of possible numbers for the final solution. Otherwise, as people noted, it'll be impossible to find an answer in the middle of so many possible combinations.

If Adam managed to tell that his number is 1691 it should be that one of the others is the number 1. So, there's a limited set of solutions:

B=1 and C=1690
B=1690 and C=1
B=1 and C=1692
B=1692 and C=1

Finding the right couple is now the tricky part. One may be able to get it with some hard work trying to tell out all that common "A didn't know, B didn't know, C didn't know, but, hey, A found it" logical mambo jambo. But I'm feeling logicsick of playing that... Could someone advance in those steps?

Posted by vj on 2006-08-17 09:02:35