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?

No wait, there are multiple solutions...

1. ...because I can't really see what is wrong with the recursive approach. As opposed to what Avin states in his response, A does not hypothesize about seeing numbers he does not actually see. In fact, he hypothesizes about player B hypothesizing about player C hypothesizing about player A seeing numbers, that A does not actually see.

2. ...because when I have look at Mat's post with a list of all puzzles solvable in round 1, 2, 3... I see that there are already 2 essentially different puzzles that Player B could solve in round 5 with the answer 7 : (2,7,5) and (4,7,3). This cleary shows that this type of puzzle does not always have a unique solution. Mats post also shows that Gamer is missing some possible solution, unless I misunderstood Gamers approach.

3. ...as to the bottom-up approach: Mat has described the bottom-up approach and it can easily be implemented in a computer program. I believe however Mat has gone wrong with his ratio calculation at the end, which is why he missed the other round-10 solutions. It would be nice if one could describe all solvable puzzles by means of a handy formula. Haven't been able to figure this out yet. Probably Avin will do that in his official solution (har, har)

4. ...because I really, really wanted to know for sure and hacked Mat's bottom-up approach into a C-program and got exactly the 7 solutions posted earlier. Agreed, an ugly solution for a nice problem. Sorry I cannot post anything better than computer solution, believe me I am embarrassed...

Posted by JLo on 2006-08-16 17:23:58