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 (nonzero) 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 bottomup approach: Mat has described the bottomup 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 round10 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 bottomup approach into a Cprogram 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 20060816 17:23:58 