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?

Here is why I think Bob = 356, Chuck = 1335 is a possible solution (The other solutions I guessed in my previous post follow the same reasoning).

1. In each step, the player who's turn it is has two possible guesses for his own number: It is either the sum or the difference of the other two hats. Plus of course he can reason based on the number of unsuccessful guesses by his predecessors, see below.
2. If the other two hats have identical numbers, the player immediately knows his own; it is the sum of the others, because the difference would be zero, which is not allowed.

Now here is why I think 1691 356 1335 could be a possible hat numbering for the described conversation. I am afraid the indentations don't format very well, so probably not quite readable, maybe copying into a text editor helps...

1. Player A thinks: Could it be that the answer is NOT 1691 356 1335? In that case, in the previous guess, Player C would have reckoned:
2.   Player C thinks: Could it be that the answer is NOT 979 356 1335? In that case, in the previous guess, Player B would have reckoned:
3.     Player B thinks: Could it be that the answer is    979 356 623? In that case, in the previous guess, Player A would have reckoned:
4.       Player A thinks: Could it be that the answer is NOT 979 356 623? In that case, in the previous guess, Player C would have reckoned:
5.         Player C thinks: Could it be that the answer is NOT 267 356 623? In that case, in the previous guess, Player B would have reckoned:
6.           Player B thinks: Could it be that the answer is NOT 267 356 89? In that case, in the previous guess, Player A would have reckoned:
7.             Player A thinks: Could it be that the answer is NOT 267 178 89? In that case, in the previous guess, Player C would have reckoned:
8.               Player C thinks: Could it be that the answer is    89 178 89? In that case, in the previous guess, Player B would have reckoned:
9.                 Player B thinks: Hey, I got it, my number must be 178! Although I dont care about it, Player A would have reckoned:
10.                  Player A thinks: Don't have a clue.
9.                 Player B concludes: OK, my number is 178.
8.               Player C concludes: OK, cannot be 89 178 89, otherwise Player B would have guessed the numbers earlier
7.             Player A concludes: OK, must be 267 178 89, otherwise Player C would have guessed the numbers earlier
6.           Player B concludes: OK, must be 267 356 89, otherwise Player A would have guessed the numbers earlier
5.         Player C concludes: OK, must be 267 356 623, otherwise Player B would have guessed the numbers earlier
4.       Player A concludes: OK, must be 979 356 623, otherwise Player B would have guessed the numbers earlier
3.     Player B concludes: OK, cannot be 979 356 623, otherwise Player A would have guessed the numbers earlier
2.   Player C concludes: OK, must be 979 356 1335, otherwise Player B would have guessed the numbers earlier
1. Player A concludes: OK, must be 1691 356 1335, otherwise Player C would have guessed the numbers earlier

Since Avin states that the solution is unique, something must be wrong with my deduction. What?

I am a little dizzy now, need rest after so much logic...

Posted by JLo on 2006-08-15 17:34:18