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 Logicians, hats, and numbers (Posted on 2006-08-14)
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 Solution Yet Submitted by Avin Rating: 3.7778 (9 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 OK, I'll have a shot at it... | Comment 1 of 20
...really not quite sure yet, but could this puzzle have multiple solutions? I believe the conversation is possible for the following values on Bob's and Chuck's hats:

Bob = 356  , Chuck = 1335
Bob = 623  , Chuck = 1068
Bob = 646  , Chuck = 1045
Bob = 712  , Chuck = 979
Bob = 979  , Chuck = 712
Bob = 1068 , Chuck = 623
Bob = 1246 , Chuck = 445

Avin, is one of these the solution you are after?

 Posted by JLo on 2006-08-14 20:50:55

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