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Logicians, hats, and numbers (Posted on 2006-08-14) Difficulty: 5 of 5
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)

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Hints/Tips Take the downs with the ups | Comment 12 of 20 |

First of all, I don't think anyone has said recursion isn't a good idea for this problem, yet anyway :)

Second of all, no turns are wasted in my solution. If they all said "I don't know" each turn until one of them knew it, it would still take 10 turns. Where as with something like 3, 2, 1  (267 178 89) A has to wait for B to say he doesn't know to say A does.

This brings the question of whether each person is saying "I don't know" because he needs to in order to rule out another possiblitiy, or just because it is his turn to say whether he knows or not. 

If each are asked in turn if they know or not, then wasted turns are a definite possiblity, and need to be included in the solution. If each turn needs to contribute something to the conversation, then the only solution possible is the one I gave as it wastes no turns.

Thus, that may be the confusion that has occurred.


  Posted by Gamer on 2006-08-17 02:00:17
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