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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)

Comments: ( Back to comment list | You must be logged in to post comments.)
Some Thoughts If only it was Chuck... | Comment 16 of 20 |
The only case when Adam could have deducted his number from the first attempt is: (2n, n, n) for Adam, Bob and Chuck (n is a natural number). The only case Bob could have deducted in the first turn is (2n, 3n, n). Below is the sequence of combinations when they could have guessed in their turn: 1A means Adams turn in the first round (skipping n for now).
 
1A: 2,1,1
1B: 2,3,1
1C: 2,3,5
2A: 8,3,5
2B: 8,13,5
2C: 8,13,21
3A: 34,13,21
3B: 34,55,21
3C: 34,55,89
4A: 144,55,89
 
Therefore, the number on Adams hat should be a multiple of 144, which is not. 
 
Note that 1691=89*19. If instead of Adam, Chuck was the one who guessed his number in the 3rd attempt, then the numbers were - Adam: 34*19=646, Bob: 55*19=1045, Chuck: 1691.
 
So what is wrong with my logic?

  Posted by Art M on 2006-08-24 21:16:43
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