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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)

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 I've solve it! | Comment 17 of 20 |

I think there are more good answers than the only one.

X is an odd number, maybe 1 or other number - we don't know, but logicians know that for sure.

This is my first solution:
Do you agree with this: if A has x, B has 2x, and C has x, we'll hear "I don't know" one time and "I know!" as second answer
(from Mr B)?

Do you agree with this: in configuration x, 2x, 3x, we'll hear "I know!" as a third answer (from Mr C)?

Do you agree with this: in 5x, 2x, 3x, we'll hear "I know!" as a fourth answer (from Mr A)?

Do you agree with this: in 5x, 2x, 7x, we'll hear "I know!" as a sixth answer?

Do you agree with this: in 5x, 12x, 7x, we'll hear "I know!" as a eight answer?

And finally do you agree with this: in 19x, 12x, 7x, we'll hear "I know!" as a tenth answer (from Mr A)?

So, we know also that: 19*89=1691. x=89. B and C have: 623 and 1068 (we don't know which number)

So, every logician knows what number he has, when Mr A says "I've 1691", but we don't know, becouse there are second
solution:
x, x, 2x
3x, x, 2x
3x, 5x, 2x
3x, 5x, 8x
3x, 11x, 8x
19x, 11x, 8x
So, B and C have 979 and 712.

 Posted by Hetman_elephant on 2006-09-06 05:12:37

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