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?

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.