Three logical people, A, B, and C, are wearing hats with positive integers painted on them. Each person sees the other two numbers, but not her own, and each person knows that the numbers are positive integers and that one of them is the sum of the other two.

They take turns (A, then B, then C, then A, etc.) in a contest to see who can be the first to determine her number.

During round one, A, B, and C pass. In round two, A and B again pass, at which point C states that she now knows all three numbers and that their sum is 144.

How did C figure this out?

As the total of the positive integers on the hats of all three logical people is 144 and that two of the numbers total the third, one of the numbers is 72 and the other two numbers total 72.

Given that one person's hat had the 72.

Let another's hat have 36, the third's hat will also have 36. The person with 72 would have been able to deduce, in the first round, that her own hat was 72, thus this can not be the case.

Let another's hat have 24, the third's hat will then be 48. The person with 72 would then then be able to deduce, in the first round, her own would be either 24 or 72; yet, if it were 24, the third would have been able to deduce, in the first round, that her own had been 48. Thus, this can not be the case.


It is therefore determined that the hats can neither be
{24, 48, 72} nor {36, 36, 72}.

Edited on May 3, 2009, 6:37 pm

Posted by Dej Mar on 2009-05-01 23:49:20