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?

I reckon  that A, B and C have hats numbered (18, 54, 72) respectively.
Then, from the start, C can deduce that the 3 numbers must be either (18, 54, 72) or (18, 54, 36).

Assuming that they are (18, 54, 36)  (Proposition)
Then, from the start, B would deduce that  the numbers are either (18, 54, 36) or (18, 18, 36).
However, B would know that if (18, 18, 36) is true then C would win the contest in round 1. So, when C passes, B would deduce that the numbers must be (18, 54, 36), and would be able to state this and win in round 2.
However, B passes in round 2, so the proposition given above is false and the correct numbers are (18, 54, 72).

I think we need to check that A couldn't spoil things by winning in round 2. From the start she would deduce that the numbers are either (18, 54, 72) or (126, 54, 72), and the three passes in round 1 would not give her reason to eliminate either of these possibilities, as far as I can see.

If this is indeed a possible solution, are there any more...? 

Posted by Harry on 2009-05-03 15:18:34