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?

(In reply to re(3): Possible solution (spoiler) by pcbouhid)

That C concluded the sum as 144 is fairly obvious...

1) The numbers on each hat are positive integers...no negative numbers, no zero, no fractions....
2) One of the numbers is the sum of the other two....in other words, the sum of all three numbers would be double that of the largest number.

C's number must either be the absolute difference between the two numbers she sees OR be the sum of the two numbers she sees.  As C has determined the sum to equal 144, the largest number of the three numbers must be 72.

If C sees 16 and 24, then her own number would be either 8 or 40, and the sum of the three numbers would then be 48 or 80.  If she sees 24 and 36, then her own number must either be 12 or 60, and the sum of the three numbers would be 72 or 120.  As we are given that she determined the total to be 144, she neither saw 16 and 24 nor 24 and 36.

As all three were unable to deduce their own number in the first round none of the numbers were duplicated as the absolute difference between such would be 0, which is not a positive integer, leaving the third number to be the sum of the other two and only a single choice for the person who should see the two duplicate numbers.  Similarly it is true if one of the numbers is half of the other; the absolute difference between two numbers would be the duplicate of the smaller number, leaving, again, only one possible number for the third. 

As it was possible for C to determine the three numbers in the second round, it may be the three numbers need be (x, 3x, 4x), such that x was the smallest number.  And, given the sum was 144, this would give the numbers to be (18, 54, 72).

Posted by Dej Mar on 2009-05-04 02:37:38