A puzzle from MIT Technology Review, July/August 2008:
Each of three logicians, A, B, and C, wears a hat that displays a positive integer. The number on one of the hats is the sum of the numbers on the other two.
All the 3 know of the above information and they know that the others also know the above information. Also, they can all hear what the others say.
They make the following statements:
A: “I don’t know my number.”
B: “My number is 15.”
What numbers appear on hats A and C?
By the opening conditions
B has to see two numbers whose sum or difference is 15
What are the circumstances where A would know his number right away?
If you see two of the same number. Since your number is a positive integer, neither of the other two numbers could be a sum and they both be equal. So B and C cannot be the same number.
Given that, what does B know after A speaks that he doesn't know before?
The only way A's info is helpful is if B saw a situation where he and C having the same number was a possibility--that is, if A's number was double C's. Which also means that A's number must be bigger than C's, and therefore that C's number is not the sum.
The only way that happens, that also meets the condition that A and C add to 15 is
If A is 10 and C is 5
