In the October 2003 issue of MIT Technology Review, Donald Aucamp offered this conundrum:

Three logicians, A, B, and C, are wearing hats. Each hat displays a positive integer, and each logician can see their companions’ numbers but not their own.

All of them know that the numbers are positive integers and that one of the numbers is the sum of the other two. The three then take turns(in the order of A, B and C) to see who can determine his number first. In the first round, all three pass, but in the second round, A correctly states their number is 50. What are the other two numbers, and how did A know that theirs was 50?

P.S: Please show your thinking process also and how you arrived at the answer. I am a teacher and have to present the answer to this question to 8th graders.

All the answers here and on the internet are just stating the final answer without much details. For example, everywhere I have read on the internet simply state the following without providing any explanation why this is: A only wins if the numbers are in ratio (2,1,1).