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Yet another logic puzzle

Posted By: Maik Stiebler
Date: Friday, 21 August 2009, at 6:19 p.m.

Three players A, B and C play a logic game. Players A and B secretly choose one positive integer number each and pass them on to player C. Player C chooses two positive integer numbers and announces them. He also announces the sum of one of his numbers and the numbers of players A and C.

Example: Player A chooses 10, player B chooses 20, player C announces "My numbers are 7 and 9 and the sum of one of my numbers and your numbers is 39."

Players A and B, who are both logic geniuses and know it (and know that they know and so on...), are asked to deduce what the other player's secret number is and, beginning with player A, take alternating turns in saying if they are able to deduce it or not. The game ends as soon as one player says that he can deduce it.

The question is: Does the game always end? And why?

And don't expect an answer from me. After nearly twenty years of pondering this puzzle, I think I'm close, but the explanation is still shaky.

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