Arthur and Bert each writes down a positive integer on a piece of paper and then shows it to Charles. Charles then writes two numbers on a blackboard, visible to Arthur and Bert: one of them is the sum of Arthur's and Bert's numbers, and the other is a random number.

After this Charles asks Arthur if he knows Bert's number. If Arthur says he doesn't know, then he asks Bert if he knows Arthur's number. If Bert says he doesn't know, Charles continues with Arthur, then if necessary with Bert and so on... until he gets a positive answer.

When will Charles get a positive answer?

Starting from FK reasoning, letīs say that A = 4, B = 9, X = 13 and Y = 16, WLOG.

They both know, after the first turn that A and B are less than 13.

In the second turn, Arthur knows that Bertīs number is (9, 10, 11 or 12). And answer "NO". Bert reasons that Arthurīs could be (4, 5, 6 or 7), and answer "NO".

In the third turn, Arthur knows that Bert is in doubt about (4, 5, 6 or 7) and reasons: "I know that his number is (9, 10, 11 or 12). But if he had a 12, he, knowing that my number is (4, 5, 6 or 7), would answered "4" (12+4=16, unique). So his number is (9, 10 or 11). My answer is still "NO"". Bert knows now that Arthur knows that his number is (9, 10 or 11) and reasons: "If Arthur (4, 5, 6 or 7) had a 7, he would answer "9". Since he answered "NO", his number is not a 7, that must be (4, 5 or 6). So my answer is "NO".

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Proceeding like this, they will narrowing the possibilities, and, no matter the numbers writen in the board, at some finite time, one will arrive at a unique possibility, and will answer "YES".
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Edited on October 30, 2005, 11:47 am

Edited on October 30, 2005, 11:50 am

Edited on October 30, 2005, 11:53 am

Posted by pcbouhid on 2005-10-30 11:44:09