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?

Here is my take on the example given earlier of A=4, B=9, X=13, Y=16.

Q1: Ask Arthur – Do you know Bert’s number?
Arthur’s thoughts: My number (4) plus Bert’s number must equal 13 or 16 (X or Y). So, Bert’s number must be 9 or 12, but I can’t be sure which. NO

Q2: Ask Bert – Do you know Arthur’s number?
Bert’s thoughts: My number (9) plus Arthur’s number must equal 13 or 16 (X or Y). So, Arthur’s number must be 4 or 7, but I can’t be sure which. NO

Q3: Ask Arthur – Do you know Bert’s number?
Arthur’s thoughts: I know Bert’s number is a 9 or 12. By now, Bert must have also narrowed my number down to two possibilities, but exactly what would Bert think I have? Well if Bert’s number is 9, he would think I have a 4 or 7, and if his number were 12, he would think I have a 1 or 4. Hmmm, if I did have a 1, I would know Bert must have a 12 and would be able to answer “yes”, but I still don’t have enough. NO

Q4: Ask Bert – Do you know Arthur’s number?
Bert’s thoughts: I know Arthur’s number is a 4 or 7. By now, Arthur must have also narrowed my number down to two possibilities, but exactly what would Arthur think I have? Well if Arthur’s number is 4, he would think I have a 9 or 12, and if his number were 7, he would think I have a 6 or 9., but I still don’t have enough. NO

Q5: Ask Arthur – Do you know Bert’s number?
Arthur’s thoughts: I know Bert’s number is a 9 or 12. If Bert’s number is a 12, he would think I must have a 1 or 4. But I would have been able to answer “yes” at Q3 if I had a 1, and Bert would know that. So Bert would reason that because I answered “no” at Q3, I must have a 4. Which means that if Bert’s number is 12, he would have been able to answer “yes” at Q4. But he answered “no” at Q4, which means Bert’s number can’t be 12, it must be 9. YES

Posted by Bender on 2005-11-02 19:10:03