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?

(In reply to re: General formula by Steve Herman)

Actually, I think my formula does work for this particular case.  The way I counted rounds, Bert does not have a chance to say anything until round 2.  I guess the word "round" is a little misleading.  When I say "round number" I mean the cumulative number of responses including Arthur and Bert.  Arthur responds on odd-numbered rounds and Bert responds on even-numbered rounds.

If I were to instead define each "round" as a pair of responses, one from Arthur and one from Bert, then the formula would be modified as follows:

If A+B=x
N =
   MIN(
   ceil( A / (x-y) ),       [Arthur responds]
   ceil( B / (x-y) ) - 1   [Bert responds]
   )

If A+B=y
N =
   MIN(
   ceil( A / (x-y) ),       [Bert responds]
   ceil( B / (x-y) )        [Arthur responds]
   )

Our formulas look a little more similar with these changes.
Of course, it's still possible that I have made other mistakes.

Posted by Tristan on 2005-10-31 21:22:43