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?

As I understand it, Arthur & Bert are being asked if they KNOW the number, not to guess the other's number.

Let's call the numbers on the blackboard X and Y, with X<Y. (If X=Y, the problem would be trivial.) Arthur's and Bert's numbers are A and B.

If X≤A, Arthur would know that B=Y-A, in the first round.

If Arthur doesn't know, then Bert knows that A<X. If X≤B, Bert would know that A=Y-B, in his first round.

If Bert doesn't know, then both know that A and B are <X... but I don't see know how can they progress beyond that, without making guesses.