Sum & Product Puzzle

X and Y are two different integers, greater than 1, with sum less than 100. S and P are two mathematicians; S knows the sum X+Y, P knows the product X*Y, and both know the information in these two sentences. The following conversation occurs.

* P says "I cannot find these numbers."
* S says "I was sure that you could not find them."
* P says "Then, I found these numbers."
* S says "If you could find them, then I also found them."





What are these numbers?
Solution

The solution has X and Y as 4 and 13 (or vice versa), with P initially knowing the product is 52 and S knowing the sum is 17.

Initially P does not know the solution, since

52 = 4 × 13 = 2 × 26

and S knows that P does not know the solution since all the possible sums to 17 within the constraints produce similarly ambiguous products. However, each can work out the solution by eliminating other possibilities following the other's statements and that is enough for the reader to find the solution given the constraints.