Solution
The Sum and the Product
Show the problem again
Two integers x and y satisfy 1 < x < y and x + y ≤ 100. S is told their sum and P is told their product. P: "I don't know the numbers." S: "I knew you didn't." P: "Now I know them." S: "Now I know them too." What are x and y?
Worked solution
The answer is x = 4, y = 13. S's claim means every decomposition of the sum yields an ambiguous product, restricting the sum to a small set (11, 17, 23, 27, 29, 35, 37, 41, 47, 53). P then finds exactly one of its factorizations has a sum in that set, identifying the pair; S in turn finds exactly one split of its sum that would have let P do this. Systematically eliminating candidates leaves only sum 17, product 52: the numbers 4 and 13. (Freudenthal's "Impossible Puzzle," genuinely brutal without a computer or great bookkeeping.)
Source: The Impossible Puzzle, posed by Hans Freudenthal (1969). Statement written for AxiomIQ.