The sum and product puzzle

Question

Two integers x and y satisfy 2 \le x \le y \le 99. Mr. P is told only their product xy; Ms. S is told only their sum x + y. They are truthful perfect logicians and have this conversation: 1. P: "I don't know the two numbers." 2. S: "I knew you didn't. I don't know them either." 3. P: "Now I know the two numbers." 4. S: "Now I know them too." Find x and y.

Accepted Answer

The unique answer is \boxed{x = 4,\ y = 13} (product 52, sum 17). Here is the deduction chain; each line prunes the candidate set. Statement 1, "P doesn't know." If the product had a unique factorization into two numbers in [2,99], P would know immediately. So xy must have multiple valid factorizations. In particular x,y are not both prime, and the pair isn't otherwise uniquely determined by its product. Statement 2, "S knew P couldn't know." S is certain, from the sum alone, that every factorization consistent with that sum leaves P ambiguous. The critical consequence: the sum s = x+y cannot…

The sum and product puzzle

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