Two mathematicians who can't see each other's clue somehow figure out the same two numbers
X and Y are whole numbers with 1 < X < Y and X + Y ≤ 100. Mr. S is told only their sum; Mr. P is told only their product. Mr. P says: 'I don't know X and Y — but I know that you don't know them either.' Mr. S replies: 'Now I know X and Y.' Mr. P says: 'Now I know them too.' What are X and Y?
Reveal the answer
X = 4 and Y = 13. Devised by mathematician Hans Freudenthal around 1969, the puzzle is solved by successive elimination: Mr. P's first statement rules out any product whose factor-pairs include one with a sum unique enough that Mr. S might already know it; each later statement narrows the surviving candidates further, until only one pair of numbers satisfies the entire dialogue.
— Hans Freudenthal, Sum and Product Puzzle — Nieuw Archief voor Wiskunde, 1969