The impossible problem (Q1343288)

The problem attributed to in the title is the following: ``Two numbers (not necessarily different) are chosen from the range of positive integers greater than 1 and not greater than 20. Only the sum of the two numbers is given to mathematician \(S\). Only the product of the two is given to mathematician \(P\). On the telephone \(S\) says to \(P\), `I see no way you can determine my sum.' An hour later \(P\) calls him back to say, `I know your sum.' Later \(S\) calls \(P\) again to report, `Now I know your product.' What are the two numbers?'' This problem appeared in Martin Gardner's ``Mathematical Games'' department in the Scientific American in 1979. By carefully examining all possible cases, the author shows that this problem has a unique solution, a fact that was missed by Gardner and the readers of his column. Also, a few related riddles are discussed.