The Impossible Puzzle, also named Sum and Product Puzzle is a puzzle called "impossible" because it seems to lack sufficient information for a solution. It was first published in 1969,[1] and the name Impossible Puzzle was coined by Martin Gardner.[2] The puzzle is solvable, though not easily. There exist many similar versions of puzzles.
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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.
What are these numbers?
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
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.