This puzzle was coined the “Impossible Puzzle” by Martin Gardner, a famous math and science writer that liked to create and write about math games and puzzles. The puzzle is named as such because it *appears* to provide insufficient information to solve, but it is in fact solvable! Read on for the puzzle and the (very difficult) solution.

There are two distinct whole numbers greater than 1, we can call them *x* and *y* (where *y* > *x*). We know the sum of *x* and *y* is no more than 100.

Sam and Prada are perfect logicians. Sam (“sum”) is told *x* + *y* and Prada (“product”) is told *x* * *y*, and both of them know all the information provided so far.

Sam and Prada have this conversation in which they truthfully deduce the numbers:

- Sam: I know Prada does not know
*x*and*y*. - Prada: Well now I know
*x*and*y*. - Sam: Ah, now I also know
*x*and*y*.

Can you figure out *x* and *y* using this information?