I choose two different whole numbers, both greater than 1, whose sum is less than 100. I whisper their sum to Sam and their product to Paula. They then have the following conversation:

Paula: I don’t know the numbers.
Sam: I knew you didn’t. I don’t either.
Paula: Ah, now I know them.
Sam: Now I know them too.

Which two numbers had I chosen?

Note: assume Paula and Sam were told all the relevant information above; when their conversation begins the only thing they don't know (besides the numbers themselves) is exactly which number was whispered to the other. And, of course, assume they are actually correct about the above. :-)


This "simple" little puzzle, concocted by Hans Freudenthal in the 1960s, is now commonly (and appropriately) known by the name The Impossible Problem. In fact, you can Google that very phrase to learn more, but be careful: spoilers abound.

Amazingly, it really is solvable!
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