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

~~~~~

This "simple" little puzzle, concocted by Hans Freudenthal in the 1960s, is now commonly (and appropriately) known by the name

Amazingly, it really is solvable!

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!

View 11 previous comments

- +Kevin Bourrillion - really? you must be better at math than I am :) Even knowing the correct solution (I googled my numbers and found the problem :) I have no idea how I would verify this without a program.Feb 5, 2013
- +Kevin Bourrillion - still waiting for an explanation :) How did you find it without a program? Even the first step of getting all sums that have at least one partition whose product has only one possible factorization within the rules seems hard to get to manually. Did you find some mathematical approach?Feb 6, 2013
- +Manuel Klimek Well, here's how I started. With Sam's first statement Paula knows that all number pairs with this sum (S) yield a product (P) with at least 3 prime factors. So S can't be expressible as the sum of two primes. Ergo S can't be even (Goldbach), so it has to be 2 greater than an odd composite (11, 17, 23, 27, 29, 35, 37, 41...), and the two numbers must be an even and an odd.Feb 9, 2013
- Thanks to citing Goldback I can follow you until the step where you exclude even numbers. I don't understand the next logical step (the last one is clear again). Where do you get "2 greater than an odd composite" from? (thanks for explaining btw, this is fun :)Feb 9, 2013
- +Manuel Klimek Because (a) it has to be odd, and (b) it can't be 2 greater than a prime, or it would again be the sum of two primes! Yes, it is fun.Feb 9, 2013
- /me facepalms. Of course. Heh, ok, I give up. 20 lines of python FTW, no thinking necessary :PFeb 9, 2013