An impossible problem
OK, here's to every geek out there (and I know there are many of you):
Two different integers between 1 and 100 are picked out of an opaque green hat. A person (call him P) is given the product of the two numbers. A person (call him S) is given the sum of these two numbers. After this, they have this following, rather stilted conversation:
P: I don't know that those two numbers were.
S: Yeah, I knew you didn't know. I don't know them either.
P: Ah, in that case I know the numbers.
S: Yes, now I know them too.
What were those two numbers?
At first, it seems to be impossible to answer this question, but in fact, it is solvable. Once you've baked your noodle long enough, check out the correct answer, or another explanation here. (Via Tangra on IRC.)
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