Alan and Bob are trying to figure out two numbers. They know that both numbers are integers between 1 and 100 (but not 1 or 100). Alan knows the product of the numbers, and Bob knows the sum. Their conversation goes as follows:
Alan: I can't tell what the two numbers are.
Bob: I knew you couldn't.
Alan: Ok, now I know the numbers.
Bob: Now I know them, too.
What are the two numbers?
If Alan can't tell what the numbers are, then they are not a prime and 1 (or 1 and 1). For Bob to be certain that Alan can't make the determination, they can't add up to a prime plus 1. For Alan to determine the numbers based on what Bob knows, the product must have two factorizations, all but one of which have factors that add up to one plus a prime. Finally, for Bob to be able to figure out the numbers, all possible pairs of numbers that add up to his sum must only yield one pair that has those qualities (which fortunately means the numbers should be small).
1 and 4 seem to fit the bill. Alan knows the product is 4. That means either (1 4) or (2 2). Bob knows the sum is 5, which means either (2 3) or (1 4). Bob's first statement eliminates (2 2) for Alan (because they add up to a prime plus 1). Alan's second statment elminates (2 3) for Bob (since he wouldn't have been able to make a unique determination based on Bob's statement).
I don't know if 1 and 4 are the only solution, but they do seem to work.
My guess
