Here’s one:
Alice and Bob, two perfect logicians, are captured by an evil logician, who believes that the world should be rid of all those who are not perfect logicians.
He puts them in seperate prison cells, with no communication to one another. However, he offers a way to escape.
He gives Alice the sum of two distinct positive integers greater than 2 and less than 100, and Bob the product of them.
He explains:
“I gave Alice the sum, and Bob the product, of two distinct positive integers strictly between 2 and 100. Every night, at 8:00, I will come to each of you, first Alice, then Bob, and ask if you know what the two numbers are. You may pass, or guess. If one of you guesses correctly, you will both be freed. If one of you guesses incorrectly, you will both be stuck here forever! Good luck!”
The first night, Alice passes, and then Bob passes. Bob wasn’t sure if Alice would be able to figure it out, but the next night, Alice does get it right and both are freed.
Bob says, “So the sum was odd, right?” Alice says yes.
Assuming Alice and Bob would escape at the first possible opportunity, and do not take any chances, what are the numbers? What would they be if the sum was even (2 solutions)?
(edited for clarification)
(another edit: I forgot to state that Bob wasn’t sure whether Alice would know. That’s quite an important point, otherwise there would be 2 more solutions for both the odd riddle and the even riddle.)