An email from a friend arrived just as I was cramming myself into a middle seat for a cross-country flight:
I am thinking of two whole numbers greater than 1 whose sum is less than 100. I tell this to Jason and Brendan, and give Jason the product and Brendan the sum.
Jason: I don't know what the two numbers are.
Brendan: I knew Jason wouldn't know what the two numbers are.
Jason: Ah, now I know what the two numbers are.
Brendan: Ah, now I know what they are as well.
What are the two numbers? Assume Jason and Brendan are perfectly logical, honest, smart, etc.
Talk about good timing! The hours of the flight passed quickly as I scribbled figures in my notebook. (I had the answer by the time we landed, but it was not easy.)
This beautiful puzzle is justly famous; you can read about it online by searching 'Freudenthal's Impossible Problem.'