Inspired by agardenchair, I thought I would post some brain teasers too.

Rather easy one:

How big is the angle between the minute hand and the hour hand when the time is exactly a quarter past three (3:15)?

Slightly tougher one:

Define p to be a prime number bigger than 3. Show that (p^2) - 1 is always divisible by 24.

Close to impossible one:

Two distinct numbers were picked between 2 and 99 inclusive. The sum of the two numbers was given to Sam, and the product was given to Pat. Both of Pat and Sam are perfect logicians. They had the following conversation:

Pat: I don't know the two numbers.
Sam: Neither do I, but I already knew that you didn't know.
Pat: Ah, I know the numbers now.
Sam: Yeah, me too.

What are the two numbers? (Note: it's finding the solution that's interesting here, you don't have to show that it's the only solution.)