This game will be familiar to many mathematicians, and it is always good fun to play.
I am looking to find a list of good questions with short, when-you-see-it solutions. The kind of question one could solve pretty much anywhere: over dinner, while taking a walk - essentially, without pen & paper, and where the solution lies in spotting a clever trick or fact (rather than via some monstrously power theorem).
To get an idea of what I mean, here is an example:
Q. A real number is called repetitive if its decimal expansion contains arbitrarily long blocks which are the same. Prove that the square of a repetitive number is repetitive.
(please do not post a solution, since I am sure anyone can figure it out given enough time)
Does anyone have similar chestnuts to offer?