We all know the classic puzzle of trying to tile a mutilated chessboard with $2\times1$ dominoes. Let's imagine that, instead of dominoes, we have $n^2$ L-shaped tetrominoes and we want to tile a $2n\times2n$ chessboard using them (in any orientation, i.e. we're allowed to turn and flip tetrominoes if required). For which values of $n$ is this possible?
I got the idea for this from a question in the Danish Mathematical Olympiad.