We remove the four corners of an $n \times n$ rectangle. Is it possible to cover it with L-tetrominoes?
Answer: $n=4k+2$. We need to pave $n^2-4$ squares with tiles of 4 squares, so $n$ must be even. We color the rectangle in alternating black and white lines. Then each tile covers either 3 white squares and 1 black, or the opposite. Thus, we must have an even number of tiles to cover the same number of white and black squares. Therefore, $n^2-4 \equiv 0 \bmod 8$, so $n \equiv 2 \bmod 4$. It is easy to construct a solution for all $n$ of this type.