Remove two opposite corners from a chessboard. Can dominoes still cover it?
Take a standard 8x8 chessboard and cut off two opposite corner squares, leaving 62 squares. You have 31 dominoes, each covering exactly two adjacent squares. Can you tile the whole mutilated board with them, with no gaps and no overlaps?
Reveal the answer
No, and the proof needs no trial and error at all. A chessboard's 64 squares alternate colours, 32 light and 32 dark, and any single domino always covers one of each colour. But the two opposite corners removed are always the same colour, leaving 32 of one colour and only 30 of the other. Since every domino must cover one of each, 31 dominoes can never tile a board with an unequal colour count, no matter how they're arranged.