Domino tiling and the mutilated board

A domino covers two adjacent squares. Tiling a region means covering it with dominoes, no gaps and no overlaps. A region can be tiled only if it has an even number of squares, but that is not enough. Colour the squares like a chessboard: every domino covers one light and one dark square, so a tileable region must hold equal numbers of each colour.

This proves the famous mutilated-board result. Remove two opposite corners of the $8 \times 8$ board and you delete two squares of the same colour, leaving $32$ of one colour and $30$ of the other, so no domino tiling can exist.