The Mutilated Chessboard
Remove two opposite corners from an 8x8 board and try to cover it exactly with 31 dominoes. Counting the black and white squares shows why it cannot be done. You can also choose which squares to remove.
Squares: light 30, dark 32 / 0 dominoes
Tap a square, then a square next to it, to place a domino. Tap a placed domino to take it off.
How to play
Choose a board under "Board". The first one is 8x8 with two opposite corners removed. Tap a square, then a square next to it, to place a domino; tap a placed domino to take it off. On the other boards you first choose two squares to remove (tap a removed square to put it back). The light and dark counts are always shown under the board. "Hint" goes from the idea, to counting colours, to the next place to put a domino. "Auto-solve" places as many dominoes as it can. A coverable board is finished; an impossible one stops at a dead end, and the computer explains why by counting the colours.
Learn more
On a chessboard, neighbouring squares always have different colours, so a domino always covers one light and one dark square. Remove two opposite corners from an 8x8 board and both are the same colour, leaving 30 light and 32 dark squares. 31 dominoes would cover 31 light and 31 dark squares, so the numbers do not match, and no arrangement can ever work. Counting proves it is impossible without trying every arrangement, which is a lovely trick in mathematics. On the other hand, a board missing one light and one dark square can always be covered. This site checked every such pair on boards from 2x2 to 8x8.