Overview
The mutilated chessboard is a landmark example of a colouring argument: a hard-looking tiling question that collapses once you notice something invariant about the board.
How to solve The Mutilated Chessboard
- Colour the board like a real chessboard: 32 dark and 32 light squares, alternating.
- The two removed corners are diagonally opposite, so they share the same colour — leaving 30 of one colour and 32 of the other.
- Every domino, wherever it lands, covers exactly one dark and one light square, so 31 dominoes would need 31 of each colour.
The key insight
Count a conserved quantity — colour balance — rather than trying tilings. Each domino preserves the 1-to-1 colour ratio, so an unbalanced board can never be tiled.
Variations & echoes
- Colouring/parity arguments prove many impossibility results in combinatorics.
- Remove two squares of opposite colour instead and the board can always be tiled (Gomory's theorem).
Frequently asked questions
Does it matter which two corners?
Diagonally opposite corners always share a colour, so tiling is impossible. Removing two adjacent corners takes one of each colour and can be tiled.