The Mutilated Chessboard
Take an ordinary 8 × 8 chessboard — 64 squares — and cut away the square in the top-left corner and the square in the bottom-right corner. Sixty-two squares are left.
You have a pile of dominoes. Each domino covers exactly two squares that share an edge, lying either flat across or straight down. Dominoes may not overlap and may not hang off the board.
Two corners gone. Sixty-two squares to cover, one domino at a time.
Thirty-one dominoes would cover 62 squares exactly. Try it for a while — and then ask yourself whether trying harder is going to help.
Answer to type: the largest number of dominoes you can place on this board without overlapping.
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Colour is not decoration on a chessboard. Look at the colour of the two squares you removed, and the colours any single domino must cover.
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Answer: 30
- On a chessboard, every square touches only squares of the opposite colour. So a domino, which always covers two neighbours, covers exactly one light square and one dark square — every time, without exception.
- Therefore any number of dominoes covers equal numbers of light and dark squares.
- Now look at the corners you cut off. On a chessboard, the two ends of a long diagonal are always the same colour. You removed two squares of the same colour.
- The board started with 32 of each. It now has 32 of one colour and 30 of the other — and those two extra squares can never be paired with anything.
- So at most 30 dominoes fit, one for each square of the scarcer colour. And 30 really do fit: cover the board row by row, leaving the two odd squares out. The answer is 30 — and covering all 62 is impossible, not merely difficult.
The trick: nobody solves this by trying arrangements — there are billions. You solve it by finding a quantity that never changes no matter what you do: every domino, always, one of each colour. Then you show the target breaks that rule. This is called a parity argument, and it is one of the most powerful ideas in mathematics: proving something is impossible without examining a single case.