Covering a Chessboard 1

You’re given an 8x8 chessboard with 2 opposite corner tiles removed. You can place dominoes on the board, with each one covering exactly 2 adjacent tiles (you can’t place them diagonally to cover 2 tiles touching at a corner. The dominoes can be rotated by 90° increments, but cannot be overlapped. Can you cover all the empty tiles with 31 dominoes? If yes, show an arrangement of such a covering. If not, prove why it cannot be done.

Hint

There’s an interesting property of tiles on a chessboard that could point you in the right direction.

Hint

Let’s try colouring the tiles like a chessboard. Does this shine the light on anything?

Hint: Interactive board

Try playing around with the interactive board. Click in between any 2 tiles to place a domino over it. You may click on a domino to remove it.

Solution

It is impossible to cover the chessboard with dominoes.

If you had a go with the interactive board, you might’ve noticed an interesting property: you’ll end up with 2 tiles of the same colour left over. This hints at a property that we can use to prove it cannot be done.

Let’s colour the chessboard with alternating black and white tiles. Notice that each domino covers a black and white tile, so 31 dominoes would cover 31 black tiles and 31 white tiles.

However, with opposite corners removed, a chessboard will have 30 black tiles and 32 white tiles (or vice versa, depending on which corner you cut off). Thus, there can be no arrangement of dominoes that will cover the entire chessboard.

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