The Mutilated Chessboard

The Mutilated Chessboard


It’s Mathsy Mouse Here’s a chessboard. It has 8 squares in length and 8 squares in height. Let’s say we want to cover up the board with dominoes. Each domino covers two squares, and it can be placed horizontally or vertically. I have 64 squares on my chessboard, so 32 dominoes are needed to completely cover it. That was a little bit too easy ! What if we removed the two white corners from the chessboard? Do you think it will be any harder? This was a challenge proposed by the philosopher Max Black in 1946. There are certainly many ways to try and cover up the board. However, one crucial fact makes this task impossible. Every time we put a domino down, we cover exactly one white square and one black square. If we keep doing this, we will eventually be left with two black squares. These black squares will not be next to each other and it won’t be possible to cover them with a single domino. We can visualise this much easier on a smaller 4 by 4 chessboard. The two black corner squares have been removed. In this example there were 8 white squares and 6 black squares. The two extra white squares are left over. Because we removed two squares of the same colour, it is not possible to cover the board I wonder if there’s a way that we can mutilate our chess board, but still be able to cover it completely with dominoes. The only way that might work is if we remove squares that are different colours. Let’s also mix things up and remove squares which are near the middle of the board rather than the corner. This is really like a game of Tetris. We now have an equal number of white and black squares. Each domino covers one white and one black square. So I expect we’ll be able to completely cover the chessboard. You still have to be careful how you arrange the dominoes. What do you think will happen if you remove all four corners of the chessboard? Give it a try and let me know in the comments.

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