The Mutilated Chessboard: Part II

Shadows and chessboard

Last post, we looked at a tiling of a complete chessboard:

Chess Board

Dominoes tiling 8x8

Then we cut off the corners of the board:

Mutilated checkerboard 3

The puzzle was to figure out whether or not there is a tiling of this board, and to explain either how to do it or why it can’t be done.

Don’t read further unless you’re ready to hear the solution!

Domino Spiral

Okay, here it is.

There is no way to tile the mutilated chessboard. It’s impossible.

One explanation is this.

Each domino covers exactly two squares: a white square and a black square. Try it yourself: imagine placing a domino on the board. You’ll see that no matter where you put your imaginary domino, it has to cover exactly one white and one black.

Mutilated checkerboard 3Since each domino covers exactly one white square and one black square, any board that can be tiled must have the same number of black squares as white squares. Otherwise, the dominos can’t match up with the board.

The mutilated chessboard has 32 black squares and 30 white squares. So it cannot be tiled by dominos. No matter how hard you try, there will always be two black squares left over.

That’s the reason it can’t be done.

I hope this explanation was clear. This is kind of a surprising answer. I for one would never have thought to look at the colors of the squares on the board. That’s a pretty creative way to go about solving the puzzle.

Keep in mind that there is more than one solution to this problem. The board can’t be tiled: that’s a fact. But there are many ways to explain why. I like this reasoning. You might not. Whether or not you find this interesting is a matter of taste.

Hope you enjoyed thinking about this!

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