# Mutilated Checkerboard and Dominoes Problem

The way you go about solving this problem makes a big difference in its difficulty. You are presented with a standard 8×8 checkerboard or chessboard that has two squares on opposite corners removed. You have 31 dominoes. Each domino can cover two adjacent squares. Is it possible to arrange them to cover every square? If it is possible, how do you do it? If it is impossible, how do you know?

Think about how you would go about solving this problem before reading on for the answer.

Answer and Explanation

One way of solving this problem is to begin placing dominoes on the checkerboard and trying to find a way to make them fit. This is a scientific approach. But unless you have tried every possible combination, you cannot be sure that the task is impossible.

The other approach is mathematical. Note that the opposite corners have been removed, and they were of the same color (red, in this case). Since the colors on the board alternate, each domino must cover a red square and a black square. However, there are 32 black squares and only 30 red squares. Therefore it is impossible to cover all 62 squares, because you would always be left in the end with two black squares.

This problem was proposed by Max Black in his 1946 book Critical Thinking. It is also discussed in Simon Singh’s book, Fermat’s Enigma, which contains many more mathematical puzzles.