## 1000.91 – Chessboard Dominoes

A chessboard has sixty-four squares on it. Imagine that you have some dominoes that are just the right size so that one domino will cover two squares on a chessboard. This means that you can put thirty-two dominoes on your chessboard. Now suppose that you cut off a1 and h8, two opposite corners, from your chessboard. You are left with sixty-two squares. Is it possible to arrange thirty-one dominoes to cover all the remaining squares? If so, show how you did it.

Solution

No. If you try this with an actual chessboard and dominoes, you will notice that a domino, when placed, will always cover one black square and one white square. This means that, when placing dominoes onto the board, the number of black squares covered will always equal the number of white squares covered. Because opposite corners of a chessboard are the same color, removing a1 and h8, which are both black squares, leaves us with thirty-two white squares and thirty black squares. The fact that the number of black squares does not equal the number of white squares means that this problem is impossible.

Interestingly, while this is impossible when removing any two squares of the same color—not necessarily the opposite corners—if the two squares removed are of different colors, it will always be possible. This is known as Gomory’s Theorem.