# Is it possible to cover with dominos a chess-board with opposite corners removed?

This question was in my course book but without an answer.

From a chessboard, two boxes each located on an opposite corner, get cut away. So for example the most top-left and bottom-right box get cut away. Is it possible to cover the whole chess-board with Domino-stone's, when an half of the Domino equals the size of one box on the chess-board (1 domino-stone covers two boxes).

My solution (math): Possible

Amount of boxes on this chessboard? 64 - 2 = 62 (Two cut away)

Amount of Domino's needed to cover the whole board: 31 (62 / 2)

Amount of horizontal boxes: 7 + 7 + (8 * 6) - 14

=> #horizontalFirstRow + #horizontalLastRow + #allOtherBoxes - #verticalBoxes = 48

How many Domino's fit on 48 horizontal boxes? 48/2 = 24

31 - 24 = 7 (7 Domino's left)

We have 14 vertical boxes, so 14/2 = 7 (Used the 7 left domino's), so 0 domino's left and all boxes covered.

My solution (graphical): Impossible You will have to zig-zag the domino-stones, but in the 7th row you won't be able to do the zig-zag because the bottom-right corner is cut-away.

Question: Is it possible or not? And how to prove that mathematically?

• It's always worth trying it with a smaller example - take a $4\times 4$ board instead, and remove the top left and bottom right, then try to cover it. Nov 1, 2017 at 17:29
• This is a classing problem. Search for covering chessboard dominoes Nov 1, 2017 at 17:30

## 2 Answers

The issue with your first approach is that it only counts the squares without taking into account where the dominoes are placed.

The simplest approach here is to notice that each domino must cover one black square and one white square, but the two squares removed are the same color, so we cannot cover the board with dominoes.

It is impossible! Look at a chess board: The two corners you cut away, have the same color. However you use the dominos to cover the board, each domino will cover one white and one white square. So for the total of covered squares, there will be equal number of whites and blacks. So impossible ...

By the way, this problem is mentioned in the preface to the following book: https://www.amazon.com/Arnolds-Problems-Vladimir-I-Arnold/dp/3540207481/ref=sr_1_1?s=books&ie=UTF8&qid=1509557229&sr=1-1&keywords=arnold%27s+problems where Arnold says that in Moscow, mathematics training starts before school, and so he gives two problems he says Moscow 4-5-year olds solve in half an hour, and your problem is one of them.