Logic Puzzles

11. Chessboard and Dominoes

You have a chessboard with two diametrically opposite squares cut out. How can you cover the remaining 62 squares completely with 31 domino pieces ?(without breaking the dominoes or the board of course). Each domino covers exactly two squares.

Submitted by dedo · Added 8 February 2008 · Updated 5 July 2026

Solution:

It is impossible to cover the remaining 62 squares with 31 dominoes after cutting out two opposite corners of the chessboard. Each domino covers one black and one white square, and removing two opposite corners (which are the same color) leaves an imbalance of colors, resulting in 32 squares of one color and 30 of the other, making it impossible to cover the board completely.

Why 31 dominoes can never cover this board

A normal chessboard has:

32 light squares
32 dark squares

The two opposite corner squares are the same color. Removing them leaves:

32 light squares
30 dark squares

But every domino placed on a chessboard must cover exactly one light square and one dark square:

31 dominoes would need
31 light squares + 31 dark squares

But the board has
32 light squares + 30 dark squares

Therefore, the board cannot be covered completely.


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Comments (6)

tlmarjot 15 April 2008

the chess board can be folded in half like a lot of chess boards.
so half of the 62 squares is 31 same as the amount of dominos. each domino covers one square completely but when the board is folded over it covers all 62squares and remains true to the fact that each domino covers two squares

dedo 3 May 2008

tlmarjot, I didn't get it - can you cover it, or not? If yes - how? If not - why?

steinjim 10 June 2008

If you cut out opposite corners, that will be two of the same color. Since each domino must cover one black and one white square, you will NOT be able to cover the remaining 62 squares with dominos (there will be 32 of one color and 30 of the other)

alexonfyre 22 December 2008

I took it to mean that one square of each color was cut out, at random...though it would be impossible to show a solution without the actual coordinates and/or a board showing such that we could draw on.

porcelina 1 August 2009

If I'm not mistaken, the opposite squares could also be squares directly across the middle from one another. In which case, you'd simply lay the dominoes horizontally, filling up each row EXCEPT the two middle rows. There would be one square left uncovered on each. You would then place the last domino vertically, thereby covering both.

fadhlarnold 18 September 2013

The blank dominoe can be left out

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