Answer to Puzzle #16: Covering a Chess Board With Dominoes

No, it is not possible to cover the remaining 62 squares of the chessboard using the dominoes without any of them sticking out over the edge of the board and without any of them overlapping.

The reason is that there are 32 black squares and 30 white squares on the chessboard, and each domino covers one black square and one white square. This means that there is one more black square than white square, and it is impossible to cover all of the squares with 31 dominoes.

Here is a proof by contradiction. Suppose that it is possible to cover the remaining 62 squares of the chessboard using the dominoes without any of them sticking out over the edge of the board and without any of them overlapping. Then there must be a way to cover the 32 black squares with 31 dominoes. But this is impossible, because each domino covers one black square and one white square. Therefore, it is not possible to cover the remaining 62 squares of the chessboard using the dominoes without any of them sticking out over the edge of the board and without any of them overlapping.

No, it is not possible to cover the remaining 62 squares of the chessboard using the dominoes without any of them sticking out over the edge of the board and without any of them overlapping.

A domino always covers two adjacent squares on the chessboard, and the chessboard has an equal number of white and black squares. Since each domino covers one white and one black square, it follows that the number of dominoes needed to cover the 62 remaining squares must be equal to each other.

However, the two diagonally opposing corners (1, 1) and (8, 8) that have been marked with an "X" are of the same color (both black or both white). Therefore, there are 31 squares of one color and 31 squares of the other color remaining on the chessboard. This means that the number of squares that need to be covered by dominoes of one color is one more than the number of squares that need to be covered by dominoes of the other color. Therefore, it is impossible to cover the remaining 62 squares of the chessboard using dominoes without any of them sticking out over the edge of the board and without any of them overlapping.

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