Problem
COM-B2-M05-P002 Two Same-Coloured Corner Squares
#2
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From an \(8\times8\) board, two corner squares of the same colour are removed. Prove that the remaining board cannot be tiled by \(1\times2\) dominoes.
Colour the board like a chessboard.
Under chessboard colouring, each \(1\times2\) domino covers one black and one white square. Therefore any region tiled by dominoes contains equal numbers of black and white squares.
The full \(8\times8\) board has equal numbers of black and white squares. Removing two squares of the same colour destroys this equality. Hence a domino tiling is impossible.
A classic colouring invariant; the statement is used as a technique, not as source copying.