Problem
COM-B1-M12-P022 Set 6. Tetrominoes
#22
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The four corners are removed from an \(8\times8\) board. Prove that the remaining region cannot be tiled by straight \(1\times4\) tetrominoes.
Color cells by \(i+j\pmod4\).
Every straight tetromino covers one cell of each of the four colors \(i+j\pmod4\). On the full board each color occurs \(16\) times, and the corners have colors \(2,1,1,0\). After removal the counts are \(15,14,15,16\), not equal. Hence no tiling exists.
Strong coloring argument.