Problem
COM-B1-M07-P018 Main Diagonal and Tetrominoes
#18
★★★☆☆ Level 3 of 5
All cells on the main diagonal of an \(8\times8\) board are removed. Can the remaining region be tiled by straight \(1\times4\) tetrominoes?
Check not only chessboard coloring, but also \(i+j\pmod4\).
Color cell \((i,j)\) by \(i+j\pmod4\). A straight tetromino covers one cell of each color. On the full \(8\times8\) board each color occurs \(16\) times. On the main diagonal the colors alternate only between \(0\) and \(2\): four cells of color \(0\) and four of color \(2\) are removed, while colors \(1\) and \(3\) are not removed. The remaining counts are \(12,16,12,16\), not all equal. Hence no tiling exists.
Teaches changing the coloring modulus to match the tile length.