Problem
COM-B1-M11-P022 Four Corners
#22
★★★★☆ Level 4 of 5
The four corners are removed from an \(8\times8\) board. Prove that the remaining region cannot be tiled by straight \(1\times4\) tetrominoes.
Use coloring \(i+j\pmod4\).
Color cell \((i,j)\) by \(i+j\pmod4\). Any straight tetromino covers one cell of each of the four colors. On the full \(8\times8\) board each color occurs \(16\) times. The corners have colors \(2,1,1,0\), so after removal the color counts become \(15,14,15,16\), not equal. Hence no tiling exists.
Strong coloring in mixed practice.