Problem
COM-B1-M07-P021 Four Corners Do Not Help
#21
★★★★☆ Level 4 of 5
The four corner cells are removed from an \(8\times8\) board. The remaining area is divisible by \(4\), and the numbers of black and white cells are equal. Prove that it still cannot be tiled by straight \(1\times4\) tetrominoes.
The usual chessboard coloring is too crude. Use four colors.
Color cell \((i,j)\) by \(i+j\pmod4\). Each straight tetromino covers one cell of each color. On the full \(8\times8\) board each color occurs \(16\) times. The corners have colors \(2,1,1,0\), so after removal the counts are \(15,14,15,16\). These are not equal, so the region cannot be split into pieces each taking one cell of each color.
Repeats the key idea at a more olympiad level: area and two colors are not always enough.