Problem
COM-B1-M07-P010 A Cell Adjacent to a Corner
#10
★★☆☆☆ Level 2 of 5
The cell \((1,2)\) is removed from a \(5\times5\) board. Prove that the remaining region cannot be tiled by straight \(1\times4\) tetrominoes.
Color cell \((i,j)\) by \(i+j\pmod4\).
Under coloring by \(i+j\pmod4\), each straight tetromino covers one cell of each of the four colors. On the \(5\times5\) board, colors \(0,1,2,3\) occur \(6,6,7,6\) times. The cell \((1,2)\) has color \(3\), so after removing it the counts are \(6,6,7,5\). They are not equal, so no tiling exists.
The area \(24\) is divisible by \(4\), so coloring is needed.