Problem
COM-B1-M07-P005 A Corner of a \(5\times5\) Board
#5
★☆☆☆☆ Level 1 of 5
The cell \((1,1)\) is removed from a \(5\times5\) board. Can the remaining region be tiled by straight \(1\times3\) trominoes?
Color cell \((i,j)\) by the residue of \(i+j\pmod3\).
Under the coloring by \(i+j\pmod3\), every straight tromino covers one cell of each color. On the \(5\times5\) board, colors \(0,1,2\) occur \(9,8,8\) times. The cell \((1,1)\) has color \(2\), so after removing it the counts become \(9,8,7\). They are not equal, so no tiling exists.
First contact with diagonal three-coloring.