We color some cells of an $8 \times 8$ chessboard gray. What is the maximum number of cells that can be colored if we want to avoid having a gray $3 \times 1$ tromino?
We can color 43 cells as shown in the left figure.
![](https://cdn.mathpix.com/cropped/2024_05_10_ae651857c6c918274013g-68.jpg?height=418&width=886&top_left_y=1512&top_left_x=628)

We cannot do better because if there were strictly fewer than 21 white cells, one of the triominoes in the right figure would not contain one and would therefore be entirely gray.