We have 21 pieces of type $\Gamma$ (formed of three small squares). We are allowed to place them on an $8 \times 8$ chessboard (without overlapping, so that each piece covers exactly three squares). An arrangement is said to be maximal if no additional piece can be added while respecting this rule. What is the smallest $k$ such that there exists a maximal arrangement of $k$ pieces of type $\Gamma$?
On pave the chessboard with 16 squares $2 \times 2$. Each square contains two cells of the arrangement, so it covers at least 32 cells, which requires a minimum of 11 pieces. Conversely, we can construct such an arrangement:

![](https://cdn.mathpix.com/cropped/2024_05_10_e97c5d5a4c19c9000d9eg-067.jpg?height=422&width=417&top_left_y=2059&top_left_x=817)