Shapovalov A.V.

On a $5 \times 5$ grid, Petya marks several cells. Vasya wins if he can cover all these cells with non-overlapping and non-protruding L-shaped pieces consisting of three cells (L-shaped pieces can only be placed "along the grid lines"). What is the minimum number of cells Petya must mark so that Vasya cannot win?
Example. Petya can mark the cells indicated in the left figure. Then Vasya will not be able to cover more than one marked cell with one corner. But nine corners without overlaps cannot be placed on the board, since 27 $>25$.
![](https://cdn.mathpix.com/cropped/2024_05_06_b41405b02b6e888d68cag-35.jpg?height=248&width=974&top_left_y=24&top_left_x=536)

Estimate. If Petya marks fewer than nine cells, then at least one of the black cells indicated in the example will not be marked. Then Vasya will be able to cover all cells of the board except for it. Indeed, the shaded cells in the right figure can be extended to corners so that only one of the cells 1, 2, or 3 remains free. If another black cell needs to be excluded, the figure needs to be rotated.

## Answer

9 cells.