5. Masha and Alina are playing on a $5 \times 5$ board. Masha can place one chip in some cells. After that, Alina covers all these cells with L-shaped pieces consisting of three cells (non-overlapping and not extending beyond the boundaries of the square, L-shaped pieces can only be placed along the grid lines). If Alina manages to cover all the cells with chips, she wins; otherwise, Masha wins. What is the minimum number of chips Masha needs to place so that Alina cannot win?
Answer: 9 chips.

Solution. Example. Masha can place chips in the cells indicated in the figure (a). Then, Alina will need nine corners, as one corner cannot cover more than one cell with a chip. However, nine corners without overlapping cannot be placed on the board, since $27>25$.

Estimate. If Masha places fewer than nine chips, then at least one of the cells indicated in the figure (a) will not have a chip. Then, Alina will be able to cover all cells of the board except for that one. For example, if there is no chip in cell 1,

![](https://cdn.mathpix.com/cropped/2024_05_06_fb99c487cce312227d62g-2.jpg?height=356&width=365&top_left_y=2035&top_left_x=1211)
a)

![](https://cdn.mathpix.com/cropped/2024_05_06_fb99c487cce312227d62g-2.jpg?height=357&width=351&top_left_y=2037&top_left_x=1618)

b) or 2, or 3 in the figure (b). Then, Alina can place 6 corners and completely cover part of the board, meaning all other chips will be covered. In the area shaded gray, two corners can be placed so that only one of the cells 1, 2, or 3 - the one without a chip - remains free. If the chip is not placed next to another side, the figure needs to be rotated.

Comment. A complete and justified solution is provided - 7 points. The correct answer and an example of the table are provided, but the estimate of the minimum value is missing or incorrect - 3 points. Only the answer is provided - 0 points.