10.3. What is the minimum number of L-shaped corners consisting of 3 cells that need to be painted in a $5 \times 5$ square so that no more L-shaped corners can be painted? (Painted L-shaped corners should not overlap.)
Answer: 4.

Solution: Let the cells of a $5 \times 5$ square be painted in such a way that no more corners can be painted. Consider the 4 corners marked in Fig. 7. Since none of these corners can be painted, at least one cell in each of these corners must be painted. Note that one corner cannot paint cells of two marked corners. Therefore, at least 4 corners must be painted.

![](https://cdn.mathpix.com/cropped/2024_05_06_0acf1b7004b6bfc557ceg-2.jpg?height=220&width=214&top_left_y=1031&top_left_x=417)

Fig. 7

![](https://cdn.mathpix.com/cropped/2024_05_06_0acf1b7004b6bfc557ceg-2.jpg?height=220&width=214&top_left_y=1031&top_left_x=867)

Fig. 8

Fig. 8 shows how to paint 4 corners so that no more corners can be painted.

Comment: It is proven that the number of painted corners is not less than $4-4$ points.

A correct example with 4 painted corners is drawn 3 points.