5. Pete wants to color several cells of an $8 \times 8$ square so that for any vertex, there is a colored square to which it belongs. What is the minimum number of squares he must color?
Answer: 25

Let's mark 25 vertices of an $8 \times 8$ square (see the figure on the right). At each marked vertex, there must be a shaded square. Each square touches only one such vertex, so there must be at least 25. The example shown below indicates that the 25 shaded squares touch all the vertices of the grid.

![](https://cdn.mathpix.com/cropped/2024_05_06_d72b6c7cd13063cd4e08g-2.jpg?height=380&width=373&top_left_y=1134&top_left_x=2294)
squares touch all the vertices of the grid.

Criteria. Example: 4 points. Evaluation: 3 points.