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

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 on the diagram. Since none of these corners can be painted, at least one cell in each of them 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_08150518f205927190b8g-2.jpg?height=1276&width=574&top_left_y=1370&top_left_x=1026)

The following diagram shows how to paint 4 corners so that no more corners can be painted.