5. What is the minimum number of L-shaped corners consisting of 3 cells that need to be painted in a $6 \times 6$ square of cells so that no more L-shaped corners can be painted? (Painted L-shaped corners should not overlap.)
5. Answer. 6.

Let the cells of a $6 \times 6$ square be painted in such a way that no more corners can be painted.

Then, in each $2 \times 2$ square, at least 2 cells are painted, otherwise, a corner in this square can still be painted. By dividing the $6 \times 6$ square into 9 $2 \times 2$ squares, we get that at least $9 \cdot 2 = 18$ cells are painted. Therefore, at least 6 corners are painted. The figure below shows how to paint 6 corners so that no more corners can be painted.

![](https://cdn.mathpix.com/cropped/2024_05_06_e5b29893916304ac2b30g-1.jpg?height=386&width=357&top_left_y=2537&top_left_x=1015)