5. There is a paper square $7 \times 7$, all cells of which are white. What is the minimum number of cells that need to be painted black so that it is impossible to cut out a rectangle from it with at least 10 cells, all of which are white?
# Answer: 4.

Solution. Divide the $7 \times 7$ square into 5 rectangles (see figure): four $3 \times 4$ and a $1 \times 1$ square. If only three cells are shaded, there will be a white rectangle of 12 cells. How to shade 4 cells is shown in the following figure:

![](https://cdn.mathpix.com/cropped/2024_05_06_604d08537f3329c35a4fg-3.jpg?height=361&width=366&top_left_y=804&top_left_x=1514)

Grading criteria.

Example of four black cells: 3 points.

Estimation that three are not enough: 4 points.