7. What is the minimum number of cells that need to be painted in a square with a side of 35 cells (a $35 \times 35$ square, with a total of 1225 cells), so that among any four of its cells forming a corner (an "L" shape), there is at least one painted cell.

#
# Solution.

You should color every 3rd cell diagonally (see the figure). Thus, $\left[\frac{N^{2}}{3}\right]$ cells will be colored. This is the minimum possible number, as within any $3 \times 3$ square, at least three cells need to be colored. $\left[\frac{35^{2}}{3}\right]=408$.

![](https://cdn.mathpix.com/cropped/2024_05_06_04a9d2f4e9a143a4e5c1g-04.jpg?height=831&width=639&top_left_y=650&top_left_x=183)

Answer: 408.