7. What is the minimum number of cells that need to be painted in a square with a side of 65 cells (65x65 - there are a total of 4225 cells in the square), so that among any four of its cells forming a corner (an "L" shape), there is at least one painted cell.

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# Solution.

Shading should be done diagonally every 3rd (see fig.). Thus, $\left[\frac{N^{2}}{3}\right]$ cells will be shaded. This is the minimum possible number, as within any $3 \times 3$ square, at least three cells need to be shaded. $\left[\frac{65^{2}}{3}\right]=1408$.

![](https://cdn.mathpix.com/cropped/2024_05_06_0d43a086e0070a45fd08g-04.jpg?height=537&width=643&top_left_y=2030&top_left_x=181)

Answer: 1408.