7.5. What is the minimum number of cells that need to be marked on a 5 by 5 board so that among the marked cells there are no adjacent ones (having a common side or a common vertex), and adding any one cell to these would violate the first condition?
Answer: 4 cells.

Solution: Estimation. Divide the board into four parts (see fig.). In each of them, a cell must be marked, otherwise the black cell contained in it can be added.

Example. The four black cells in the figure satisfy both conditions.

Criteria: Answer only - 0 points. Estimation only - 5 points. Example only - 2 points. 7 points are given if both estimation and example are present.

![](https://cdn.mathpix.com/cropped/2024_05_06_a46b30dcdeca657878e4g-2.jpg?height=288&width=279&top_left_y=744&top_left_x=1688)