7.5. Lёsha colors cells inside a $6 \times 6$ square drawn on graph paper. Then he marks the nodes (intersections of the grid lines) to which the same number of colored and uncolored squares are adjacent. What is the maximum number of nodes that can be marked?
Answer: 45.

## Solution.

Estimation. Each grid node belongs to one, two, or four squares.

The corner vertices of the original square are adjacent to only one small square, so Lёsha will not be able to mark them. Therefore, the maximum number of marked nodes does not exceed $7 \cdot 7-4=45$.

Example.

![](https://cdn.mathpix.com/cropped/2024_05_06_5015d6d24811d58ef542g-3.jpg?height=220&width=348&top_left_y=96&top_left_x=380)

With a checkerboard coloring, Lёsha will be able to mark all nodes except the points described above.

## Criteria.

2 points. A correct coloring is devised, but the number of nodes is incorrectly counted.

3 points. A correct example is provided.

3 points. The estimation is correctly proven.

7 points. A complete, correct, and justified solution.