1. On a $7 \times 7$ checkerboard, 14 cells are marked. We will call a pair of cells with a common side interesting if at least one cell in the pair is marked. What is the maximum number of interesting pairs that can be?
# Answer: 55.

Solution. Let's call two cells adjacent if they share a side. The number of interesting pairs containing a given marked cell is no more than 4, and for a boundary cell, no more than 3. Then the total number of interesting pairs does not exceed $14 \cdot 4 = 56$. At the same time, if there are two adjacent marked cells, the interesting pair containing them is counted twice. Note that among the 14 cells in a $5 \times 5$ square, there are necessarily two adjacent ones. Therefore, among the marked cells, there is either a boundary cell or two adjacent ones. Thus, the total number of interesting pairs does not exceed 55. An example of a marking with 55 interesting pairs is provided below.

![](https://cdn.mathpix.com/cropped/2024_05_06_627b343ad954e1e3a31bg-04.jpg?height=266&width=263&top_left_y=821&top_left_x=905)