10.4 Another participant in the competition for meaningless activity marked the centers of 13 cells in a grid rectangle of size $(N-1) \times(N+1)$ such that the distance between any two marked points is greater than 2. What is the smallest value that $N$ can take?
Solution: We will show that it is impossible to mark cells in a $6 \times 8$ rectangle (and thus in any smaller size) in such a way. Indeed, let's divide the rectangle into $2 \times 2$ squares. In each of them, the pairwise distances between the centers of the cells do not exceed $\sqrt{2}$, so no more than one cell is marked in each. Therefore, the total number of marked cells does not exceed 12.

Example for a $7 \times 9$ rectangle:

|  |  |  | $\mathrm{X}$ |  |  |  |  | $\mathrm{X}$ |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
|  | $\mathrm{X}$ |  |  |  |  | $\mathrm{X}$ |  |  |
|  |  |  |  | $\mathrm{X}$ |  |  |  |  |
| $\mathrm{X}$ |  | $\mathrm{X}$ |  |  |  |  | $\mathrm{X}$ |  |
|  |  |  |  |  | $\mathrm{X}$ |  |  |  |
|  | $\mathrm{X}$ |  | $\mathrm{X}$ |  |  |  |  | $\mathrm{X}$ |

Criteria:

- Only proved that $N>7-3$ points;
- Only provided an example for $N=8-3$ points;
- No points are deducted for the lack of justification in the example.