10.4 The Oddball marked the centers of 17 cells in an $N \times N$ grid 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 in an $8 \times 8$ square (and then in any smaller size), it is impossible to mark the cells in this way. Indeed, let's divide the square 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 of the four cells. Therefore, the total number of marked cells does not exceed 16.

Example for a $9 \times 9$ square:

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

## Criteria:

- Only proved that $N>8-3$ points;
- Only provided an example for $N=9-3$ points;
- No points are deducted for the lack of justification in the example.