10.7. A square wooden board is divided into $n^{2}$ unit squares by horizontal and vertical lines. Mark $n$ squares so that any rectangle with an area of at least $n$ and whose sides lie along the grid lines contains at least one marked square. Find the largest $n$ that satisfies this condition.
10.7. 7 .

Obviously, if $n$ marked cells satisfy the conditions of the problem, then in each row and each column there is exactly one marked cell. Let $n \geqslant 3$ (obviously, $n=2$ is not the maximum), take the first row with a marked cell as $A$, the row adjacent to $A$ as $B$, and take either a row adjacent to $A$ (and not coinciding with $B$) or a row adjacent to $B$ (and not coinciding with $A$) as $C$.

Let $b$ be the number of the marked cell in row $B$. If $b \leqslant n-\left[\frac{n}{2}\right]$ or $b>\left[\frac{n}{2}\right]+1$, then in rows $A$ and $B$ there exists a rectangle with an area of at least $n$ that does not contain any marked cells, so,
$$
n-\left[\frac{n}{2}\right]7$, then the area of both rectangles is not less than $n$, but row $C$ has only one marked cell, meaning one of these rectangles does not contain a marked cell.
Thus, we have proved that $n \leqslant 7$. A $7 \times 7$ board that meets the conditions is shown above.