7.43 A square blackboard is divided into $n^2$ small squares of side length 1 by a network of horizontal and vertical straight lines. For what largest natural number $n$ is it always possible to select $n$ small squares such that any rectangle of area at least $n$ (with sides along the grid lines) contains at least one of the selected small squares?
[Solution] $n=7$.
Obviously, if $n$ small squares are selected to meet the problem's conditions, then in each row and each column there is exactly one selected small square (we stipulate that $n \geqslant 3$, obviously $n=2$ cannot be the largest).

We fix a row $A$, where the first square is selected. We take another row $B$ adjacent to $A$, and then select a row $C$ ($C$ is selected such that it is either adjacent to $A$ but does not overlap with $B$, or adjacent to $B$ but does not overlap with $A$).

Assume $b$ is the position of the selected square in row $B$ (i.e., the $b$-th square is selected).
If $b \leqslant n-\left[\frac{n}{2}\right]$ or $b>\left[\frac{n}{2}\right]+1$, then in rows $A$ and $B$, we can find a rectangle with an area of at least $n$ that does not contain any selected small squares. Therefore, it must be that
$$
n-\left[\frac{n}{2}\right] < b \leqslant \left[\frac{n}{2}\right]+1.
$$
If $n > 7$, then the areas of the two rectangles are both at least $n$, but row $C$ can only have one selected small square, meaning that one of these rectangles must not contain any selected small squares, which contradicts the problem's requirements.
Thus, $n \leqslant 7$.
The figure provides an example where $n=7$ and the conditions of the problem are met.