Example 4 A square blackboard is divided into $n^{2}$ small squares of side length 1 by a grid of horizontal and vertical lines. For what maximum positive integer $n$ is it always possible to select $n$ small squares such that any rectangle of area at least $n$ contains at least one of the selected small squares (the sides of the rectangles are along the grid lines)?
From the problem, it is evident that 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.

Call the row in which the selected square in the first column is located as $A$. If $A$ is the first row, then call the second and third rows $B$ and $C$; if $A$ is the $n$-th row, then call the $n-1$ and $n-2$ rows $B$ and $C$; if $A$ is neither the first row nor the $n$-th row, then call the two rows adjacent to $A$ as $B$ and $C$.

Let the $b$-th square in row $B$ be the selected one. If $b \leqslant\left[\frac{n}{2}\right]$ or $b \geqslant n-\left[\frac{n}{2}\right]+2$ (where $[x]$ denotes the greatest integer not exceeding $x$), then the rectangle formed by the $\left[\frac{n}{2}\right]+1, \left[\frac{n}{2}\right]+2, \cdots, n$ columns in rows $A$ and $B$, or the rectangle formed by the $2, 3, \cdots, n-\left[\frac{n}{2}\right]+1$ columns, has an area of $2\left(n-\left[\frac{n}{2}\right]\right) \geqslant n$, and does not contain any selected squares. Therefore, it must be that $\left[\frac{n}{2}\right]+1 \leqslant b \leqslant n-\left[\frac{n}{2}\right]+1$. Now consider the rectangle formed by the $2, 3, \cdots, \left[\frac{n}{2}\right]$ columns in rows $A, B, C$, and the rectangle formed by the $n-\left[\frac{n}{2}\right]+2, n-\left[\frac{n}{2}\right]+3, \cdots, n$ columns. The area of each of these rectangles is $S=3\left(\left[\frac{n}{2}\right]-1\right)$, and when $n \geqslant 8$, we have $S \geqslant n$. Moreover, these two rectangles do not contain the selected squares in rows $A$ and $B$. Since there is only one selected square in row $C$, one of these two rectangles must not contain any selected squares. Therefore, when $n \geqslant 8$, the conditions of the problem are not satisfied.

On the other hand, as shown in Figure 4-6, when $n=7$, there is a selection method that meets the problem's requirements. Therefore, the largest positive integer $n$ is 7.