On the plane, nine parallel lines intersect with $n$ other parallel lines, all of which are perpendicular to the first nine. The lines together form a total of 756 rectangles. What is the value of $n$?
Solution. The problem actually involves determining the number of rectangles formed by 9, and respectively $n$ lines. We will present two possibilities for this below:

a) A rectangle is uniquely determined by its two pairs of parallel side lines. From the 9 "horizontal" lines, we can choose 2 in $\binom{9}{2}$ ways, and from the $n$ "vertical" lines, we can choose 2 in $\binom{n}{2}$ ways.

Thus, the number of rectangles is

$$
\binom{9}{2} \cdot\binom{n}{2}=18 n(n-1)
$$

b) In the drawn grid, each rectangle is uniquely determined by its diagonals. Since the number of grid points is $9 n$, we can choose one end of the diagonal in this many ways. The other end cannot be in the same row or column as the chosen point, so we can choose it in $(9-1)(n-1)=8(n-1)$ ways.

Thus, on the one hand, we counted each diagonal twice, and on the other hand, we counted each rectangle according to both of its diagonals, so the number of resulting rectangles is

$$
\frac{9 n \cdot 8(n-1)}{4}=18 n(n-1)
$$

Solving the equation $18 n(n-1)=756$ for $n$, we get two roots: -6 and 7. The former is clearly not a solution, so the value of $n$ is 7.