Example 1: An $8 \times 8$ chessboard is formed by 9 horizontal lines and 9 vertical lines, creating a total of $r$ rectangles, of which $s$ are squares. The value of $\frac{s}{r}$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are positive integers, and $\frac{m}{n}$ is a reduced fraction. Find the value of $m+n$.
(1997 15th American Invitational Mathematics Examination Problem)
Solution One: Since a rectangle must be determined by two horizontal lines and two vertical lines, now we have 9 vertical lines and 9 horizontal lines, so there are $C_{9}^{2} C_{9}^{2}=1296$ rectangles, among which there are $8^{2}$ squares with side length 1, $7^{2}$ squares with side length 2, $\cdots$. Generally, there are $(9-i)^{2}$ squares with side length $i$ $(1 \leqslant i \leqslant 8)$, so the number of squares is $8^{2}+7^{2}+\cdots+2^{2}+1^{2}=204$, thus $\frac{s}{r}=\frac{204}{1296}=\frac{17}{108}$, so $m+n=17+108=125$.