jmc

To determine the two horizontal sides of a rectangle, we have to pick two of the horizontal lines of the checkerboard, or $\left(\genfrac{}{}{0ex}{}{9}{2}\right)=36$. Similarily, there are $\left(\genfrac{}{}{0ex}{}{9}{2}\right)$ ways to pick the vertical sides, giving us $r=1296$ rectangles. For $s$, there are $8^2$ unit squares, $7^2$ of the $2\times 2$ squares, and so on until $1^2$ of the $8\times 8$ squares. Using the sum of squares formula, that gives us $s=1^2+2^2+\cdots +8^2=\frac{(8)(8+1)(2\cdot 8+1)}{6}=12\ast 17=204$. Thus $\frac{s}{r}=\frac{204}{1296}=\frac{17}{108}$, and $m+n=\boxed{125}$.