Problem: The nine horizontal and nine vertical lines on an $8\times8$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$ Solution: We count the rectangles in the following way. From $ \{0,1,\ldots, 8\}$ we pick any $x$ and $y$ to denote a position on the chess board. Then, we choose from among $8$ possible left over coordinates for $x'$ and $y'$. These two points specify a rectangle, but for each rectangle formed, there are $4$ ways of choosing it (2 orders of diagonal coordinates, and 2 orders of the opposite diagonal coordinates) so there are $\frac{81\cdot 64}{4} = 1296$ rectangles. On the other hand, we can count the number of squares in the following way. There are $8^2$ one-by-one squares, $7^2$ two-by-two squares, blah blah, and $1^2$ eight-by-eight squares. Recalling that the sum of the first $n$ squares is $\frac{n(n+1)(2n+1)}{6}$, we find there are $204$ squares. Our ratio reduces to $\frac{17}{108}$ giving us a $m+n$ value of $125$.