Problem Statement
The nine horizontal and nine vertical lines on an $8 \times 8$ 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 $m + n.$
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Solution
$$\text{Let r be the number of rectangles, s be the number of squares.}$$ $$\text{Suppose we have an } n\times m \text{ rectangle. There are } (9-n)(9-m) \text{ ways to make this rectangle.}$$ $$\therefore r = \sum_{i=1}^{8} \sum_{k=1}^{8} (9-i)(9-k)$$ $$r = (9-1)(9-1) + (9-1)(9-2) + ... + (9-1)(9-8) + (9-2)(9-1) + (9-2)(9-2) + ... + (9-8)(9-8)$$ $$r = (\sum_{i=1}^{8} 9-i)(\sum_{k=1}^{8} 9-k) = (1+2+3+...+8)(1+2+3+...+8) = 1296$$ $$s = \sum_{i=1}^{8} (9-i)(9-i) = 1^2 + 2^2 + ... + 8^2 = 204$$ $$\therefore \dfrac{s}{r} = \dfrac{204}{1296} = \dfrac{17}{108}$$ $$\text{Suppose } m/n = 17/108 \implies m + n = 125$$ 