1997 AIME Problems/Problem 2

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

To determine the two horizontal sides of a rectangle, we have to pick two of the horizontal lines of the chessboard, or ${9\choose 2} = 36$ . Similarily, there are ${9\choose 2}$ ways to pick the vertical sides, giving us $r = 1296$ rectangles.

For $s$ , there are $8^2$ unit squares, $7^2$ of the $2\times2$ squares, and so on until $1^2$ of the $8\times 8$ squares. Using the sum of squares formula, that gives us $s=1^2+2^2+\cdots+8^2=\dfrac{(8)(8+1)(2\cdot8+1)}{6}=12*17=204$ .

Thus $\frac rs = \dfrac{204}{1296}=\dfrac{17}{108}$ , and $m+n=\boxed{125}$ .

1997 AIME (Problems • Answer Key • Resources)
Preceded by Problem 1	Followed by Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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1997 AIME Problems/Problem 2

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