Question

The nine horizontal and nine vertical lines on an $8 \times 8$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $\frac{s}{r}$ can be written in the form $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Answer

Answer: 125

Concepts:counting shapes in figures combinations sum of first n squares

Difficulty rating: 1890

Small Hint:

A rectangle is determined by choosing $2$ of the $9$ horizontal lines and $2$ of the $9$ vertical lines

Big Hint:

There are $(9 - k)^2$ squares of size $k \times k;$ sum over $k = 1$ to $8$

Solution:

A rectangle is determined by choosing two of the nine horizontal lines and two of the nine vertical lines, so $r = \binom{9}{2}^2 = 36^2 = 1296.$

A $k \times k$ square can be placed in $(9 - k)^2$ positions, so $\begin{aligned} s &= \sum_{k=1}^{8} (9 - k)^2 \\ &= 8^2 + 7^2 + \cdots + 1^2 \\ &= \frac{8 \cdot 9 \cdot 17}{6} = 204. \end{aligned}$

Then $\frac{s}{r} = \frac{204}{1296} = \frac{17}{108},$ which is in lowest terms, so $m + n = 17 + 108 = 125.$

1997 AIME Problem 2

Answer: 125

Difficulty rating: 1890

Small Hint:

Big Hint:

Solution:

Problem 2 in Other Years