Discussion

Find the number of rectangles which are not squares, and the number of squares, on an 8 by 8 chessboard, respectively.

Answer: B

A rectangle on the board is fixed by choosing two of the nine vertical lines and two of the nine horizontal ones.

\[ {}^{9}C_{2} \times {}^{9}C_{2} = 36 \times 36 = 1296 \]

The squares are counted by size: an \( n \times n \) square can sit in \( (9-n)^{2} \) positions.

\[ 8^{2} + 7^{2} + \cdots + 1^{2} = 204 \]

\[ \text{rectangles that are not squares} = 1296 - 204 = 1092 \]

So the answer is 1092 and 204. Every square is a rectangle, which is why the two counts must be separated before comparing.

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