How many rectangles are on a chessboard?

Question

An 8 by 8 chessboard is drawn with 9 horizontal and 9 vertical lines.

How many rectangles, of any size and shape, have their sides along these grid lines? (Squares count as rectangles.)

Then: how many of those are squares, and what fraction of all rectangles is that?

Accepted Answer

1,296 rectangles, of which 204 are squares. Choosing the sides. Any rectangle with grid-aligned sides has a top edge and a bottom edge, which are two of the 9 horizontal lines, and a left edge and a right edge, which are two of the 9 vertical lines. Conversely, any choice of two horizontal lines and two vertical lines gives exactly one rectangle. The number of ways to choose 2 lines from 9 is 9 × 8 / 2 = 36. So the number of rectangles is 36 × 36 = 1,296. Squares. Squares need equal width and height. Count by size: there are 8 × 8 = 64 squares of side 1, 7 × 7 = 49 of side 2, and so on down t…

How many rectangles are on a chessboard?

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