How many rectangles can be made from the individual spaces of a chess board?
To specify a rectangle
it suffices to say where its left and right boundaries are, and where its top and bottom boundaries are. There are $\binom92$ choices for each pair of boundaries and therefore $\binom92^2$ rectangles. That is to say, 1296 rectangles.
Of course @Gareth_McCaughan got this well-known puzzle immediately. But for people who aren't up on their combinatorics, here's the same calculation in a way that seems easier (at least to me).
- There are 9x9 = 81 corners.
- For each of these there are 8x8 = 64 corners that are not in the same row or column.
- Each pair of these makes a rectangle.
- But then each rectangle has been counted four times (you can have top-left and bottom-right or top-right and bottom-left and both of those can be done two ways)
- So the final answer is 81 x 64/4 = 1296