Rectangles on a chessboard

Idea. Choosing a rectangle is the same as choosing two of the horizontal grid lines and two of the vertical ones.

Step 1. There are 9 horizontal lines, so $\left(\genfrac{}{}{0ex}{}{9}{2}\right)=36$ ways to choose the top and bottom sides, and likewise 36 ways for the left and right sides.

Step 2. $$36\times 36=1296.$$

Squares only. A $k\times k$ square can be placed in $(9-k{)}^2$ positions, so there are $$1^2+2^2+\cdots +8^2=204$$ squares. Interviewers often ask for the squares first, then the rectangles.

A curiosity. $1296=36^2=(1+2+\cdots +8{)}^2$, which also equals $1^3+2^3+\cdots +8^3$.