Number of rectangles in a chessboard

I was asked by an interviewer to find out the number of squares in a chessboard to start with. I simply took the iterative approach. For a 1x1 board has 1, 2x2 board has 5, 3x3 board has 14 & so on. It all easily fits into the series 1², 1² + 2², 1² + 2² + 3² etc., Hence, for a 8x8 chessboard, it would be sum of squares of the numbers from 1 to 8.

Then he asked me if the same approach would work for finding out the number of rectangles in the board. I tried something for a couple of minutes & it didn't seem to form any pattern. Later while looking up on the net for the solution, I realized that for finding the rectangles, any iterative approach would be simply futile.

The actual solution to the problem is very simple. Let's see how it works for a square first. The basic idea is to find out the number of possible types of squares & to see where all they can be fit. For example, the square of size 1x1 can be fit in 64 positions, the square of size 2x2 can be fit in 49 positions, the square of size 3x3 can be fit in 36 positions and so on.

For a rectangle, the same thing applies. We need to first identify the various possible shapes of the rectangles that can be formed. Anything from 1x1 to 1x8 and 8x1 to 8x8 is possible. We need to find out where all these would fit & that would be it.