Chessboard Squares

No 3 in a series of posts based on Thinking Mathematically (1985) by Mason, Burton, Stacey

This is a classic task for working systematically:

It was once claimed that there are 204 squares on an ordinary chessboard. 
Can you justify this claim?

 

I like this way of stating the problem, rather than just “how many squares on a chessboard?” because it gets straight to the nub of the issue – we are looking at different sized squares, some of which overlap.