Chessboard Extension

Looking at the question "How many squares in a chessboard?" is an interesting and challenging problem. At first glance, I thought to count all the black squares, count all the white squares, and then add up the two totals. Upon further thought, and upon learning that a standard chessboard is 8 x 8 squares, I realized one could just multiply 8 by 8 and get 64 individual squares.
However, it was brought to my intention that there are multiple sizes of squares on a board (1 x 1, 2 x 2, 3 x 3, 4 x 4, 5 x 5, 6 x 6, 7 x 7, 8 x 8). This posed a larger, more complicated problem. Counting is not efficient enough. I had to start look for patterns.
When looking at 8 x 8 squares, there is only one possible square on a chessboard (1^2).
When looking at 7 x 7 squares, there are four possible squares on a chessboard (2^2).
When looking at 6 x 6 squares, there are nine possible squares on a chessboard (3^2).

It is at this point that I noticed the pattern. There would then be 4^2 5 x 5 squares, 5^2 4 x 4 squares and so on.

Thus, through pattern recognition, one can surmise that there would be 1^2 + 2^2 +3^2 + ... + 7^2 + 8^2 total differently sized squares on an 8 x 8 chessboard. This adds up to 204 squares!

My thought process changed when I actually understood what the problem was asking; when I sort of opened my mind to a different understanding of "how many squares"; that there could be different sized squares that aren't as apparent at first glance.

The tools I used were a pencil and paper to draw a mini diagram of a chess board to visualize where squares were oriented. It would've helped to use different coloured pencils to identify different sized squares on the diagram. I also used a calculator to add up all the numbers.

You could extend the problem by cutting out a section of a chess board, and eliminating the occurrence of several squares. Maybe even cut out a rectangular portion of the chessboard to eliminate a number of squares that wouldn't follow a particular pattern (ie. if I just cut out the bottom left quadrant, you can easily figure out how much would be eliminated, but if I cut out a 2 x 4 rectangle, it's a little more difficult).