Chessboard Problem

When I first considered this chessboard problem, I immediately jumped to 64 as the number of squares. But once I realized I could also consider squares of bigger sizes, the problem got a bit more interesting! I am a very visual person and I always jump at a chance to draw a picture, so first thing I did was draw a chessboard in my notebook. Next, I wrote down the number of small squares (64) because that was something I already knew. Then, I tackled the next smallest square, 2x2, so I could start systematically working my way through all potential square sizes, smallest to largest, not missing any along the way. I started counting 2x2 squares along the bottom row, and part way along, I realized I was counting the centers of each 2x2 square. So I decided to count how many centers along and how many up there would be in the chessboard, and multiply them together, which happened to be 7(across)x7(up)=49. I did the same method with the 3x3 squares, which I found was 6 middles across and 6 middles up (6x6=36). At this point, I realized the pattern would continue all the way to the 8x8 square, and just wrote down the answers without counting centers.
Finally, it was time to add up all the squares I'd found! To make the mental math a bit easier for me, I added first the numbers that make multiples of five (so 1+9, 4+16, etc.) and then added up the rest. My solution, as you can see from the picture, was 204 squares!

From the above, you can see there were a few changes in my thinking:
- Considering only small, 1x1 squares, to considering many sizes
- Counting individual squares, to counting unique centers
- Counting centers all the centers, to multiplying the number of centers across and up together
- Multiplying centers, to just assuming the continuation of the pattern

Tools I used:
- Drawing a picture
- Use of the center of the squares
- Multiplication and addition

How to extend the puzzle:
- How would the solution change if it was a bigger board? Or if the board was not a square itself (ex. 8x9 board)?
- How would the puzzle change if we were instead looking for rectangles of the form Ax2A? Would there still be a pattern in the answer?
- What if we instead had a board of triangles? (below)