Chessboard Puzzle

Once I read the problem the first number immediately came to my mind was 64. Then I realized that there were different squares of dimension of 1x1, 2x2, 3x3, 4x4, 5x5, 6x6, 7x7, 8x8. The puzzle became more interesting! I drew a chessboard on a paper. I started counting the 2x2 squares on the left column and found 7 squares and I was sure that could find the same numbers on the bottom row. I came up with 7x7 for the number of the 2x2 squares. I used the same method for counting the 3x3 squares and the result was 6x6. And I realized that there was a pattern and I could continue this pattern for the remaining dimension. At this point I googled the chessboard problem to check my answer and other solutions for this problem.

The tools I used were a pencil, paper and a calculator.

I could extend this puzzle to finding number of rectangles or number of 2x3 rectangles in the chessboard.