How many total squares are there on a standard 8 by 8 checkerboard?

(It is more than 64)!
Make sure to show all your work and explain your reasoning.
Explain how you can determine the number of
total squares for any N by N board.
Bonus: How many total rectangles?

Thanks!

a web search turns up several well-explained solutions

Number of 1x1 squares = 64.

Number of 2x2 squares:
7 rows x 7 columns = 49
Number of 3x3 squares:
6 rows x 6 columns = 36
....
Number of 8x8 squares:
1 row x 1 col. = 1

So the total number of squares
=1²+2²+3²+...+8²
=n(n+1)(2*n+1)/6
where n=8.

To determine the number of total squares on an N by N board, we need to consider all possible sizes of squares on the board.

For a standard 8 by 8 checkerboard, we can begin by counting the 1x1 squares. There are 8 rows and 8 columns, so there are 8x8 = 64 1x1 squares.

Next, we count the 2x2 squares. Since there are 7 rows and 7 columns left after considering the 1x1 squares, we can have a total of 7x7 = 49 2x2 squares.

We continue this process for larger squares until we reach an N by N square. For example, for 3x3 squares, we have 6 rows and 6 columns remaining after considering the 1x1 and 2x2 squares, resulting in 6x6 = 36 3x3 squares.

Following this pattern, we can determine the number of squares for any N by N board by summing up the squares of all numbers from 1 to N:

Total number of squares = 1^2 + 2^2 + 3^2 + ... + N^2.

For an 8 by 8 board, the total number of squares would be:

1^2 + 2^2 + 3^2 + ... + 8^2 = 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 = 204.

Therefore, the total number of squares on an 8 by 8 checkerboard is 204.

To determine the number of total rectangles on an N by N board, we can use a similar approach. We start by counting all the 1x1 rectangles (which are essentially squares) as we did before.

Then, we consider the horizontal 1x2 rectangles and the vertical 2x1 rectangles. For an N by N board, there are (N-1) horizontal 1x2 rectangles and (N-1) vertical 2x1 rectangles.

We continue this process for larger rectangles of sizes 1x3, 1x4, 2x2, 2x3, and so on until we reach an N by N rectangle. This can be done by iterating through the possible lengths and widths, subtracting the dimensions from the board's size and counting the resulting rectangles.

By summing up all the counts, we can determine the total number of rectangles for any N by N board.

Unfortunately, providing the exact calculations for the total number of rectangles on an 8 by 8 checkerboard would be more complex. It would involve considering multiple dimensions and summing up the counts for each size.