Question

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!

Answer 1

a web search turns up several well-explained solutions

Answer 2

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.