Posted by **Ell** on Thursday, October 27, 2011 at 5:06pm.

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!

- Math -
**Steve**, Thursday, October 27, 2011 at 5:28pm
a web search turns up several well-explained solutions

- Math -
**MathMate**, Thursday, October 27, 2011 at 5:30pm
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.

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