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### Checkerboard Squares

```
Date: 09/29/97 at 18:40:08
From: Tasha Gibson
Subject: Checkerboard squares

My teacher gives us a problem of the week and I haven't really been
able to figure out this last one. Here's the question:

Say you have an 8x8 checkerboard. There are many other little and
big squares inside of that so counting them all how many are there?
(e.g. counting all the 1x1,2x2,3x3,4x4,5x5,6x6,7x7,8x8).

Suppose you have a square checkerboard of some other size - not 8x8
- what is the easiest way to find how many squares there are in it?

Make sure that when you are done you will be able to find the number
of squares in any size checkerboard within 2 minutes.

I tried counting the squares individually, but that wouldn't be quick
enough for 12x12. Could you please give me some tips for how to
approach this problem?

Thanks,
Tash
```

```
Date: 09/29/97 at 19:41:21
From: Doctor Anthony
Subject: Re: Checkerboard squares

Consider the lefthand vertical edge of a square of size 1 x 1.
This edge can be in any one of 8 positions. Similarly, the top edge
can occupy any one of 8 positions for a 1 x 1 square. So the total
number of 1 x 1 squares = 8 x 8 = 64.

For a 2 x 2 square the lefthand edge can occupy 7 positions and the
top edge 7 positions, giving 7 x 7 = 49 squares of size 2 x 2.

Continuing in this way we get squares of size 3 x 3, 4 x 4 and so on.

We can summarize the results as follows:

Size Of square      Number of squares
---------------     -----------------
1 x 1               8^2 = 64
2 x 2               7^2 = 49
3 x 3               6^2 = 36
4 x 4               5^2 = 25
5 x 5               4^2 = 16
6 x 6               3^2 =  9
7 x 7               2^2 =  4
8 x 8               1^2 =  1
---------------
Total   = 204

There is a formula for the sum of squares of the integers
1^2 + 2^2 + 3^2 + ...  + n^2

n(n+1)(2n+1)
Sum  = ------------
6

In our case, with n = 8, this formula gives  8 x 9 x 17/6 = 204.

As an extension to this problem, you might want to calculate the
number of rectangles that can be drawn on a chessboard.

There are 9 vertical lines and 9 horizontal lines. To form a rectangle
you must choose 2 of the 9 vertical lines, and 2 of the 9 horizontal
lines. Each of these can be done in 9C2 ways = 36 ways. So the number
of rectangles is given by 36^2 = 1296.

-Doctor Anthony,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Permutations and Combinations

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