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### Number of Squares in an NxN Square

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Date: 7/29/96 at 7:57:25
From: Anonymous
Subject: How Many Squares in an N by N square?

How many squares are there in a 8 x 8 square?
How many rectangles are there?
Find an equation that will find out the number of squares simply by
knowing the size of the square.

I can't find out the equation. Can you help? This equation has to
apply to both squares and rectangles.
```

```
Date: 7/30/96 at 15:52:27
From: Doctor Robert
Subject: Re: How Many Squares in an N by N square?

There are quite a few!  This is an interesting problem.  Let's tackle
the one about the squares first.  There is only ONE 8x8 square.  There
are FOUR 7x7 squares.  There are NINE 6x6 squares, and so on.  You can
see this by taking a square (say 6 by 6) and seeing how many squares
you can move it horizontally and how many vertically and multiplying
these two numbers together).  If you see the pattern above, the answer
to the question as to how many squares there are is

1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 = 204

The "equation" you want is  N = (9-s)^2 where N is the number of
squares of side s.  That is, for a square of N sides, the total number
of squares = 1^2 + 2^2 + 3^2 + ... + N^2.]

For rectangles, it's just a bit harder to see.  Consider a rectangle
1 by 2, that is, 1 unit high and 2 units long.  How many units can fit
in the top row?  Seven! How many rows are there?  Eight.  So there
must be 56 1 by 2 rectangles.  Of course, there are also 56  2 by 1
rectangles.

If you proceed this way to count the possible number of rectangles of
different sizes you can generate the following table.  Across the top
are the possible lengths of the rectangles.  Down the side are
possible heights of the rectangles.  In the table are the number of
possible rectangles with these dimensions.

1   2   3   4   5   6   7   8

1  64  56  48  40  32  24  16  8

2  56  49  42  35  28  21  14  7

3  48  42  36  30  24  18  12  6

4  40  35  30  25  20  15  10  5

5  32  28  24  20  16  12   8  4

6  24  21  18  15  12   9   6  3

7  16  14  12  10   8   6   4  2

8   8   7   6   5   4   3   2  1

There are quite a few interesting things about this table.  Notice
that the elements along the major diagonal represent the squares.  The
sum of all the numbers is 1296, so there must be 1296-204 = 1092
rectangles that are not squares.  The "equation" must be
N = (9-m)(9-n) where N is the number of rectangles, m is the height of
the rectangle and n is the width of the rectangle.

I hope that this helps.

-Doctor Robert,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```

```
Date: 10/02/98 at 11:24:31
From: John Pinkerton
Subject: Re: Comment on Number of Squares in NxN Square

Dear Dr. Math:

Here's a quick derivation for the number of rectangles in an NxN
chessboard. Each rectangle is defined by selecting two of the N + 1 vertical
lines and two of the N + 1 horizontal lines that are the edges of the
squares. So, the number of rectangles is the square of N + 1 "choose" 2,
which is the square of (N + 1)*N/2.

For an 8x8 square, you have ((8 + 1)*8/2)^2, or 36 squared, which is the 1296
that Dr. Robert gets by adding up his interesting table.

Keep up the good medicine,
John Pinkerton

P.S. on "choose": Recall that the number of ways to choose 2 objects out of
K is K "choose" 2, or (K)*(K - 1)/2. There are K choices for the first
object and K - 1 choices for the second, so there are (K)*(K - 1) ordered
pairs. But, order doesn't matter. So, we divide by 2 since you don't want
to double count for the two orders you might choose a given pair of
objects. In the above chessboard problem, K is N + 1.
```
Associated Topics:
High School Permutations and Combinations
High School Puzzles
Middle School Puzzles

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