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### Formula for Squares inside Rectangles

```Date: 06/29/2003 at 09:48:23
From: Jamie
Subject: Number of Squares inside rectangles

Is there an equation for how many squares there are in a rectangle
divided up into 1cm blocks? This is like the typical chessboard
puzzle.
```

```
Date: 06/29/2003 at 10:50:49
From: Doctor Jaffee
Subject: Re: Number of Squares inside rectangles

Hi Jamie,

There is a formula that will calculate the number of squares in a
rectangle. Here is how I found it.

First, let's consider only rectangles whose width is 1.

A 1x1 rectangle contains exactly one square, a 1x2 rectangle contains
two squares, a 1x3 rectangle contains three squares, etc.  In general,
if the dimensions of a rectangle are 1xn, there will be n squares.

That was easy, and I probably haven't told you anything you didn't

Now let's consider rectangles whose width is 2, starting with a 2x2
rectangle. We skip the 2x1 rectangle because we've already considered
that in the previous case.  A 2x2 rectangle consists of four squares
whose dimensions are 1x1 and one square whose dimensions are 2 x 2.
This gives us a total of five squares.

I'm going to create a table in which I list the width, length, number
of 1x1 squares, number of 2x2 squares, and total number of squares.

2 x 2     4    1     5
2 x 3     6    2     8
2 x 4     8    3    11
2 x 5    10    4    14

In general, you should see that if you have a 2xn rectangle, the total
number of squares will be 3n - 1.

I repeated this process for 3xn rectangles, 4xn rectangles, 5xn
rectangles, until I recognized that there was a pattern developing.
If the width is m and the length is n, I was able to write the total
number of squares in terms of m and n.

Give it a try and if you want to check your answer with me or if you
have difficulties or other questions, write back to me and I'll try to

Good luck,

- Doctor Jaffee, The Math Forum
http://mathforum.org/dr.math/
```
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