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### Squares in a Square

```Date: 01/23/2003 at 20:51:22
From: Curtis
Subject: Squares in a box

If you have a square that is 50x50 with small squares inside it, how
many squares will there be altogether?

I took a 4x4 box and tried to find how may squares were in it, and I
found 30 different squares. Then my teacher told me to find all the
squares in a 50x50 square box.
```

```
Date: 01/26/2003 at 02:56:44
From: Doctor Ian
Subject: Re: Squares in a box

Hi Curtis,

That was a good way to start!  In fact, try it with a few different
sizes, and see if you can spot a pattern. But don't just focus on the
total number you get for each size. Look at the number you get for
each size:

+---+
|   |     1 x 1: 1
+---+

+---+---+
|   |   |      1 x 1: 4
+---+---+      2 x 2: 1
|   |   |
+---+---+

+---+---+---+
|   |   |   |      1 x 1: 9
+---+---+---+      2 x 2: 4
|   |   |   |      3 x 3: 1
+---+---+---+
|   |   |   |
+---+---+---+

+---+---+---+---+
|   |   |   |   |       1 x 1: 16
+---+---+---+---+       2 x 2:  9
|   |   |   |   |       3 x 3:  4
+---+---+---+---+       4 x 4:  1
|   |   |   |   |
+---+---+---+---+
|   |   |   |   |
+---+---+---+---+

Once you spot the pattern, you can compute the total number of squares
for _any_ size, without having to count them.

Is this enough to get started?

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Elementary Puzzles
Middle School Puzzles

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