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/