Pattern of Squares

Date: 1/08/98 at 22:48:32
From: Neil Abrahams
Subject: pattern of squares

Do squares of integers have a pattern? I believe that they do, but I 
don't know what it is.

Date: 01/22/98 at 14:51:31
From: Doctor Sonya
Subject: Re: pattern of squares

Hi there Neil.  

There is a very definite pattern to the squares of integers. I'll show 
you how to find it.

Pick a number, any number. I picked 7. Now say that you have that 
number of dots.  I have 7 dots, and there is not much I can do with 
them. I could put them in a line:

   . . . . . . .

or arrange them in two unequal rows:

   . . .
   . . . .

but it's not too pretty.

But say I had nine dots. Then I could arrange them in a perfect 
square:

  . . .
  . . .
  . . .

If you are getting ahead of me, you'll notice that 9 is 3^2 (this is 
the way to type "3 squared"). 

In fact, if the number of dots is the square of a positive integer, 
you can always arrange them in a square.

For example, 5^2 = 25, and we can make a 5 by 5 square out of the 25 
dots.

Now that you've got that, let's look at successive squares. Our first 
square is 1 by 1, or just one dot.

  o

Our second square is 2 by 2, or four dots:

  o o
  o o

Notice the original dot in the upper right hand corner.  Our third 
square is 3 by 3, with 9 dots. 
   
  o o o
  o o o
  o o o

Our forth square is 4 by 4, with 16 = 4^2 dots.

  o o o o
  o o o o
  o o o o
  o o o o

Are you starting to see a pattern?  Notice that with each successive 
square, we have to add a new row and a new column. The number of dots 
we have to add each time is odd.  Do you see why?

Okay, now we're ready to go back and look for the patttern. Our first 
square had 1 dot in it.  

To make our second square, we had to add 3 dots. So our second square 
(which had 2^2 dots in it) has 1 + 3 dots.

  . o
  o o

Our third square (with 3^2 = 9 dots) we got by adding 5 dots to the 
2 by 2 square. Thus the total number is 1 + 3 + 5. (Check that this 
equals 9.)

  . . o
  . . o
  o o o

Likewise, the number of dots in our forth square is 1 + 3 + 5 + 7.

  . . . o
  . . . o
  . . . o
  o o o o

Do you see that pattern?  What happens in the 5 by 5 square?

Write us back if you want another hint.

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