Squares on a Checkerboard

Date: 04/26/98 at 22:12:21
From: Joanna Moore
Subject: Checkerboard

How many squares are there on a checkerboard?

Joanna Moore

Date: 04/27/98 at 11:17:57
From: Doctor Schwenoha
Subject: Re: Checkerboard

Let's see if we can get you started and then YOU can come up with the 
answer.  

This question implies "squares of any size." How about if we make a 
table and collect our data; we might find a pattern going on.  

If you look at the smallest size square on a checkerboard, we can call 
that a 1 unit by 1 unit square. There are 64 of that size.  

The next size square is a 2 unit by 2 unit square. It's okay to use 
these more than once in different new squares. The top lefthand corner 
has 4 of the 1x1's which, make a 2x2. See it? Good, then move right 
one square and find a new 2x2 made with two of the 1x1's you just used 
and two 1x1's right next to them. Likewise you can shift down just one 
row and use two that you've already used. Count all of these 2x2's.  
You should come up with 49. Our table would look like this:

                size        number  of squares
              ---------------------------------
               1 x 1              64
               2 x 2              49
               3 x 3
               4 x 4
                etc.

You go ahead and complete this table. After you're done you still have 
to add up the righthand column to get the total number of squares on a 
checkerboard.

Let's get back to the question of a pattern. Is there anything you 
notice about the numbers in the righthand column?  Is there a relation 
to the number of 1x1 squares on a side of the checkerboard?  2x2's?  
Etc.?

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