Checkerboard Squares

Date: 09/29/97 at 18:40:08
From: Tasha Gibson
Subject: Checkerboard squares

My teacher gives us a problem of the week and I haven't really been 
able to figure out this last one. Here's the question:

Say you have an 8x8 checkerboard. There are many other little and 
big squares inside of that so counting them all how many are there? 
(e.g. counting all the 1x1,2x2,3x3,4x4,5x5,6x6,7x7,8x8). 

Suppose you have a square checkerboard of some other size - not 8x8 
- what is the easiest way to find how many squares there are in it?

Make sure that when you are done you will be able to find the number 
of squares in any size checkerboard within 2 minutes.

I tried counting the squares individually, but that wouldn't be quick 
enough for 12x12. Could you please give me some tips for how to 
approach this problem?

Thanks,
Tash

Date: 09/29/97 at 19:41:21
From: Doctor Anthony
Subject: Re: Checkerboard squares

Consider the lefthand vertical edge of a square of size 1 x 1.  
This edge can be in any one of 8 positions. Similarly, the top edge 
can occupy any one of 8 positions for a 1 x 1 square. So the total 
number of 1 x 1 squares = 8 x 8 = 64.

For a 2 x 2 square the lefthand edge can occupy 7 positions and the 
top edge 7 positions, giving 7 x 7 = 49 squares of size 2 x 2.

Continuing in this way we get squares of size 3 x 3, 4 x 4 and so on.

We can summarize the results as follows:

   Size Of square      Number of squares
   ---------------     -----------------
      1 x 1               8^2 = 64
      2 x 2               7^2 = 49
      3 x 3               6^2 = 36
      4 x 4               5^2 = 25
      5 x 5               4^2 = 16
      6 x 6               3^2 =  9
      7 x 7               2^2 =  4
      8 x 8               1^2 =  1
                      ---------------
                      Total   = 204

There is a formula for the sum of squares of the integers 
1^2 + 2^2 + 3^2 + ...  + n^2

                n(n+1)(2n+1)
         Sum  = ------------
                     6

In our case, with n = 8, this formula gives  8 x 9 x 17/6 = 204.

As an extension to this problem, you might want to calculate the 
number of rectangles that can be drawn on a chessboard.

There are 9 vertical lines and 9 horizontal lines. To form a rectangle 
you must choose 2 of the 9 vertical lines, and 2 of the 9 horizontal 
lines. Each of these can be done in 9C2 ways = 36 ways. So the number 
of rectangles is given by 36^2 = 1296.

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