Quadrilateral Patterns

Date: 03/06/97 at 16:28:11
From: Michael McDonel
Subject: Quadrilaterals...how to find a pattern when increasing the 
size of a grid by one unit (each side)

We are studying quadrilaterals in 4th grade and I cannot seem to find 
a pattern when you increase a square grid by one unit on each side.  
For instance, a 1x1 grid has one quadrilateral, a 2x2 has 9, and so 
forth.  And when you start asking about 5x5's and 6x6's, it really 
gets complicated.  Please help in any way you can.  

Thank you.

Date: 04/30/97 at 03:31:01
From: Doctor Mike
Subject: Re: Quadrilaterals...how to find a pattern when increasing 
the size of a grid by one unit (each side)

Hello Michael,
   
Your problem has made me think, and the answer turns out to be pretty 
simple.  So here goes. I will show most of the details of the proof 
that my answer is right. This is not often done in 4th grade, but I 
believe you can take it.
  
By a quadrilateral, I think you mean one with horizontal and vertical
sides that go along the grid lines. These are like the rectangles you 
can make using some of the squares on a checker or chess board. I have 
a formula for the number of rectangles in an NxN grid, but I want to 
start you out thinking with me about something similar, but which is 
simpler. 
   
Take the same question as yours, but instead of a square grid use a 
grid that is 1xN, that is, just a straight string of N grid squares 
like this: 
          _____________________________________
          |     |     |     |     |     |     |
          |  1  |  2  |  3  |  4  |  5  |  6  | 
          |_____|_____|_____|_____|_____|_____| 
  
This is a 1x6 grid, and it will help to look at this in detail. How
many rectangles are there in this grid with 6 squares? Yes, just one.
How many rectangles are there in this grid with 5 squares? I count
that there are 2 of those, the one consisting of the left five squares
and the one consisting of the right five squares. Now it gets a bit
more complicated. How many rectangles are there with 4 squares?  I
will list them all: the left 4, the middle 4, and the right 4, for a
total of 3.  Do you see a trend yet?  Let's make a table of values.
   
   How many squares in a rectangle   How many different rectangles 
   --------------------------------|------------------------------
                         6         |        1 
                         5         |        2
                         4         |        3
                         3         |        4
                         2         |        5
                         1         |        6 
                      -------      |      ----
                       Total       |       21 
  
The total number of possible rectangles you can make is 1+2+3+4+5+6,
which is 21. If it were a 1xN grid the total is 1+2+3+4+....+N or
the sum of all numbers from 1 to N. There is a formula, which I will
not prove here because this e-mail is already long, that this sum is
equal to N*(N+1)/2. In our case N = 6 so the formula is 

  6*7/2 = 42/2 = 21.
   
Now, finally, let's get to YOUR problem.  We want to count the number
of rectangles we can make on an NxN grid. Let's imagine one. Let's
imagine the grid is an 8x8 chess board, and the rectangle is 5 squares
wide and 3 squares high. This 5x3 rectangle can be located anywhere
on the chess board, so there are many possible locations. How can
we possibly count them all?  
  
How? We can use what we already know from the simpler case! See if you 
can answer these two questions about rectangles on the chess board:
  
   Q1 -- For an AxB rectangle on the chess board, how many ways are
         there to choose both the number "A", and also the location
         (left to right) on the chess board?
   
   Q2 -- For an AxB rectangle on the chess board, how many ways are
         there to choose both the number "B", and also the location
         (up and down) on the chess board? 
   
The answers to these two questions are the same, since you can count 
all the possibilities in exactly the same way as we did the simple 
1xN example.  Both answers are 1+2+3+4+5+6+7+8 = (8*9)/2 = 72/2 = 36.  
Now for the final step.  Because there are 36 ways to choose A and 
place the AxB rectangle left/right, and because there are 36 ways to 
choose B and place the AxB rectangle up/down, that is a total of 36*36 
or 36 squared ways to choose the size and exact location of a 
rectangle. 
  
Now for the general version of the formula: If you have an NxN grid of
squares, then the number of possible rectangles you can draw in that
grid is the square of the number N*(N+1)/2. You have waited a really
long time for this answer, so I will give you a bonus, namely, the
more general formula for when the whole grid itself is a rectangle.
  
If you have an NxM grid of squares, then the number of rectangles that
can be drawn on that grid is given by the following expression: 
     
          N*(N+1)     M*(M+1)
         --------- * ---------  =  N*(N+1)*M*(M+1)/4 
             2           2
  
For example, if N = 5 and M = 3, then there are 90 possibilities!  
  
I will leave you with two thoughts.  First, the formula for adding up
the first N counting numbers can be proven by a method known as
mathematical induction.  You might want to look that up in a few
years. Second, this e-mail would probably be considered complicated
by most high school students. You may have to read it several times
before it starts to come together for you.  Please do that.  I think
you will enjoy thinking about this problem and solution.  Also, I
suggest you get out a real physical chess (or checker) board, and some 
paper and scissors to cut out rectangles of different shapes and move 
them around the grid of the board.  Math may seem mental, but some of 
the best mathematicians have a good understanding of how numbers 
connect with the physical real world.  This is especially true for 
geometry and for applying math to solve real world problems.
  
I hope this helps.  

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