Checkerboard Rectangles

Date: 03/24/2003 at 12:26:32
From: Veronica
Subject: Rectangles

A checkerboard has 8 horizontal boxes and 8 vertical boxes. How many 
rectangles are possible inside that board?

I can't figure out what the right size is for the rectangles. If you 
split the board in two you get two rectangles.

Date: 03/25/2003 at 14:45:21
From: Doctor Ian
Subject: Re: Rectangles

Hi Veronica,

That's not quite what the question is asking. Instead of a checker
board, let's use a tic-tac-toe board, which is smaller and easier to
think about:

   A---B---C---D
   |   |   |   |
   E---F---G---H
   |   |   |   |
   I---J---K---L 
   |   |   |   |
   M---N---O---P

Now, each square is a rectangle, right?  And there are 9 small (1x1)
squares: 

  ABEF  BCGF  CDHG

  EFJI  FGJK  GHKL

  IJNM  JKON  KLPO

And there are also four larger (2x2) squares:

  ACKI  BDLJ

  EGOM  FHPN

And the whole thing is a (3x3) square:

  ADPM

Now, what about non-square rectangles?  There are some 1x2 rectangles:

  ACGE  BDHF

  EGKI  FHJL

  IKOM  JLPN

and some 2 x 1 rectangles:

  ABJI  BCKJ  CDLK

  EFNM  FGON  GHPO

There are also 1 x 3 rectangles, and 3 x 1 rectangles, and 2 x 3 
rectangles, and 3 x 2 rectangles. I'll leave those for you to find. 

In other words, in a 3x3 square, you can find rectangles with these 
sizes:

  1x1  1x2  1x3

  2x1  2x2  2x3

  3x1  3x2  3x3

In a 4x4 square, you could find some additional sizes:

  1x1  1x2  1x3  1x4

  2x1  2x2  2x3  2x4

  3x1  3x2  3x3  3x4

  4x1  4x2  4x3  4x4

But the question isn't asking how many _sizes_ of rectangles there
are, but the _number_ of rectangles. For the 3x3 case, that's 

  1x1  1x2  1x3
  (9)  (6)  (3)

  2x1  2x2  2x3   Total = 9 + 6 + 3 + 6 + 4 + 4 + 3 + 4 + 1
  (6)  (4)  (4)
                        = 40
  3x1  3x2  3x3
  (3)  (4)  (1)

For some explanations of how to go about figuring out the total number
of rectangles for a square of any size, see

   Rectangles on a Chessboard
   http://mathforum.org/library/drmath/view/55431.html 

But if you feel like finding the answer on your own, a good way to do
it is to start looking at how things change as you go from a 2x2
board, to a 3x3, to a 4x4, and so on.  

The idea is to spot a pattern that lets you express the number of
rectangles of each size as a function of N, where NxN is the size of
the board.  

For example, in the 3x3 case, we could replace the numbers with

   1x1   1x2   1x3
   (N^2) (2N)  (N)

   2x1   2x2   2x3
   (2N)  (N+1) (N+1)
 
   3x1   3x2   3x3
   (N)   (N+1) (1)

Does that pattern work for the 2x2 and 4x4 cases as well? Or would you 
need a different pattern to describe all three cases? I'll leave that 
for you to think about. 

I hope this helps. Write back if you'd like to talk more. 

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