Dividing a Square

Date: 07/31/99 at 02:00:10
From: Lulu
Subject: Square Share

Dear Dr Math:

Hello! I am studying exponents and I have a project called "Square 
Share." I am really confused about the questions posed by the project. 
Can you please help me?

The questions are:

A line can divide a square into a number of equal-sized smaller 
squares. The table below shows the number of lines required to divide 
a square into a certain number of equal smaller squares. Find a 
pattern that would tell you how many lines would be required to divide 
a square into 100 smaller squares, into 400 smaller square, into "n" 
smaller squares.

Number of Smaller Squares    Number of Lines
           4                       2
           9                       4
         100                       ?
           n                       ?

1. Does your pattern (rule) work with no lines/no smaller squares? 
   Why or why not?

2. Explain any limits on the number of smaller squares you can make.


3. Is it possible to use lines to divide a square into smaller squares 
   that are not equal in size?

4. What is the general pattern (rule) for the number of smaller 
   squares formed if you are given the number of lines? Explain any 
   restrictions on the number of lines you can use in the pattern 
   (Rule).

I worked out the rule as (Line/2+10)^2. How can I use this to answer 
the questions?

I am looking forward your help.
Thanks a lot!

Date: 09/02/99 at 09:58:16
From: Doctor Dwayne
Subject: Re: Square Share

Hey Lulu,

Tough project! However, it may turn out to be a very interesting one 
because of the neat properties that you'll discover about squares. So 
before looking at this project from a purely mathematical angle, let's 
just make a few observations which will enhance our intuition on the 
topic.

Take a square and draw a perpendicular line down from one of its 
horizontal sides to the opposite horizontal side. The result is two 
rectangles. Now draw a similar line from one vertical side to the 
opposite vertical side. The first rule we'll learn is that the same 
number of lines (in the same relative position) is needed in the 
horizontal as in the vertical to ensure squares. Why? Well the 
horizontal lines divide the length of the square up into segments, and 
to divide the width into the segments of the same length (necessary 
for tiny squares) you need the same number of vertical lines.

     +-----+-----+          +-----+-----+
     |     |     |          |     |     |
     |     |     |          |     |     |
     |     |     |          +-----+-----+
     |     |     |          |     |     |
     |     |     |          |     |     |
     +-----+-----+          +-----+-----+

The second thing of note is a trend: one vertical line divides the 
square into 2 parts, and two such lines divide the square into three 
parts, and three such lines into four parts and the trend continues.

     +---+---+---+          +--+--+--+--+          
     |   |   |   |          |  |  |  |  |
     |   |   |   |          |  |  |  |  |
     |   |   |   |          |  |  |  |  |
     |   |   |   |          |  |  |  |  |
     |   |   |   |          |  |  |  |  |
     +---+---+---+          +--|--+--+--+

But remember that the same number of horizontal lines is always drawn 
as well (from our earlier discovery). Consequently, if we have one 
vertical line we have 2 by 2 sections, if we have 2 vertical lines we 
have 3 by 3 sections (since we have two horizontals as well), or for 3 
verticals we have 4 by 4 and so on. (It may help if you draw this.)

     +---+---+---+          +--+--+--+--+          
     |   |   |   |          |__|__|__|__|
     +---+---+---+          |  |  |  |  |
     |   |   |   |          |--+--+--+--|
     +---+---+---+          |__|__|__|__|
     |   |   |   |          |  |  |  |  |
     +---+---+---+          +--|--+--+--+

Now we convert our observations into mathematical expressions. One 
vertical line leads to 2 by 2 (2 squared) smaller squares, two 
vertical lines lead to 3 by 3 (3 squared) smaller squares, etc. As a 
result we realize that 4 vertical lines would lead to 5^2 squares. But 
note that 4 verticals are only half the total number of lines, since 
there are 4 horizontals as well. So if we call the total number of 
lines 'x', then the number of verticals equals x/2. And the number of 
small squares can be calculated as well. See if you can now write a 
formula using x to describe the number of smaller squares (call that 
number n). This formula will be the basis of the project.

Here are some hints:

To answer #1, try x = 0 and see if the formula gives you 1 as the 
answer.

Question #2 is hard if you don't immediately see what they mean by 
limitations. To begin, see if there's any way to get 64 squares, 
then see if there's any way to get 65 squares. See the difference? 
Explain why, and the limitations will be clear.

We've already answered #3, but see if you can state it clearly here.

And #4 is all yours.

Good luck and I hope this all helped.  If not write back.

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