What is the Four-Color Theorem?

Date: 04/03/98 at 08:45:49
From: Dan Luczak
Subject: Color Theorem

What is the color theorem?

Date: 04/03/98 at 10:44:59
From: Doctor Daniel
Subject: Re: Color Theorem

Hi there,

You asked what the "color theorem" is.  I think you mean the Four 
Color Theorem, although there may be other things you mean here.

Basically, here's what this Theorem states: 

Suppose I give you any map that is on a flat piece of paper. (That 
is, it doesn't look like a doughnut, or have 2 sides, or anything like 
that. It's two-dimensional.)  

Now, on that map, suppose that there are countries that are "simple," 
which means that no country has a part that's not connected to the 
rest of the country. (So something like Alaska being part of the rest 
of the U.S. isn't allowed.)  

So if I give you one of these maps, with the boundaries between 
countries drawn, the Four Color Theorem says that it's possible to 
give each country a color and have no two countries that touch along 
an edge with the same color.  

I always think of those puzzles of the lower 48 states that I put 
together when I was very small; the Theorem says essentially that no 
matter what the states are shaped like, there's some way the pieces 
can be in only 4 colors and when you are done there'll be no 
adjacent pieces the same color.

The Theorem is also very important because it was the first major 
theorem that was partially proven by a computer. Basically, the 
authors used a computer to find roughly 1000 small maps and showed 
that if the Theorem was false, one of them had to require 5 colors. 
Then they made the computer color them with 4 colors, showing that 
none required 5 colors. So the theorem is true.

A lot of mathematicians argued for quite a while about whether it's a 
good proof, since it's somehow less beautiful than a lot of other math 
out there. But most, at this point, accept that the theorem is 
actually true.

Try making some maps of your own and seeing how to 4-color them. You 
might also try seeing what you need to make sure it needs 4 colors and 
not 3.  

Good luck!

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