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### What is the Four-Color Theorem?

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

What is the color theorem?
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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

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/
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Associated Topics:
High School Discrete Mathematics

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