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/ |
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