The Four-Color Map Problem

By Hillary

     This most unusual problem is a exciting piece of mathematical history.  At the beginning I did not know what a interesting topic I had. As a project in our 8th grade math class we are required to make a web page about a math game. I hope you find the four-color map problem as amazing a mathematical mystery as I do.   

Image from: Four Color Theorem                                                                                                           

 Guthrie's Find

The Four Color Map Problem is something that was unsolved until 1976. Up until then it was believed that to color a map where no touching countries were the same color required five colors. Now mathematicians know that only four colors are needed to make a map where none of the countries that touch another are the same color. In 1852 Francis Guthrie was the first to propose that only four colors were needed. Guthrie was a part-time mathematician who, one day, while drawing a map of the counties of Britain noticed he needed only four colors to color the map with no county touching others with its same color.

Cartographers were the first people to except that only four colors were needed to color a map. They knew this because no one had ever come across a map that required five colors. Francis Guthrie figured this out and tried to prove why it was. He could not, so he sent a letter to his brother. His brother mentioned it to mathematicians, such as William Hamilton and Augustus De Morgan, who told other mathematicians.  

In 1879 Alfred Bray Kempe published proof of the theorem. Many mathematicians accepted this proof.  However, in 1890 Percy John Heawood found a flaw in that proof. A great breakthrough in the mystery of the Four-Color Map Problem was uncovered in 1922. Philip Franklin proved that a map with 25 or less regions could be colored with only four colors. By 1970 the proof showed 39 regions could be colored with four colors.

Kenneth Appel and Wolfgang Haken finally proved the Four-Color Map Problem in 1976. With a computer they analyzed 1476 configurations. The analysis took 1200 hours on the computer. The theorem was the first to be proven using a computer. Some mathematicians did not accept the Four-Color Map Solution. However, many programs have rechecked the proof.

In a time where society heavily depends on the aid of computers, the question asked is whether the work of a computer is believable.  Whatever the views on this question, the Four-Color Map Problem has been solved. A map only needs four colors to be colored with no two regions that are adjacent the same color.

Image from manipulative I created

Hyper Links

In this site I have mainly covered the history of the Four Color Map and how it works on a two dimensional surface. For a look at the more mathematical side of the Four Color problem and how it works on a three dimensional surface visit Four Color Theorem.

At The Mathematics Behind the Maps you learn more about coloring the maps and can print out your own maps to color using the Four Color Map rules.

Bibliography

Four color theorem. Wikipedia. 26 Nov. 2003
    <http://en.wikipedia.org/wiki/Four_color_theorem>. 

Maps of Many Colors. 6 Jan. 1997. MAA Online. 26 Nov. 2003
    <http://www.maa.org/mathland/mathland_1_6.html>. 

The Four Color Map problem. 11 Nov. 1997. The Math Forum. 29 Nov. 2003
    <http://mathforum.org/library/drmath/view/52466.html>.

Four Color Theorem. MegaMath. 30 Nov. 2003
    <http://www.c3.lanl.gov/mega-math/gloss/math/4ct.html>. 

The Four-Color Map Problem. Mappa Mundi Magazine. 8 Dec. 2003
    <http://mappa.mundi.net/locus/locus_014/>. 

The Colossal Book of Mathematics. New York: W.W. Norton & Company Inc,
    2001. 674-685

Last up dated by Hillary on February 19, 2004