What is the four-colour map problem?

The four-color map problem, originally posed in the 1850s, asks whether four colors are always enough to color the regions on any map so that no two adjacent regions sharing a common boundary segment have the same color. It was demonstrated that three colors would not suffice, but it was proved that five colors would. In 1976, mathematicians Kenneth Appel and Wolfgang Haken solved the problem, with the proof requiring more than 1,000 hours of computer calculations. The proof has generated debate because it relied substantially on computer analysis that could not be verified by hand.

Why do people want to colour maps using as few colours as possible?

The desire to color maps using as few colors as possible originates from the four-color map problem, which asks whether every map can be colored using only four colors in such a way that regions sharing a border are different colors. The problem stems from an empirical observation that, in geographical maps, no more than four colors are needed to color the regions such that two regions sharing a common boundary are always colored differently. The formalization of this observation constitutes the four-color theorem. The four-color map problem translates to determining whether the vertices of a planar graph can always be colored using only four colors in such a way that vertices joined by an edge have different colors.

What does it mean for two regions on a map to be considered 'adjacent'?

In the context of the four-color map problem , two regions on a map are considered 'adjacent' if they share a common boundary. This means the regions must border each other along a line, not just meet at a single point. For example, Colorado and Arizona in the United States, which only meet at a corner, are not considered adjacent. The four-color theorem states that any map can be colored using no more than four colors, with no adjacent regions sharing the same color.

How was the four-colour map problem first discovered?

The four-color map problem originated in 1852 when Francis Guthrie, while coloring a map of England, noticed that only four colors were needed to ensure no adjacent regions shared the same color. He inquired with his brother, a student of Augustus De Morgan, about whether this observation could be proven mathematically. This question intrigued mathematicians for over a century. The problem asks whether four colors are always sufficient to color any map such that regions sharing a common border have different colors. The empirical observation was formalized into the four-color theorem.

How was the four-colour map problem finally solved?

The four-colour map problem, dating back to the 1850s, asks what is the minimum number of colours needed to colour a map so that no two adjacent regions share the same colour. In 1977, mathematicians Kenneth Appel and Wolfgang Haken at the University of Illinois solved the problem using a computer-assisted proof that was several hundred pages long. Appel and Haken created a catalog of 1,936 unavoidable configurations, one of which must be present in any graph. They demonstrated how each of these configurations could be reduced to a smaller one, such that if the smaller one could be coloured with four colours, so could the original configuration. The computer required over 1,000 hours to perform the task. The reliance on computers in the proof sparked debate within the mathematics community. In 1997, other mathematicians reduced the number of configurations to 633 and simplified the argument, but the computer portion of the proof was not eliminated.

Why was the solution to the four-colour map problem important or unique in mathematics?

The four-colour map problem, which originated in the 1850s and was solved in 1976, asks for the minimum number of colours needed to colour a map so that no two adjacent regions share a common border segment. Consideration of this problem has spurred discoveries in topology and combinatorics . The solution, involving an unprecedented synthesis of computer search and theoretical reasoning, was reached by Kenneth Appel and Wolfgang Haken in 1977. They created a catalog of 1,936 unavoidable configurations and showed how each could be reduced. The complete proof required over 1,000 hours of computer calculations. The reliance on computers in the proof sparked debate among mathematicians as to whether it should be considered a valid proof.

four-colour map problem

Introduction References & Edit History Related Topics

Quick Facts

Related Topics:: map-colouring problem

See all related content

four-colour map problem

Written by

Robert Osserman

Fact-checked by

Britannica Editors

History

Top Questions

What is the four-colour map problem?
The four-color map problem, originally posed in the 1850s, asks whether four colors are always enough to color the regions on any map so that no two adjacent regions sharing a common boundary segment have the same color.

It was demonstrated that three colors would not suffice, but it was proved that five colors would. In 1976, mathematicians Kenneth Appel and Wolfgang Haken solved the problem, with the proof requiring more than 1,000 hours of computer calculations. The proof has generated debate because it relied substantially on computer analysis that could not be verified by hand.
4 Britannica Sources
Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.
Why do people want to colour maps using as few colours as possible?
The desire to color maps using as few colors as possible originates from the four-color map problem, which asks whether every map can be colored using only four colors in such a way that regions sharing a border are different colors.

The problem stems from an empirical observation that, in geographical maps, no more than four colors are needed to color the regions such that two regions sharing a common boundary are always colored differently. The formalization of this observation constitutes the four-color theorem.

The four-color map problem translates to determining whether the vertices of a planar graph can always be colored using only four colors in such a way that vertices joined by an edge have different colors.
2 Britannica Sources
1. graph theory
2. combinatorics: The four-colour map problem
Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.
What does it mean for two regions on a map to be considered 'adjacent'?
In the context of the four-color map problem, two regions on a map are considered 'adjacent' if they share a common boundary. This means the regions must border each other along a line, not just meet at a single point. For example, Colorado and Arizona in the United States, which only meet at a corner, are not considered adjacent.

The four-color theorem states that any map can be colored using no more than four colors, with no adjacent regions sharing the same color.
1 Britannica Source
1. combinatorics: The four-colour map problem
Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.
How was the four-colour map problem first discovered?
The four-color map problem originated in 1852 when Francis Guthrie, while coloring a map of England, noticed that only four colors were needed to ensure no adjacent regions shared the same color. He inquired with his brother, a student of Augustus De Morgan, about whether this observation could be proven mathematically. This question intrigued mathematicians for over a century. The problem asks whether four colors are always sufficient to color any map such that regions sharing a common border have different colors. The empirical observation was formalized into the four-color theorem.
2 Britannica Sources
1. four-colour map problem
2. combinatorics: The four-colour map problem
Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.
How was the four-colour map problem finally solved?
The four-colour map problem, dating back to the 1850s, asks what is the minimum number of colours needed to colour a map so that no two adjacent regions share the same colour. In 1977, mathematicians Kenneth Appel and Wolfgang Haken at the University of Illinois solved the problem using a computer-assisted proof that was several hundred pages long.

Appel and Haken created a catalog of 1,936 unavoidable configurations, one of which must be present in any graph. They demonstrated how each of these configurations could be reduced to a smaller one, such that if the smaller one could be coloured with four colours, so could the original configuration. The computer required over 1,000 hours to perform the task.

The reliance on computers in the proof sparked debate within the mathematics community. In 1997, other mathematicians reduced the number of configurations to 633 and simplified the argument, but the computer portion of the proof was not eliminated.
2 Britannica Sources
1. four-colour map problem
2. combinatorics: The four-colour map problem
Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.
Why was the solution to the four-colour map problem important or unique in mathematics?
The four-colour map problem, which originated in the 1850s and was solved in 1976, asks for the minimum number of colours needed to colour a map so that no two adjacent regions share a common border segment. Consideration of this problem has spurred discoveries in topology and combinatorics.

The solution, involving an unprecedented synthesis of computer search and theoretical reasoning, was reached by Kenneth Appel and Wolfgang Haken in 1977. They created a catalog of 1,936 unavoidable configurations and showed how each could be reduced. The complete proof required over 1,000 hours of computer calculations.

The reliance on computers in the proof sparked debate among mathematicians as to whether it should be considered a valid proof.
4 Britannica Sources
Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

four-colour map problem, problem in topology, originally posed in the early 1850s and not solved until 1976, that required finding the minimum number of different colours required to colour a map such that no two adjacent regions (i.e., with a common boundary segment) are of the same colour. Three colours are not enough, since one can draw a map of four regions with each region contacting the three other regions. It had been proved mathematically by the English attorney Alfred Bray Kempe in 1879 that five colours will always suffice; and no map had ever been found on which four colours would not do. As is often the case in mathematics, consideration of the problem provided the impetus for the discovery of related results in topology and combinatorics. A similar problem had been solved for the seemingly more complicated situation of a map drawn on a torus (doughnut-shaped surface), where seven colours were known to be the minimum.

The four-colour problem was solved in 1977 by a group of mathematicians at the University of Illinois, directed by Kenneth Appel and Wolfgang Haken, after four years of unprecedented synthesis of computer search and theoretical reasoning. Appel and Haken created a catalog of 1,936 “unavoidable” configurations, at least one of which must be present in any graph, no matter how large. Then they showed how each of these configurations could be reduced to a smaller one so that, if the smaller one could be coloured with four colours, so could the original catalog configuration. Thus, if there were a map that could not be coloured with four colours, they could use their catalog to find a smaller map that also could not be four-coloured, and then a smaller one still, and so on. Eventually this reduction process would lead to a map with only three or four regions that, supposedly, could not be coloured with four colours. This absurd result, which is derived from the hypothesis that a map requiring more than four colours might exist, leads to the conclusion that no such map can exist. All maps are in fact four-colourable.

The strategy involved in this proof dates back to the 1879 paper of Kempe, who produced a short list of unavoidable configurations and then showed how to reduce each to a smaller case. Appel and Haken replaced Kempe’s brief list with their catalog of 1,936 cases, each involving up to 500,000 logical options for full analysis. Their complete proof, itself several hundred pages long, required more than 1,000 hours of computer calculations.

four-colour map problem

Thank you for signing up!