# 4 Colour Theorem

Topics: Four color theorem, Mathematics, Mathematical proof Pages: 2 (578 words) Published: January 15, 2013
The four color theorem is a mathematical theorem that states that, given a map, no more than four colors are required to color the regions of the map, so that no 2 regions that are touching (share a common boundary) have the same color. This theorem was proven by Kenneth Appel and Wolfgang Haken in 1976, and is unique because it was the first major theorem to be proven using a computer. This proof was first proposed in 1852 by Francis Guthrie when he was coloring the counties of England and realized he did not need more than four colors to color the map. Either he or his brother published this theorem (you only need four colors to color a map) in The Athenaeum in 1854. Many people had tried to solve this and had failed, two notables who had tried were, Alfred Kempe (1879) and Peter Guthrie Tait (1880). Many mathematicians kept failing until around the 1960s – 1970s when German mathematician Heinrich Heesch developed a way to use computers to solve proofs. And by 1976 Kenneth Appel and Wolfgang Haken, at the University of Illinois had stated that they had proven the theorem. They had used two technical concepts to prove that there was no map that had the smallest possible regions that required five colors. The two concepts were: 1. An unavoidable set contains regions such that every map must have at least one region from this collection. 2. A reducible configuration is an arrangement of countries that cannot occur in a minimal counterexample. If a map contains a reducible configuration, then the map can be reduced to a smaller map. This smaller map has the condition that if it can be colored with four colors, then the original map can also. This implies that if the original map cannot be colored with four colors the smaller map can't either and so the original map is not minimal. What they had done was use mathematical rules and procedures to prove that a minimal counterexample to the four color conjecture could not exist. They had to check around 1900...
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