The Four color Theorem
Have you ever paint a design and have wondered how many colors should you use?
How many different colors you need to paint the block?
What is the minimum number of colors you need to paint the block?
As far quite two colors
• We set only one rule: two areas that share a common edge can not be painted with the same color! A common tip (node) does not matter.
• How about the following figure? Here you can, applying the rule we set, we cannot use less than three colors
• And now a more complex shape: How many colors will need less? Six or four or three?
The less you will need four colors. By restriction we set you can not use fewer colors
The four-color theorem
Few words about history The problem of coloring maps arose in 1852, when Francis Guthrie trying to paint a map of England noted that four colors suffice that two bordering counties do not have the same color.
• MÜbius mentioned the problem in his lectures as early as 1840.[3] The conjecture was first proposed on October 23, 1852 [4] when Francis Guthrie, while trying to color the map of counties of England, noticed that only four different colors were needed. At the time, Guthrie's brother, Frederick, was a student of Augustus De Morgan (the former advisor of Francis) at University College London
• De morgan did not knew the fact, as he posed the question again in the Athenaeum magazine in 1860. • Since then Mathematicians try to prove the theory of Four colors. One alleged proof was given by Alfred Kempe in 1879, another was given by Peter Guthrie Tait in 1880.
• It was not until 1890 that Kempe's proof was shown incorrect by Percy Heawood. In 1943, Hugo Hadwiger formulated the Hadwiger conjecture (Hadwiger 1943), a far-reaching generalization of the four-color problem that still remains unsolved.
• Kenneth Appel and Wolfgang Haken at the University of Illinois announced, on June 21, 1976, [9] that they had proven the theorem. They were assisted in some algorithmic work by John A. Koch • the math department at the University of Illinois used a postmark stating "Four colors suffice." At the same time the unusual nature of the proof—it was the first major theorem to be proven with extensive computer assistance—and the complexity of the human-verifiable portion, aroused considerable controversy
• In 1996, Neil Robertson, Daniel P. Sanders, Paul Seymour, and Robin Thomas created a quadratic time algorithm, improving on a quartic algorithm based on Appel and Haken’s proof. Both the unavoidability and reducibility parts of this new proof must be executed by computer and are impractical to check by hand
http://en.wikipedia.org/wiki/Four_color_theorem
• In 2005, Benjamin Werner and Georges Gonthier formalized a proof of the theorem inside the Coq proof assistant. This removed the need to trust the various computer http://www.mathsisfun.com/activity/coloring. programshtml used to verify particular cases; it is only necessary to trust the Coq kernel ( Gonthier 2008).
http://www.mathsisfun.com/activity/coloring.html http://en.wikipedia.org/wiki/Four_color_theorem
• Like any kind of brain-teasing puzzle so the problem of the four colors can provide many hours of entertainment for many people. Furthermore it is a challenge. So just worded that can not be proven is for mathematicians an intellectual challenge with enormous complexity