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By: Megan Duke Jason Holman Eli Morris Aaron Wagner

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Color your map http://www.gamedesign.jp/flash/fourcolor/f ourcolor.html

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The Four Color Problem came about when Francis Guthrie was working with coloring a map of England and discovered that you could color the map using only 4 colors. Francis was trying to show that any map can be colored using only four colors so that no regions sharing a common boundary (other than a point) share the same color. Francis’s brother was a student of De Morgan, and Francis asked his brother to show the maps to him to prove if this is always true.

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De Morgan studied the problem in 1852 and could not figure out a solution. He wrote to another mathematician (Hamilton) about the problem, saying “A student of mine asked me today to give him a reason for a fact which I did not know was a fact - and do not yet. He says that if a figure be anyhow divided and the compartments differently colored so that figures with any portion of common boundary line are differently colored - four colors may be wanted, but not more - the following is the case in which four colors are wanted. Query cannot a necessity for five or more be invented....... If you retort with some very simple case which makes me out a stupid animal, I think I must do as the Sphynx did....”

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Many people continued to work on it throughout the years. In 1879 Alfred Bray Kempe announced that he had solved the Four-Color Problem. 2

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In 1890 Percy John Heawood proved that Kempe’s proof was incorrect, and then proved that every map can be colored using 5 colors. Heawood also proved that if the number of edges around each region is divisible by 3 then the regions are 4-colorable. 1 George Birkhoff then worked on the problem for many years. His work led Phillip Franklin to prove that the four color problem is true for maps with at least 25 regions. 1

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In 1976 Kenneth Appel and Wolfgang Haken then used a computer to prove the Four Color Problem. They went about their proof by using a proof by contradiction. They assumed a map that needed more than 4 colors and then developed a formula to look at maps using numbers and algorithms to express neighboring countries. 3

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Their program looked at almost 2,000 maps and proved that it must be true since no same colored region touched. It took the program almost 1,200 hours to receive the results. While many believe this is not true, no one has been able to disprove what they found. 3 Many do not accept this proof because it uses a computer and the math involved is too tedious and difficult to prove by hand.

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Edge- the line between two boundaries Vertex- a point where two or more lines meet Graph- a visual representation of edges and vertices Vertices

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http://www-groups.dcs.st- and.ac.uk/~history/HistTopics/The_four_colo ur_theorem.html http://www-groups.dcs.st- and.ac.uk/~history/HistTopics/The_four_colo ur_theorem.html http://mathworld.wolfram.com/Four- ColorTheorem.html http://unhmagazine.unh.edu/sp02/mathpion eers.html http://unhmagazine.unh.edu/sp02/mathpion eers.html http://www.mathsisfun.com/basic-math- definitions.html

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Let G be a pseudograph with vertex set V, edge set E, and incidence mapping f. Let n be a positive integer. A path of length n between vertex v and vertex.

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