1. Four Color Theorem - Wikipedia, The Free Encyclopedia O'Connor; Robertson (1996), The Four Colour Theorem, MacTutor archive, http//wwwgroups.dcs.st-and.ac.uk/~history/HistTopics/The_four_colour_theorem.html http://en.wikipedia.org/wiki/Four_color_theorem

Four color theorem From Wikipedia, the free encyclopedia Jump to: navigation search Example of a four-colored map A four-coloring of an actual map of the states of the United States (ignoring water, other countries and text color). In mathematics , the four color theorem , or the four color map theorem states that, given any separation of a plane into contiguous regions, producing a figure called a map , no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. Two regions are called adjacent only if they share a border segment, not just a point. For example, Utah and Arizona are adjacent, but Utah and New Mexico , which only share a point, are not. Despite the motivation from coloring political maps of countries, the theorem is not of particular interest to mapmakers. According to an article by the math historian Kenneth May Wilson 2002 , 2), "Maps utilizing only four colours are rare, and those that do usually require only three. Books on cartography and the history of mapmaking do not mention the four-color property." Three colors are adequate for simpler maps, but an additional fourth color is required for some maps, such as a map in which one region is surrounded by an odd number of other regions that touch each other in a cycle. The

2. Four Colour Theorem Ugliness 80x60cm 2005 Higher resolution Back to Gallery. THE FOUR COLOUR THEOREM. In Greek mythology, the Sirens were sisters whose song lured sailors to their deaths http://www.justinmullins.com/four_colour_theorem.htm

Ugliness Higher resolution THE FOUR COLOUR THEOREM In Greek mythology, the Sirens were sisters whose song lured sailors to their deaths on a treacherous reef. The Four Colour Theorem is to mathematicians what the song of the Sirens was to the sailors of ancient times. Many a mathematician has foundered in attempting its proof . The problem is deceptively simple. Take a map of the world, a few coloured pens and start colouring in the countries. There's one ruleadjacent countries cannot be the same colour. The Four Colour Theorem is the hypothesis that the maximum number of colours needed to fill the map is four. It's the kind of problem that, for a mathematician, has the smell of simplicity and elegance about it. You just know that somebody somewhere will have come up with an elegant method that makes the answer apparent. Except they haven't. In place of an elegant proof is a monstrous machine created by Kenneth Appel and Wolfgang Haken, both mathematicians at the University of Illinois, who cracked the problem in 1976. But instead of insight and panache, they used force. Appel and Haken first worked out that they could reduce the number of possible maps from infinity to just 1,936. They then checked each one against the theory. The task turned out to be so intensive that they needed a computer to do it. Even then, it took hundreds of hours. In a sense, they squeezed the life out of the problem until it gave up its answer, a method known as brute force.

3. The Four Colour Theorem The Four Colour Conjecture first seems to have been made by Francis Guthrie. He was a student at University College London where he studied under De Morgan. http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_four_colour_theorem.h

The four colour theorem Geometry and topology index History Topics Index Version for printing The Four Colour Conjecture first seems to have been made by Francis Guthrie . He was a student at University College London where he studied under De Morgan . After graduating from London he studied law but by this time his brother Frederick Guthrie had become a student of De Morgan . Francis Guthrie showed his brother some results he had been trying to prove about the colouring of maps and asked Frederick to ask De Morgan about them. De Morgan was unable to give an answer but, on 23 October 1852, the same day he was asked the question, he wrote to Hamilton in Dublin. De Morgan wrote:- 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 coloured so that figures with any portion of common boundary line are differently coloured - four colours may be wanted, but not more - the following is the case in which four colours 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.... Hamilton replied on 26 October 1852 (showing the efficiency of both himself and the postal service):- I am not likely to attempt your quaternion of colour very soon.

4. Math G Mission College Santa Clara September 1996 http//wwwhistory.mcs.st-andrews.ac.uk/history/HistTopics/The_four_colour_theorem.html (4/25/99) The Most Colorful Math of All MegaMath http://www.missioncollege.org/depts/math/beard2.htm

Math Department, Mission College, Santa Clara, California Go to Math Dept Main Page Mission College Main Page This paper was written as an assignment for Ian Walton's Math G - Math for liberal Arts Students - at Mission College. If you use material from this paper, please acknowledge it. To explore other such papers go to the Math G Projects Page. The Four Color Theorem Michelle Beard Math G, Spring 1999 Mission College, Santa Clara How many colors are required to color any map so that no countries with common borders are the same color? It is generally held that four colors, for any flat map, will suffice. But a belief that is commonly held and easily observed, is not a mathematical certainty. Nor does the simplicity of a question reflect the ease with which the answer can be proven. The mathematical evidence to create a valid proof that four colors are all that is required had evaded mathematicians for nearly 140 years. What became known as the Four Color Conjecture has been the cause of great fascination and frustration. It has also been the stimulus for new ideas in topology, knot theory, and the concept of mathematical proof. Historical Overview: The question was originally posed by Francis Guthrie, a former student of the famous mathematician Augustus De Morgan, in 1852. Although Francis moved on to study law, his brother Frederick Guthrie had become a student of De Morgan. Francis Guthrie presented his work on the idea to his brother asking that he pass it along to De Morgan.

5. The Four Colour Theorem Tait addressed the Royal Society of Edinburgh on the subject and published two papers on the (what we should now call) Four Colour Theorem. http://www-history.mcs.st-and.ac.uk/HistTopics/The_four_colour_theorem.html

The four colour theorem Geometry and topology index History Topics Index Version for printing The Four Colour Conjecture first seems to have been made by Francis Guthrie . He was a student at University College London where he studied under De Morgan . After graduating from London he studied law but by this time his brother Frederick Guthrie had become a student of De Morgan . Francis Guthrie showed his brother some results he had been trying to prove about the colouring of maps and asked Frederick to ask De Morgan about them. De Morgan was unable to give an answer but, on 23 October 1852, the same day he was asked the question, he wrote to Hamilton in Dublin. De Morgan wrote:- 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 coloured so that figures with any portion of common boundary line are differently coloured - four colours may be wanted, but not more - the following is the case in which four colours 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.... Hamilton replied on 26 October 1852 (showing the efficiency of both himself and the postal service):- I am not likely to attempt your quaternion of colour very soon.

6. The Four Colour Theorem References Books. N L Biggs, E K Lloyd and R J Wilson, Graph Theory 17361936 (Oxford, 1986). Articles N L Biggs, E K Lloyd and R J Wilson, C S Peirce and De Morgan on the four-colour http://www.gap-system.org/~history/HistTopics/References/The_four_colour_theorem

References for: The four colour theorem Version for printing Books N L Biggs, E K Lloyd and R J Wilson, Graph Theory 1736-1936 (Oxford, 1986). Articles: N L Biggs, E K Lloyd and R J Wilson, C S Peirce and De Morgan on the four-colour conjecture, Historia Mathematica G D Birkhoff, Collected mathematical papers Vol. III : The four color problem, miscellaneous papers (New York 1968). L Arias C, A chronological account of the various theories developed for solving the four-color problem (Spanish), Rev. Integr. Temas Mat. D A Holton and S Purcell, The four colour theorem-a short history, Austral. Math. Soc. Gaz. P C Kainen, Is the four color theorem true?, Geombinatorics Proceedings of the Seminar on the History of Mathematics (Paris, 1982), 43-62. T L Saaty, The four-color problem : assaults and conquest (New York, 1977). J Wilson, New light on the origin of the four-color conjecture, Historia Mathematica JOC/EFR September 1996 The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/The_four_colour_theorem.html

7. Four Color Theorem - Math Images Add teaching materials. References. History of the four color theorem http//www.gapsystem.org/~history/HistTopics/The_four_colour_theorem.html http://mathforum.org/mathimages/index.php/Four_Color_Theorem

Four Color Theorem From Math Images Jump to: navigation search Add to NSDL View in NSDL ... Edit Metadata Four Color Theorem Field: Graph Theory Image Created By: Brendan John Four Color Theorem This image shows a four coloring and graph representation of the United States. Contents Basic Description A More Mathematical Explanation Coloring maps on other figures Teaching Materials ... References Basic Description Suppose we have a map in which no single territory is made up of disconnected regions. How many colors are needed to color the territories of this map, if all the territories that share a border segment must be of different colors? It turns out that only four colors are needed to color such a two-dimensional map. It has taken over a century for a correct proof of this fact to emerge, and currently known proofs are so long that they can only be checked with the aid of computers. An example of a map colored with only 4 colors is the map of The United States in this page's main image. Example of a planar graph (top) and a non-planar graph (bottom) Map coloring is an application of Graph Theory, the study of

8. Four Color Theorem By Ashay Dharwadker. Includes profile, research papers for other algorithms, lecture notes, and student database. http://www.dharwadker.org/

A NEW PROOF OF THE FOUR COLOUR THEOREM A NEW PROOF OF THE FOUR COLOUR THEOREM

9. Four-Color Theorem -- From Wolfram MathWorld Oct 11, 2010 The fourcolor theorem states that any map in a plane can be colored using four- colors in such a way that regions sharing a common boundary http://mathworld.wolfram.com/Four-ColorTheorem.html

10. Four Colour Theorem The famous Four Colour Theorem is concerned with mathematics as well as geography it was first noted by August Ferdinand M bius in 1840. In 1852 a student of De Morgan, then http://www.mathsisgoodforyou.com/conjecturestheorems/fourcolour.htm

The Four Colour Theorem home courses topics theorems ... timeline The famous Four Colour Theorem is concerned with mathematics as well as geography: it was first noted by in 1840. In 1852 a student of De Morgan , then professor of mathematics at the University College London (who was also founder of the London Mathematical Society ), conjectured with his brother, that any map or any figure divided in any way should be able to be coloured with four colours alone. This means that if you have a map of any number of states drawn on it, if you colour them with four colours, that would be enough to distinguish all the states and at no border on such a map will there be two states filled with the same colour. Type of the problem can be studied through drawing of different diagrams like this one - see on the right-hand side of this page for a worksheet on this topic. The problem persisted throughout the next twenty years or more and British as well as some American mathematicians tried to either prove or disprove this conjecture The colouring of geographical maps is a topological problem of a kind - it depends on the position of the countries, not on their shape, size, political systems or any other geographical, social or cultural features!

11. A Computer-checked Proof Of The Four Colour Theorem Microsoft Word A computer-checked proof of the four color theorem.doc http://research.microsoft.com/en-us/um/people/gonthier/4colproof.pdf

12. Metaphors Various theorems related to the colouring of planar graphs are studied and a counter example for Kempe’s incorrect proof of four colour theorem is also provided. http://sandeep-rb.livejournal.com/

You are viewing 's journal Log in Create a LiveJournal Account Learn more Recent Entries ... A thing of beauty is a joy for me unless I possess it. Oct. 26th, 2010 at 8:34 PM PS: Yes, Keats was right. Tags: beauty happiness possessiveness Leave a comment ... Crime Oct. 14th, 2010 at 10:40 AM If I do a crime knowingly, then the criminal is my weakness. If I do a crime unknowingly, then the criminal is my ignorance. Then, when I do a crime why do they prison me rather than imparting strength and knowledge to eradicate my weakness and ignorance? Or is it that they are more powerful than me and the act they judge as crime is not so in my perspective? Tags: crime ignorance knowledge strength ... One more classification Oct. 8th, 2010 at 1:31 PM There are people who want to solve their problems immediately. Extremists of this kind won't eat and sleep till they solve their problems. They are happy that they solve problems. There is another kind of people who cherish their problems and wonder about the existence of problems. They appreciate the process of solving problem rather than the solution itself. They are happy that they have problems. Tags: classification problem Leave a comment Add to Memories ... Share this!

13. The Four Color Theorem Computer aided proof of the four color theorem by Neil Robertson, Daniel P. Sanders, Paul Seymour and Robin Thomas. http://www.math.gatech.edu/~thomas/FC/fourcolor.html

The Four Color Theorem This page gives a brief summary of a new proof of the Four Color Theorem and a four-coloring algorithm found by Neil Robertson Daniel P. Sanders , Paul Seymour and Robin Thomas Table of Contents: History. Why a new proof? Outline of the proof. Main features of our proof. ... References. History. The Four Color Problem dates back to 1852 when Francis Guthrie, while trying to color the map of counties of England noticed that four colors sufficed. He asked his brother Frederick if it was true that any map can be colored using four colors in such a way that adjacent regions (i.e. those sharing a common boundary segment, not just a point) receive different colors. Frederick Guthrie then communicated the conjecture to DeMorgan. The first printed reference is due to Cayley in 1878. A year later the first `proof' by Kempe appeared; its incorrectness was pointed out by Heawood 11 years later. Another failed proof is due to Tait in 1880; a gap in the argument was pointed out by Petersen in 1891. Both failed proofs did have some value, though. Kempe discovered what became known as Kempe chains, and Tait found an equivalent formulation of the Four Color Theorem in terms of 3-edge-coloring. The next major contribution came from Birkhoff whose work allowed Franklin in 1922 to prove that the four color conjecture is true for maps with at most 25 regions. It was also used by other mathematicians to make various forms of progress on the four color problem. We should specifically mention Heesch who developed the two main ingredients needed for the ultimate proof - reducibility and discharging. While the concept of reducibility was studied by other researchers as well, it appears that the idea of discharging, crucial for the unavoidability part of the proof, is due to Heesch, and that it was he who conjectured that a suitable development of this method would solve the Four Color Problem.

14. Re: Four Color Theorem I am not close Steinberg's Conjecture file yet. coloring in settleing that conjecture as well. It follows from the four colour theorem. Any triangulation with minimum http://sci.tech-archive.net/Archive/sci.math/2006-07/msg01673.html

Re: Four Color Theorem From Date : 9 Jul 2006 23:39:17 -0700 I am not close Steinberg's Conjecture file yet. I hope we will again working on it again. I hope our lemma (corrected by yourself) on the triangulated ring would some way be used together with the spiral-chain coloring in settleing that conjecture as well. By the way have you heard anything new on Steinberg's conjecture newly? Cahit Proginoskes wrote: I must be careful when I give an answer to your question. Yes my proof has not yet published but I think it has been passed through extensive public review in the last two years. I have given several seminars in the mathematical departments. The bottom line is that, no one yet and I trust, no one will be given serious comments against its correctness. I have my doubts, especially in Cahit's paper which "settles" Steinberg's Conjecture. In particular, I derived the conditions under which a "triangulated ring" is 3-colorable, and they do not match Cahit's. That, and my increased workload during the academic year, have kept me from looking at more of the problem.

15. The Journey Of The Four Colour Theorem Through Time A paper on the implications of Appel and Haken s computer-aided proof by Andreea Calude. http://www.calude.net/andreea/4CT.pdf

16. Proof By Exhaustion - Definition The shortest known proof of the four colour theorem today still has over 600 cases. Mathematicians prefer to avoid proofs with large numbers of cases because these proofs feel http://www.wordiq.com/definition/Proof_by_exhaustion

Proof by exhaustion - Definition Proof by exhaustion , also known as the brute force method or case analysis , is a method of mathematical proof in which the statement to be proved is split into a finite number of cases, and each case is proved separately. A proof by exhaustion contains two stages: A proof that the cases are exhaustive; i.e., that each instance of the statement to be proved matches the conditions of (at least) one of the cases. A proof of each of the cases. In contrast, the method of exhaustion of Eudoxus of Cnidus was a geometrical and essentially rigorous way of calculating mathematical limits Example To prove that every cube number is either a multiple of 9 or is 1 more or 1 less than a multiple of 9. Proof Each cube number is the cube of some integer n. This integer is either a multiple of 3, or is 1 more or 1 less than a multiple of 3. So the following 3 cases are exhaustive: Case 1: If n is a multiple of 3 then the cube of n is a multiple of 27, and so certainly a multiple of 9. Case 2: If n is 1 more than a multiple of 3 then the cube of n is 1 more than a multiple of 9.

17. Nrich.maths.org :: Mathematics Enrichment :: The Four Colour Theorem The Four Colour Conjecture was first stated just over 150 years ago, and finally proved conclusively in 1976. It is an outstanding example of how old ideas can be combined with http://nrich.maths.org/6291

Skip over navigation Search by Topic Tech Help Thesaurus ... Contact Us Be part of our research! Are you really good at maths? Are you the parent of someone who is really good at maths? Are you the teacher of someone who is really good at maths? If so we'd love to involve you in our research. Case studies Are you aged between 10 and 17? Read about Dr Yi Feng's research which involves talking to students over the next two years about their mathematical experiences in and out of school, and respond to the call for participants. AskNRICH Do you use the ASKNRICH site, or have you previously? Libby Jared's research involves looking at what goes on on the ASKNRICH site and why. To help with this research please respond to the ASKNRICH survey Read more about the research we are doing here This month: The Four Colour Theorem Article Teachers' Notes Printable page Article by Leo Rogers Stage: 3 and 4 The Four Colour Conjecture was first stated just over 150 years ago, and finally proved conclusively in 1976. It is an outstanding example of how old ideas combine with new discoveries and techniques in different fields of mathematics to provide new approaches to a problem. It is also an example of how an apparently simple problem was thought to be 'solved' but then became more complex, and it is the first spectacular example where a computer was involved in proving a mathematical theorem. 1. In the Beginning

18. The Four Color Theorem Obviously the above graph is not 3colorable, but it is 4-colorable. The Four Color Theorem asserts that every planar graph - and therefore every map on http://www.mathpages.com/home/kmath266/kmath266.htm

The Four Color Theorem How many different colors are sufficient to color the countries on a map in such a way that no two adjacent countries have the same color? The figure below shows a typical arrangement of colored regions. Notice that we define adjacent regions as those that share a common boundary of non-zero length. Regions which meet at a single point are not considered to be "adjacent". The coloring of geographical maps is essentially a topological problem, in the sense that it depends only on the connectivities between the countries, not on their specific shapes, sizes, or positions. We can just as well represent each country by a single point (vertex), and the adjacency between two bordering countries can be represented by a line (edge) connecting those two points. It's understood that connecting lines cannot cross each other. A drawing of this kind is called a planar graph. A simple map (with just five "countries") and the corresponding graph are shown below. A graph is said to be n-colorable if it's possible to assign one of n colors to each vertex in such a way that no two connected vertices have the same color. Obviously the above graph is not 3-colorable, but it is 4-colorable. The Four Color Theorem asserts that every planar graph - and therefore every "map" on the plane or sphere - no matter how large or complex, is 4-colorable. Despite the seeming simplicity of this proposition, it was only proven in 1976, and then only with the aid of computers. Notice that the above graph is "complete" in the sense that no more connections can be added (without crossing lines). The edges of a complete graph partition the graph plane into three-sided regions, i.e., every region (including the infinite exterior) is bounded by three edges of the graph. Every graph can be constructed by first constructing a complete graph and then deleting some connections (edges). Clearly the deletion of connections cannot cause an n-colorable graph to require any additional colors, so in order to prove the Four Color Theorem it would be sufficient to consider only complete graphs.

19. BBC - H2g2 - The Four Colour Theorem h2g2 is the unconventional guide to life, the universe and everything, a guide that's written by visitors to the website, creating an organic and evolving encyclopedia of life http://www.bbc.co.uk/dna/h2g2/A57240092

@import '/h2g2/skins/brunel/brunel.css'; @import '/dnaimages/bbcpage/v3-0/toolbar.xhtml-transitional.css'; Skip to main content Text Only version of this page Access keys help Home Explore the BBC 31st October 2010 Accessibility help Text only Guide ID: A57240092 (Edited) Edited Guide Entry SEARCH h2g2 Edited Entries only Advanced Search Sign in or register to join or start a new conversation. BBC Homepage The Guide to Life The Universe and Everything 3. Everything Mathematics Created: 26th October 2009 The Four Colour Theorem Front Page What is h2g2? Who's Online Write an Entry ... Contact Us Like this page? Send it to a friend! There's nothing that occupies a child like a colouring book and a box of crayons. They can spend hours happily turning the dullest of outline drawings into something which rivals a Jimi Hendrix Colouring Maps This is another of those problems which has interested mathematicians for many years. Now, there's not much benefit to mankind in knowing how many crayons you might need to colour a picture of a rabbit, but there are some slightly more relevant applications, one being the colouring of maps need island Ireland Sometimes you need four colours. Take the land-locked state of Nevada, USA (pictured in red to the right). This has borders with 5 other states, which, clockwise, are California, Oregon, Idaho, Utah and

20. Four Colour Theorem The four colour theorem (also known as the four colour map theorem) states that given any plane separated into regions, such as a political map of the states of a country, the http://schools-wikipedia.org/wp/f/Four_color_theorem.htm

Four colour theorem 2008/9 Schools Wikipedia Selection . Related subjects: Mathematics Example of a four-colored map The four colour theorem (also known as the four colour map theorem ) states that given any plane separated into regions, such as a political map of the states of a country, the regions may be colored using no more than four colors in such a way that no two adjacent regions receive the same colour. Two regions are called adjacent only if they share a border segment, not just a point. Each region must be contiguous: that is, it may not have exclaves like some real countries such as Angola Azerbaijan , the United States , or Russia It is often the case that using only three colors is inadequate. This applies already to the map with one region surrounded by three other regions (although with an even number of surrounding countries three colors are enough) and it is not at all difficult to prove that five colors are sufficient to colour a map. The four colour theorem was the first major theorem to be proven using a computer, and the proof is not accepted by all mathematicians because it would be unfeasible for a human to verify by hand (see computer-assisted proof). Ultimately, in order to believe the proof, one has to have faith in the correctness of the

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