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	Four color theorem - Wikipedia, the free encyclopedia
		The four color theorem (also known as the four color map theorem) states that given any plane separated into regions, such as a political map of the counties of a state, the regions may be colored using no more than four colors in such a way that no two adjacent regions receive the same color.
		The four color 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-aided proof).
		O'Connor and Robertson, The Four Colour Theorem, at the MacTutor archive, 1996.
	en.wikipedia.org /wiki/Four_color_theorem (2214 words)

	four
		He successfully investigated the number of colours needed for maps on other surfaces and gave what is known as the Heawood estimate for the necessary number in terms of the Euler characteristic of the surface.
		Reynolds increased it to 27 in 1926, Winn to 35 in 1940, Ore and Stemple to 39 in 1970 and Mayer to 95 in 1976.
		The Four Colour Theorem was the first major theorem to be proved using a computer, having a proof that could not be verified directly by other mathematicians.
	library.thinkquest.org /C006364/ENGLISH/problem/four.htm (1618 words)

	Four color theorem : Four colour theorem
		The four color theorem states that every possible geographical map can be colored with at most four colors in such a way that no two adjacent regions receive the same color.
		This theorem was conjectured in 1853 by Francis Guthrie[?].
		The four color theorem was the first major theorem to be proven using a computer, and the proof was not accepted by all mathematicians because it could not directly be verified by a human.
	www.mik.fastload.org /fo/Four_colour_theorem.html (591 words)

	Four Colour Theorem (Site not responding. Last check: 2007-10-13)
		The four colour theorem states that any planar (flat 2d) map needs only 4 colours to be coloured in.
		The first colouring is created using an agressive technique - a colour is applied to a random region, and then, again randomly, to as many regions as possible.
		A simple proof of this theorem has eluded mathematicians for centuries, but here is a proof that the four colour theorem is true, and that there is no planar map that needs 5 or more colours.
	www.users.globalnet.co.uk /~perry/maths/fourcolour/fourcolour.htm (658 words)

	'Four Colours Suffice'
		The Four Colour Theorem - the statement that four colours suffice to fill in any map so that neighbouring countries are always coloured differently - has had a long and controversial history.
		Robin Wilson's book is a clear and well-written description of the history of the theorem; the people who worked on it, the mathematics they invented in their attempts to solve it, and the controversy that followed when it was finally proved.
		Where they differ is in the manner of proof; in the case of Fermat's Last Theorem, the "onslaught" was Andrew Wiles' solitary tour-de-force, in that of the Four Colour Theorem, it was a computer's brute force.
	plus.maths.org /issue25/reviews/book2 (578 words)

	Claim of Proof to Four-Color Theorem
		At this stage we had little doubt that the four colour conjecture was true, ard various mathematicians with whom we corresponded, including Bertrand Russell, were aware that we had a method that was almost certainly capable of proving it.
		The nub of the proof lies in two theorems, the first essential to the method, and the second a crucial restatement of the fourcolour conjecture in a way that renders it susceptable to proof.
		This elegant theorem I have now proved, and the proof will have been published by the time you are able to print this communication.
	www.lawsofform.org /gsb/nature.html (688 words)

	The Four Color Theorem
		The graph of a set of three mutually adjoining regions is simply a topological triangle, and if we add a fourth region, it is represented by a fourth vertex in the graph, which must be located either inside or outside the triangle formed by the graph of the original three vertices.
		Likewise the four vectors assigned to the vertices of a tetrahedral graph would be a complete set of four mutually orthogonal vectors (a tetrad), but still with arbitrary orientation.
		In this context the four color theorem tells us that a space of four dimensions is sufficient to enable us to assign one of the four basis vectors to each vertex of a planar graph in such a way that the vectors of every pair of adjacent vertices are orthogonal.
	www.mathpages.com /home/kmath266/kmath266.htm (4081 words)

	Four Colour Theorem: A Small Historical Insight.
		For purposes of proper colouring it is equivalent to consider maps on the plane and furthermore, only maps which have exactly three edges meeting at each vertex.
		Lemma 1 proves the six colour theorem using Euler’s formula, showing that any map on the plane may be properly coloured by using at most six colours.
		Define N to be the minimal number of colours required to properly colour any map from the class of all maps on the plane.
	student.adams.edu /~verderaimedj/finalEssay (1552 words)

	four colour theorem
		The famous Four Colour Theorem is concerned with mathematics as well as geography: it was first noted by August Ferdinand Möbius in 1840.
		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.
		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!
	www.mathsisgoodforyou.com /conjecturestheorems/fourcolour.htm (437 words)

	Amazon.ca: Four Colors Suffice: How the Map Problem Was Solved: Books: Robin Wilson (Site not responding. Last check: 2007-10-13)
		It is, simply stated: four colors are all that is needed to fill in any map so that neighboring countries are always colored differently.
		The history of the four color problem is one that illuminates much of what makes mathematics such a great topic to explore and was the first instance of a whole new movement in mathematics.
		To be sure, the theorem does have practical interest, if not to actual mapmakers, then to road, rail, and communications networks, but it has mainly inspired other aspects of pure mathematics like graph theory and algorithms.
	www.amazon.ca /Four-Colors-Suffice-Problem-Solved/dp/0691120234 (2266 words)

	The four colour theorem
		The Four Colour Conjecture first seems to have been made by Francis Guthrie.
		Heawood was to work throughout his life on map colouring, work which spanned nearly 60 years.
		However the final ideas necessary for the solution of the Four Colour Conjecture had been introduced before these last two results.
	www-history.mcs.st-andrews.ac.uk /history/HistTopics/The_four_colour_theorem.html (1703 words)

	4-Colour Theorem.
		Above: Map of the United States of America with the states coloured using just four colours, the minimum number that are necessary to colour all planar maps.
		The oldest surviving reference to the 4-colour Theorem is in a letter written on the 23rd October 1852 from Augustus De Morgan, Professor of Mathematics at University College London, to the famous Irish mathematician Sir William Rowan Hamilton.
		Nobody doubts that the computer demonstrated the truth of the four colour theorem, however, some mathematicians remain uncomfortable about a proof that is so cumbersome that it requires a computer to check numerous exceptional cases that a human mathematician could not reasonably be expected to check.
	www.btinternet.com /~connectionsinspace/Mapping/4-Colour_Theorem_/body_4-colour_theorem_.html (579 words)

	Four Colour Theorem - Main
		The famous four colour theorem seems to have been first proposed by Möbius in 1840, later by DeMorgan and the Guthrie brothers in 1852, and again by Cayley in 1878.
		In colouring a geographical map it is customary to give different colours to any two countries that have a segment of their boundaries in common.
		Recall the definition from section I that N is the minimal number of colours required to properly colour any map from the class of all maps on the sphere and m(N) is a specific map which requires all of the N colours to properly colour it.
	www.geocities.com /dharwadker/main.html (4317 words)

	sci.math FAQ: The Four Colour Theorem
		An equivalent combinatorial interpretation is Theorem 3 [Four Colour Theorem] Every loopless planar graph admits a vertex-colouring with at most four different colours.
		This theorem was proved with the aid of a computer in 1976.
		A recent simplification of the Four Colour Theorem proof, by Robertson, Sanders, Seymour and Thomas, has removed the cloud of doubt hanging over the complex original proof of Appel and Haken.
	www.faqs.org /faqs/sci-math-faq/fourcolour (312 words)

	The Cellular Automata pages
		One theorem is featured, the remainder can be accessed through a list on the main page.
		Bezout’s theorem is simple enough when you state it: Two polynomial curves of degree m and n respectively intersect in nm points.
		Of course, Godel’s Theorem is never proved but there’s a nice demonstration of a non-computable function (marred by typographical accidents) and an important link to a translation of Godel’s original paper.
	www.math.uwaterloo.ca /navigation/ideas/reviews/theoremmonth.shtml (359 words)

	The four colour theorem
		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.
		The Four Colour Theorem returned to being the Four Colour Conjecture in 1890.
		The year 1976 saw a complete solution to the Four Colour Conjecture when it was to become the Four Colour Theorem for the second, and last, time.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/The_four_colour_theorem.html (1703 words)

	The Four Color Theorem (Site not responding. Last check: 2007-10-13)
		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.
		This was confirmed by Appel and Haken in 1976, when they published their proof of the Four Color Theorem [1,2].
		It follows from our proof of Theorem 2 that either a good configuration appears in the second neighborhood of v (it which case it can be found in linear time), or a k-ring violating the definition of internal 6-connection can be found in linear time.
	www.math.gatech.edu /~thomas/FC/fourcolor.html (1878 words)

	[No title]
		The four-colour theorem (briefly, the 4CT) asserts that every loopless plane graph admits a 4-colouring, that is, a mapping $c:V(G)\to \{0,1,2,3\}$ such that $c(u)\not =c(v)$ for every edge of $G$ with ends $u$ and $v$.
		The way we handle this problem is that we only consider reducers that are obtained from $K$ by contracting at most four edges, for which the safety check is easy.
		Those two theorems are just stated in \cite{14} as having been proved by a computer.
	www.math.psu.edu /era-mirror/1996-01-003/1996-01-003.tex.html (2773 words)

	POTTERY MODELS AND COMPLETE...
		In addition, all these models are related to the theory of oriented matroids, the study of abstract mathematical objects, the paradigm of which are equivalence classes of matrices.
		For the former longstanding four colour problem: "are the countries of each given map on the sphere colourable with 4 colours?" we have an affirmative answer since 1977 due to Appel and Haaken.
		Again we can ask: "what is the minimal number n of colours for such a surface such that any given map on the surface can be coloured with n colours?" We find the classical contributions to this topic in Gerhard Ringel’s book about the map colour theorem from 1974.
	www.mi.sanu.ac.yu /vismath/visbook/sydbok (1109 words)

	Oxford University Press
		The four-colour theorem is one of the famous problems of mathematics, that frustrated generations of mathematicians from its birth in 1852 to its solution (using substantial assistance from electronic computers) in 1976.
		The theorem asks whether four colours are sufficient to colour all conceivable maps, in such a way that countries with a common border are coloured with different colours.
		In Part III we return to the four-colour theorem, and study in detail the methods which finally cracked the problem.
	www.oup.com /ca/isbn/0-19-851061-6 (327 words)

	Talk:Four color theorem - Wikipedia, the free encyclopedia
		Trouble is, each blue region touches all four other colours, the regions touching the inside blue one are disjoint from the regions touching the outside blue one.
		Thus in either case you have surrounded one of the four colours, freeing it up to be used again.
		Following the references for such things as the five colour theorem I notice there is a whole science of colouring edges and nodes of graphs.
	en.wikipedia.org /wiki/Talk:Four_color_theorem (4531 words)

	Four Colour Theorem
		The Four Colour Theorem is to mathematicians what the song of the Sirens was to the sailors of ancient times.
		The Four Colour Theorem is the hypothesis that the maximum number of colours needed to fill the map is four.
		And while computer-aided proofs have begun to gain acceptance, largely thanks to the Four Colour Theorem, there remains the feeling that beauty, elegance and insight should triumph over the horror of a computer-generated proof.
	www.justinmullins.com /four_colour_theorem.htm (376 words)

	Question Corner -- The Four Fours Problem
		I would like to know if you know the answer to the four number problem that mathematicians have done in previous years and all mathematicians know of.
		In its most basic form, the puzzle is to combine four copies of the number 4, through the basic operations of negation, addition, subtraction, multiplication, division, and exponentiation, to come up with different integers.
		One way is to allow the square root symbol, so that you can take square roots without using up any additional fours (instead of having to raise something to the power of 4/(4+4) which is how you'd have to do it under the original rules).
	www.math.toronto.edu /mathnet/questionCorner/fourfours.html (522 words)

	University of Wuppertal - Dep. of Mathematics - Poster
		In 1852, while colouring a map representing the english counties, the british mathematician Francis Guthrie realized that only four colours where necessary to satisfy the criterion that neighbouring counties should have different colours.
		Consequently, proving the four colour theorem correct also means proving that the program is implemented correctly and that the computer works correctly.
		The two other posters are: "the four colour problem" and "the bridges of Königsberg".
	www.math.uni-wuppertal.de /org/Poster/TextE.htm (1032 words)

	Ideas, Concepts, and Definitions (Site not responding. Last check: 2007-10-13)
		The Four Color Problem was famous and unsolved for many years.
		The basic rule for coloring a map is that no two regions that share a boundary can be the same color.
		Mapmakers are not mathematicians, so the assertion that only four colors would be necessary for all maps gained acceptance in the map-making community over the years because no one ever stumbled upon a map that required the use of five colors.
	www.c3.lanl.gov /mega-math/gloss/math/4ct.html (541 words)

	The Four Colour Theorem as a possible corollary of binomial summation - Matiyasevich (ResearchIndex) (Site not responding. Last check: 2007-10-13)
		Abstract: The Four Colour Conjecture is reformulated as a statement about non-divisibility of certain binomial coefficients.
		This reformulation opens a (hypothetical) way of proving the Four Colour Theorem by taking advantage of recent progress in finding closed forms for binomial summations.
		Matiyasevich, The Four Colour Theorem as a possible corollary of binomial summation, manuscript.
	citeseer.ist.psu.edu /462498.html (603 words)

	four colour theorem worksheet
		Try to colour them by four colours so that there are not two adjacent areas that are the same colour.
		The Four Colour Theorem states that any number of points and lines reduces itself to a map which only needs four colours.
		The Four Colour Theorem states that there is no graph which contains any set of five mutually connected vertices.
	www.mathsisgoodforyou.com /worksheets/fourcolourtheorem.htm (193 words)

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