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100 Statements to Prove for CS 2233 The idea of this
100 Statements to Prove for CS 2233 The idea of this

Foundations of Geometry
Foundations of Geometry

... standard results regarding transformations of the plane are explored. A complete proof of the classification of the rigid motions of both the Euclidean and hyperbolic planes is included. There is a discussion of how the foundations of geometry can be reorganized to reflect the transformational point o ...
The arithmetic mean of the divisors of an integer
The arithmetic mean of the divisors of an integer



Chapter 4
Chapter 4

http://waikato.researchgateway.ac.nz/ Research Commons at the
http://waikato.researchgateway.ac.nz/ Research Commons at the

Automatic construction of quality nonobtuse boundary and/or
Automatic construction of quality nonobtuse boundary and/or

Congruent Figures
Congruent Figures

A New 3n − 1 Conjecture Akin to Collatz Conjecture
A New 3n − 1 Conjecture Akin to Collatz Conjecture

Reference
Reference

Chapter 2: Reasoning and Proof
Chapter 2: Reasoning and Proof

Lesson 3.5 The Polygon Angle
Lesson 3.5 The Polygon Angle

13(4)
13(4)

A rhombus is a parallelogram
A rhombus is a parallelogram

L. ALAOGLU AND P. ERDŐS Reprinted from the Vol. 56, No. 3, pp
L. ALAOGLU AND P. ERDŐS Reprinted from the Vol. 56, No. 3, pp

Math 780: Elementary Number Theory
Math 780: Elementary Number Theory

Elementary Number Theory
Elementary Number Theory

... PMI or, simply, induction. A sample proof is given below. The rest will be given in class hopefully by students. A sample proof using induction: I will give two versions of this proof. In the first proof I explain in detail how one uses the PMI. The second proof is less pedagogical and is the type of ...
Elementary Number Theory
Elementary Number Theory

Triangles
Triangles

... We have medians and altitudes intersecting in a common point and it seems that the angle bisectors also have a common point of intersection. Use paper folding with patty paper to investigate this idea. a) Begin by drawing a large triangle on a sheet of patty paper. b) Use scissors to cut out the tri ...
Polygons and Quadrilaterals
Polygons and Quadrilaterals

Explicit Estimates in the Theory of Prime Numbers
Explicit Estimates in the Theory of Prime Numbers

Information for Students in MATH 348 2003 09
Information for Students in MATH 348 2003 09

GETE0301
GETE0301

Triangles
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Constructive Geometry and the Parallel Postulate
Constructive Geometry and the Parallel Postulate

< 1 2 3 4 5 6 7 ... 153 >

Four color theorem



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 if they share a common boundary that is not a corner, where corners are the points shared by three or more regions. For example, in the map of the United States of America, Utah and Arizona are adjacent, but Utah and New Mexico, which only share a point that also belongs to Arizona and Colorado, 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 2014, 2), “Maps utilizing only four colors 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 five color theorem, which has a short elementary proof, states that five colors suffice to color a map and was proven in the late 19th century (Heawood 1890); however, proving that four colors suffice turned out to be significantly harder. A number of false proofs and false counterexamples have appeared since the first statement of the four color theorem in 1852.The four color theorem was proven in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major theorem to be proved using a computer. Appel and Haken's approach started by showing that there is a particular set of 1,936 maps, each of which cannot be part of a smallest-sized counterexample to the four color theorem. (If they did appear, you could make a smaller counter-example.) Appel and Haken used a special-purpose computer program to confirm that each of these maps had this property. Additionally, any map that could potentially be a counterexample must have a portion that looks like one of these 1,936 maps. Showing this required hundreds of pages of hand analysis. Appel and Haken concluded that no smallest counterexamples exist because any must contain, yet do not contain, one of these 1,936 maps. This contradiction means there are no counterexamples at all and that the theorem is therefore true. Initially, their proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand (Swart 1980). Since then the proof has gained wider acceptance, although doubts remain (Wilson 2014, 216–222).To dispel remaining doubt about the Appel–Haken proof, a simpler proof using the same ideas and still relying on computers was published in 1997 by Robertson, Sanders, Seymour, and Thomas. Additionally in 2005, the theorem was proven by Georges Gonthier with general purpose theorem proving software.
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