Mathematical vistas : from a room with many windows /

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Bibliographic Details
Main Author: Hilton, Peter
Other Authors: Holton, Derek Allan, Pedersen, Jean
Format: Printed Book
Published: New York : Springer, c2002.
Series:Undergraduate texts in mathematics
Subjects:
Mathematics
Table of Contents:
  • Machine generated contents note: 1 Paradoxes in Mathematics 1
  • 1.1 Introduction: Don't Believe Everything You See and Hear 1
  • 1.2 Are Things Equal to the Same Thing Equal to One
  • Another? (Paradox 1) 4
  • 1.3 Is One Student Better Than Another? (Paradox 2)6
  • 1.4 Do Averages Measure Prowess? (Paradox 3)8
  • 1.5 May Procedures Be Justified Exclusively by Statistical
  • Tests? (Paradox 4)11
  • 1.6 A Basic Misunderstanding -and a Salutary Paradox
  • About Sailors and Monkeys (Paradox 5)14
  • References20
  • 2 Not the Last of Fermat 23
  • 2.1 Introduction: Fermat's Last Theorem (FLT)23
  • 2.2 Something Completely Different24
  • 2.3 Diophantus26
  • 2.4 Enter Pierre de Fermat-27
  • 2.5 Flashback to Pythagoras28
  • 2.6 Scribbles in Margins32
  • 2.7 n = 433
  • 2.8 Euler Enters the Fray36
  • 2.9 I Had to Solve It40
  • References46
  • 3 Fibonacci and Lucas Numbers: Their Connections and
  • Divisibility Properties 49
  • 3.1 Introduction: A Number Trick and Its Explanation 49
  • 3.2 A First Set of Results on the Fibonacci and Lucas Indices 54
  • 3.3 On Odd Lucasian Numbers56
  • 3.4 A Theorem on Least Common Multiples62
  • 3.5 The Relation Between the Fibonacci and Lucas Indices .63
  • 3.6 On Polynomial Identities Relating Fibonacci and
  • Lucas Numbers64
  • References69
  • 4 Paper-Folding, Polyhedra-Building, and Number Theory 71
  • 4.1 Introduction: Forging the Link Between Geometric
  • Practice and Mathematical Theory71
  • 4.2 What Can Be Done Without Euclidean Tools73
  • 4.3 Constructing All Quasi-Regular Polygons93
  • 4.4 How to Build Some Polyhedra (Hands-On Activities)95
  • 4.5 The General Quasi-Order Theorem114
  • References124
  • 5 Are Four Colors Really Enough? 127
  • 5.1 Introduction: A Schoolboy Invention127
  • 5.2 The Four-Color Problem127
  • 5.3 Graphs130
  • 5.4 Touring with Euler136
  • 5.5 Why Graphs?138
  • 5.6 Another Concept142
  • 5.7 Planarity144
  • 5.8 The End148
  • 5.9 Coloring Edges149
  • 5.10 A Beginning?153
  • References157
  • 6 From Binomial to Trinomial Coefficients and Beyond 159
  • 6.1 Introduction and Warm-Up159
  • 6.2 Analogues of the Generalized Star of Da,id Theorems .177
  • 6.3 Extending the Pascal Tetrahedron and the
  • Pascal m-simplex188
  • 6.4 Some Variants and Generalizations190
  • 6.5 The Geometry of the 3-Dimensional Analogue of the
  • Pascal Hexagon193
  • References 198
  • 7 Catalan Numbers 199
  • 7.1 Introduction: Three Ideas About the Same Mathematics199
  • 7.2 A Fourth Interpretation208
  • 7.3 Catalan Numbers215
  • 7.4 Extending the Binomial Coefficients218
  • 7.5 Calculating Generalized Catalan Numbers220
  • 7.6 Counting p-Good Paths223
  • 7.7 A Fantasy- and the Awakening227
  • References 233
  • 8 Symmetry 235
  • 8.1 Introduction: A Really Big Idea235
  • 8.2 Symmetry in Geometry239
  • 8.3 Homologues 254
  • 8.4 The P61ya Enumeration Theorem257
  • 8.5 Even and Odd Permutations263
  • References269
  • 9 Parties 271
  • 9.1 Introduction: Cliques andAnticliques 271
  • 9.2 Ramsey and Erd6s 275
  • 9.3 Further Progress277
  • 9.4 N (r, r) 281
  • 9.5 Even More Ramsey283
  • 9.6 Birthdays and Coincidences285
  • 9.7 Come to the Dance287
  • 9.8 Philip Hall290
  • 9.9 Back to Graphs292
  • 9.10 Epilogue295
  • References297.

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