Mathematical vistas : from a room with many windows

Includes bibliographical references and index.

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.