Table of contents for Mathematical vistas : from a room with many windows / Peter Hilton, Derek Holton, Jean Pedersen.


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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
References ..............           ...........     20
2  Not the Last of Fermat                                   23
2.1 Introduction: Fermat's Last Theorem (FLT) ........ 23
2.2  Something Completely Different .............. 24
2.3  Diophantus ........... 26
2.4  Enter Pierre de Fermat- ...................        27
2.5  Flashback to Pythagoras ..................         28



2.6  Scribbles in Margins ................         . 32
2.7 n = 4 ...       ............... 33
2.8 Euler Enters the Fray ................          . 36
2.9 I Had to Solve It ...........  .......            40
References ..............                      . 46
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 Numbers .    .   .......       .  56
3.4  A Theorem on Least Common Multiples .......... 62
3.5 The Relation Between the Fibonacci and Lucas Indices . .63
3.6  On Polynomial Identities Relating Fibonacci and
Lucas Numbers ..            ........       ....  64
References .............          ..............  69
4  Paper-Folding, Polyhedra-Building, and Number Theory  71
4.1 Introduction: Forging the Link Between Geometric
Practice and Mathematical Theory ..........      71
4.2 What Can Be Done Without Euclidean Tools ....... 73
4.3 Constructing All Quasi-Regular Polygons ....... . 93
4.4 How to Build Some Polyhedra (Hands-On Activities) . .. 95
4.5 The General Quasi-Order Theorem ............. 114
References .......             .........         124
5  Are Four Colors Really Enough?                       127
5.1 Introduction: A Schoolboy Invention ........... 127
5.2 The Four-Color Problem  .................. 127
5.3  Graphs ...........................              130
5.4 Touring with Euler .................... 136
5.5 Why Graphs? .......................            . 138
5.6  Another Concept .  ..................         . 142
5.7  Planarity ......................                144
5.8 The End .........       ......   ........      . 148



5.9  Coloring Edges .....................    ..     . 149
5.10 A Beginning? ..................    ............. 153
References .......................... 157
6  From Binomial to Trinomial Coefficients and Beyond     159
6.1 Introduction and Warm-Up ................. 159
6.2  Analogues of the Generalized Star of Da,id Theorems  . .177
6.3  Extending the Pascal Tetrahedron and the
Pascal m-simplex .....................          . 188
6.4  Some Variants and Generalizations ............. 190
6.5  The Geometry of the 3-Dimensional Analogue of the
Pascal Hexagon .......................          . 193
References... .   ...............         ..... 198
7  Catalan Numbers                                        199
7.1 Introduction: Three Ideas About the Same Mathematics .. 199
7.2  A Fourth Interpretation .................      . 208
7.3  Catalan Numbers ................... ... 215
7.4  Extending the Binomial Coefficients ............... 218
7.5  Calculating Generalized Catalan Numbers ......... 220
7.6  Counting p-Good Paths ..........       .  ..... . 223
7.7  A Fantasy- and the Awakening .............. 227
References .........................233
8  Symmetry                                              235
8.1 Introduction: A Really Big Idea .................. 235
8.2  Symmetry in Geometry ................... 239
8.3  Homologues .........................254
8.4  The P61ya Enumeration Theorem .............. 257
8.5  Even and Odd Permutations ................ 263
References .......................... 269
9  Parties                                               271
9.1 Introduction: Cliques andAnticliques  ...........271
9.2  Ramsey and Erd6s ......................275
9.3 Further Progress ................... ... 277



9.4  N (r, r)  .. . .. . . . .. .. . . . . . .. .. . . . . ..  281
9.5  Even More Ramsey  ..................... 283
9.6  Birthdays and Coincidences  .. .... ...... .  285
9.7  Come to the Dance ............... .  287
9.8  Philip Hall ... ..... ..... ....... . 290
9.9  Back to Graphs  ................... 292
9.10  Epilogue ................ .   ....... .  295
References  ...............   297








Library of Congress subject headings for this publication: Mathematics Popular works