## Mathematical Vistas: From a Room with Many WindowsFocusing YourAttention We have called this book Mathematical Vistas because we have already published a companion book MathematicalRefiections in the same series;1 indeed, the two books are dedicated to the same principal purpose - to stimulate the interest ofbrightpeople in mathematics.Itis not our intention in writing this book to make the earlier book aprerequisite, but it is, of course, natural that this book should contain several references to its predecessor. This is especially - but not uniquely- true of Chapters 3, 4, and 6, which may be regarded as advanced versions of the corresponding chapters in Mathematical Reflections. Like its predecessor, the present work consists of nine chapters, each devoted to a lively mathematical topic, and each capable, in principle, of being read independently of the other chapters.' Thus this is not a text which- as is the intention of most standard treatments of mathematical topics - builds systematically on certain common themes as one proceeds 1Mathematical Reflections - In a Room with Many Mirrors, Springer Undergraduate Texts in Math ematics, 1996; Second Printing 1998. We will refer to this simply as MR. 2There was an exception in MR; Chapter 9 was concerned with our thoughts on the doing and teaching of mathematics at the undergraduate level. |

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### Contents

Paradoxes in Mathematics | 1 |

THING EQUAL TO ONE ANOTHER? PARADOX 1 | 4 |

13 IS ONE STUDENT BETTER THAN ANOTHER? PARADOX 2 | 6 |

14 DO AVERAGES MEASURE PROWESS? PARADOX 3 | 8 |

MAY PROCEDURES BE JUSTIFIED EXCLUSIVELY BY STATISTICAL TESTS? PARADOX 4 | 11 |

PARADOX ABOUT SAILORS AND MONKEYS PARADOX 5 | 14 |

REFERENCES | 20 |

Not the Last of Fermat | 23 |

Are Four Colors Really Enough? | 127 |

53 GRAPHS | 130 |

54 TOURING WITH EULER | 136 |

55 WHY GRAPHS? | 138 |

56 ANOTHER CONCEPT | 142 |

57 PLANARITY | 144 |

58 THE END | 148 |

59 COLORING EDGES | 149 |

22 SOMETHING COMPLETELY DIFFERENT | 24 |

23 DIOPHANTUS | 26 |

24 ENTER PIERRE DE FERMAT | 27 |

25 FLASHBACK TO PYTHAGORAS | 28 |

26 SCRIBBLES IN MARGINS | 32 |

27 n 4 | 33 |

28 EULER ENTERS THE FRAY | 36 |

29 I HAD TO SOLVE IT | 40 |

REFERENCES | 46 |

Fibonacci and Lucas Numbers Their Connections and Divisibility Properties | 49 |

THE FIBONACCI AND LUCAS INDICES | 54 |

33 ON ODD LUCASIAN NUMBERS | 56 |

34 A THEOREM ON LEAST COMMON MULTIPLES | 62 |

35 THE RELATION BETWEEN THE FIBONACCI AND LUCAS INDICES | 63 |

36 ON POLYNOMIAL IDENTITIES RELATING FIBONACCI AND LUCAS NUMBERS | 64 |

REFERENCES | 69 |

PaperFolding Polyhedra Building and Number Theory | 71 |

42 WHAT CAN BE DONE WITHOUT EUCLIDEAN TOOLS | 73 |

43 CONSTRUCTING ALL QUASIREGULAR POLYGONS | 93 |

44 HOW TO BUILD SOME POLYHEDRA HANDSON ACTIVITIES | 95 |

45 THE GENERAL QUASIORDER THEOREM | 114 |

REFERENCES | 124 |

510 A BEGINNING? | 153 |

REFERENCES | 157 |

From Binomial to Trinomial Coefficients and Beyond | 159 |

62 ANALOGUES OF THE GENERALIZED STAR OF DAVID THEOREMS | 177 |

REFERENCES | 184 |

63 EXTENDING THE PASCAL TETRAHEDRON AND THE PASCAL mSIMPLEX | 188 |

64 SOME VARIANTS AND GENERALIZATIONS | 190 |

65 THE GEOMETRY OF THE 3DIMENSIONAL ANALOGUE OF THE PASCAL HEXAGON | 193 |

REFERENCES | 198 |

Catalan Numbers | 199 |

72 A FOURTH INTERPRETATION | 208 |

73 CATALAN NUMBERS | 215 |

74 EXTENDING THE BINOMIAL COEFFICIENTS | 218 |

75 CALCULATING GENERALIZED CATALAN NUMBERS | 220 |

76 COUNTING pGOOD PATHS | 223 |

76 Counting pGood Paths 225 | 225 |

233 | |

Symmetry | 235 |

96 Birthdays and Coincidences | 285 |

Selected Answers to Breaks | 299 |

325 | |

### Other editions - View all

Mathematical Vistas: From a Room with Many Windows Peter Hilton,Derek Holton,Jean Pedersen Limited preview - 2013 |

Mathematical Vistas: From a Room with Many Windows Peter Hilton,Derek Holton,Jean Pedersen No preview available - 2010 |

### Common terms and phrases

4-colorable Algebra algorithm angle argument bad path binomial coefficients bipartite graph BREAK calculate Catalan numbers Chapter conjecture construct convex coprime cosets course crease line cube cycle index diagonals dodecahedron dual graph equation Erdos Euler example faces fact FAT-algorithm Fermat Fermat's Last Theorem Fibonacci and Lucas folding procedure formula Four Color Four-Color Theorem gcd(a geometry girls given graph of Figure Hence Hilton homologues integer isomorphic Jean Pedersen linear look Lucas numbers Lucasian mathematician mathematics multinomial coefficients notation Notice odd number odd permutations pair paradox Pascal Identity permutation Petersen graph planar planar graphs player Polya polygon polynomial positive integer Pr(A prime problem proof prove Pythagorean triples quasi-order quasi-regular Ramsey numbers red edges regular of degree Section sequence Shimura-Taniyama shown in Figure snarks solution strips subgraph subset Suppose symbol symmetry group tape tetrahedron triangle trinomial vertex vertices Wiles