Mathemagical Buffet

　　Mathemagical Buffet offers a delectable feast to everyone with a basic facility in secondary-school mathematics. Every topic reflects the incomparable excitement, beauty, and joy of mathematics; they present a wealth of ingenious insights and marvelous ideas at the fundamental level.

　　The chapters are independent and can be read in any order. Everyone who enjoys elementary mathematics will truly delight in the following gems:

　　l． Pythagorean Triples via Geometry
　　l．New proofs of Generalizations of the Theorems of Ptolemy and Simson
　　l． Mind Reading Tricks, Ladder Lotteries, Mazes, Lattice Points, Round Robin Competitions, An Elementary Fixed Point Theorems and More
　　l．Simple proofs of the lovely Theorems of Pick and of Jung
　　l．The Constructibility of a Regular 17-gon
　　l．Open Problems on Egyptian Fractions and on Primes

　　Moreover, the reader is gently encouraged to participate actively by responding to a line of questions that are thoughtfully sprinkled throughout the developments of the expositions.

作者簡介

　　Liong-shin Hahn was born in Tainan, Taiwan. He obtained his B.S. from the National Taiwan University, and his Ph.D. from Stanford University. He authored Complex Numbers and Geometry (Mathematical Association of America, 1994), New Mexico Mathematics Contest Problem Book (University of New Mexico Press, 2005), Honsberger Revisited (National Taiwan University Press, 2012), and co-authored with Bernard Epstein Classical Complex Analysis (Jones and Bartlett, 1996). He was awarded the Citation for Public Service from the American Mathematical Society in 1998.

Preface　ix

1 Sums of Consecutive Integers　1

2 Galilean Ratios　7

3 The Pythagorean Theorem　11
　3.1 Proofs　11
　3.2 A Puzzle　17
　3.3 Pythagorean Triples　18
　3.4 Generalizations of the Pythagorean Theorem　20

4 Japanese Temple Mathematics　25
　4.1 Problems　25
　4.2 Solutions　27

5 Mind Reading Tricks　41
　5.1 Trick 1　41
　5.2 Trick 2　43

6 Magic Squares　47
　6.1 New Year Puzzle 2010　47
　6.2 Magic Squares　49

7 Fun with Areas　53
　7.1 Two Theorems of Newton　53
　7.2 A Charming Construction Problem　59
　7.3 A Generalization of the Simson Theorem　62

8 The Tower of Hanoi　67

9 Ladder Lotteries　71

10 Round Robin Competitions　79

11 Egyptian Fractions　83

12 The Ptolemy Theorem　89
　12.1 The Ptolemy Theorem　89
　12.2 Applications　90
　12.3 A Generalization of the Ptolemy Theorem　92

13 Convexity 95
　13.1 Introduction　95
　13.2 The Theorems of Jung and Helly　96

14 The Seven Bridges of Konigberg　99
　14.1 Unicursal Figures　99
　14.2 Mazes　101

15 The Euler Formula　105
　15.1 The Euler Formula　105
　15.2 Regular Polyhedra108

16 The Sperner Lemma　111
　16.1 The Sperner Lemma　111
　16.2 The Brouwer Fixed Point Theorem　117
　16.3 An Elementary Fixed Point Theorem　119

17 Lattice Points　125
　17.1 The Pick Theorem　125
　17.2 Lattice Equilateral Triangle　133
　17.3 Lattice Equiangular Polygons　135
　17.4 Lattice Regular Polygons　138

18 The Sums of Special Series　139
　18.1 The Sum of the Powers　139
　18.2 The Binomial Coe cients　141
　18.3 Faulhaber Polynomials　143
　18.4 The Sums of the Reciprocals of Sp(n)　150
　18.5 The Sums of Trigonometric Functions　153

19 The Morley Theorem　157

20 Angle Trisection　163
　20.1 Rules of Engagement　163
　20.2 The Trisection Equation　167
　20.3 Computations by Straightedge and Compass　169
　20.4 Fields and their Extensions　172
　20.5 Impossibility Proofs　176
　20.6 Bending of the Rules　180
　20.7 Regular Polygons　181
　20.8 Regular 17-gon　185

21 Conics　195

22 Primes　203
　22.1 Number of Primes　203
　22.2 An Open Problem　205

23 Gaussian Integers　209
　23.1 Gaussian Primes　209
　23.2 An Application to Real Primes　216

24 Calculus with Complex Numbers　219

Appendix Determinants　223
　A.1 Genesis　223
　A.2 Properties　232
　A.3 The Laplace Expansion Theorem　235