Description
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
书名简译 : Mathemagical Buffet
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