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.
Throughout my many years of teaching at the university level, I frequently enjoyed giving stimulating little talks to secondary-school students (in the U.S. and Taiwan). This book contains detailed accounts of these talks. Because each topic is aimed at a particular grade level, chapters are independent and may be read in any order. It is unlikely that more than a small fraction of our students will actually use mathematics in their careers. Therefore, we would do well to foster healthy attitudes and worthwhile habits of mind while they are in our care. In particular, I stress(a) an appreciation of the beauty of mathematics,(b) acquiring the habit of deep thinking,(c) the ability to reason logically.For this purpose, I can't think of a better choice than the revival of Euclidean geometry in our secondary school curriculum. Euclidean geometry contains a wealth of not only wonderful theorems for students to enjoy the beauty of mathematics, but also intriguing and challenging problems to lure students into deep thinking and explorations; not to mention that it provides a fertile ground for training in logical reasoning. In fact, I have heard many people in my generation and earlier who claimed that they enjoyed Euclidean geometry immensely, even though they were not good at algebra. As Albert Einstein said, "If Euclid failed to kindle your youthful enthusiasm, then you were not born to be a scientific thinker." Mathematics books are not to be "read". They should be worked through with pencil and paper. Looking back on my own experience, I learned mathematics not by reading books, nor by attending lectures. I learned mathematics mainly by solving (or trying to solve) challenging problems; by trying to explore what happens when part of the assumption of the theorem is altered or deleted; by trying to find what it means in simple particular cases; by investigating the converse. In short, by playing around with problems. Furthermore, my experience convinces me that the crux of a great theorem lies often in a simple concrete special case. Consequently, in teaching, I try to emphasize the important particular cases rather than the most general case. And I try my best to expose the motivation behind each move; at the minimum, I try to avoid presenting solutions and proofs as beautiful but "static" finished artifacts. I believe that mathematics textbooks should emphasize ideas; they should not be mere collections of the facts. My teaching motto is: "Don't try to teach everything. Always leave something for students to explore." Hence I tell my students, "I do the easy part and you do the hard part." Consequently, I am allergic to overweight textbooks trying to include everything. Does anyone really believe students are interested in reading 5-pound, 800-page textbooks with most of the pages filled with repetitions of simple routine drills, ad nauseam? I get the impression that authors of overweight textbooks are more concerned with encyclopedic coverage of topics at the expense of discussing how topics relate to each other. Consequently, students think that mathematics is just a collection of facts, facts to be remembered in order to pass multiple-choice tests. How tragic it is to starve millions of eager young minds by depriving them of being exposed to the beauty and excitement of mathematics! This book goes against the current trend. I try to present something intriguing that does not yield to well-worn standard approaches, something that involves a spark of ingenuity. The book is now presented to be judged by readers. I cherish the hope that you will enjoy the feast in my Mathemagical Buffet.L.-s. Hahn
Preface ix1 Sums of Consecutive Integers 12 Galilean Ratios 73 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 204 Japanese Temple Mathematics 25 4.1 Problems 25 4.2 Solutions 275 Mind Reading Tricks 41 5.1 Trick 1 41 5.2 Trick 2 436 Magic Squares 47 6.1 New Year Puzzle 2010 47 6.2 Magic Squares 497 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 628 The Tower of Hanoi 679 Ladder Lotteries 7110 Round Robin Competitions 7911 Egyptian Fractions 8312 The Ptolemy Theorem 89 12.1 The Ptolemy Theorem 89 12.2 Applications 90 12.3 A Generalization of the Ptolemy Theorem 9213 Convexity 95 13.1 Introduction 95 13.2 The Theorems of Jung and Helly 9614 The Seven Bridges of Konigberg 99 14.1 Unicursal Figures 99 14.2 Mazes 10115 The Euler Formula 105 15.1 The Euler Formula 105 15.2 Regular Polyhedra10816 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 11917 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 13818 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 15319 The Morley Theorem 15720 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 17620.6 Bending of the Rules 18020.7 Regular Polygons 18120.8 Regular 17-gon 18521 Conics 19522 Primes 203 22.1 Number of Primes 203 22.2 An Open Problem 20523 Gaussian Integers 209 23.1 Gaussian Primes 209 23.2 An Application to Real Primes 21624 Calculus with Complex Numbers 219Appendix Determinants 223 A.1 Genesis 223 A.2 Properties 232 A.3 The Laplace Expansion Theorem 235