Mathemagical Buffet

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.

Contents

Preface　

1. Sums of Consecutive Integers　
2. Galilean Ratios　
3. The Pythagorean Theorem　
　3.1 Proofs　
　3.2 A Puzzle　
　3.3 Pythagorean Triples　
　3.4 Generalizations of the Pythagorean Theorem　
4. Japanese Temple Mathematics　
　4.1 Problems　
　4.2 Solutions　
5. Mind Reading Tricks　
　5.1 Trick 1
　5.2 Trick 2　
6. Magic Squares　
　6.1 New Year Puzzle 2010　
　6.2 Magic Squares　
7. Fun with Areas　
　7.1 Two Theorems of Newton　
　7.2 A Charming Construction Problem　59
　7.3 A Generalization of the Simson Theorem　
8. The Tower of Hanoi　
9. Ladder Lotteries　
10. Round Robin Competitions　
11. Egyptian Fractions
12. The Ptolemy Theorem　
　12.1 The Ptolemy Theorem　
　12.2 Applications
　12.3 A Generalization of the Ptolemy Theorem　
13. Convexity
　13.1 Introduction　
　13.2 The Theorems of Jung and Helly　96
14. The Seven Bridges of Konigberg　
　14.1 Unicursal Figures
　14.2 Mazes　
15. The Euler Formula　
　15.1 The Euler Formula　
　15.2 Regular Polyhedra
16. The Sperner Lemma　
　16.1 The Sperner Lemma　
　16.2 The Brouwer Fixed Point Theorem　
　16.3 An Elementary Fixed Point Theorem　
17. Lattice Points　
　17.1 The Pick Theorem　
　17.2 Lattice Equilateral Triangle　
　17.3 Lattice Equiangular Polygons　
　17.4 Lattice Regular Polygons　
18. The Sums of Special Series　
　18.1 The Sum of the Powers　
　18.2 The Binomial Coe cients　
　18.3 Faulhaber Polynomials　
　18.4 The Sums of the Reciprocals of Sp(n)　
　18.5 The Sums of Trigonometric Functions　
19. The Morley Theorem　
20. Angle Trisection　
　20.1 Rules of Engagement　
　20.2 The Trisection Equation　
　20.3 Computations by Straightedge and Compass　
　20.4 Fields and their Extensions　
　20.5 Impossibility Proofs　
    20.6 Bending of the Rules　
    20.7 Regular Polygons　
    20.8 Regular 17-gon　
21. Conics　
22. Primes　
　22.1 Number of Primes　
　22.2 An Open Problem　
23. Gaussian Integers　
　23.1 Gaussian Primes　
　23.2 An Application to Real Primes
24. Calculus with Complex Numbers　

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