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.

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　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

From 17th century to early 20th century, people of various walks in Japan offered their mathematical discoveries on wooden tablets to shrines and temples. Most of the time, it was done for pure enjoyment or to challenge others (but certainly not for college entrance or job promotion). Majority of them were in geometry. To me that is a strong indication that geometry had (and still has) popular appeal to the general public. Thus I find it very tragic that geometry is being de-emphasized in school mathematics nowadays.
--p. 25

Suppose we have 12 table tennis players. Instead of a tournament, during which many pairs of players do not have a chance for a match, the organizer decides on a round robin competition so that every pair of players has a chance to meet. He wants every player to have one match every day so that six matches are played every day. Naturally, it takes eleven days to play all the matches, because each one has 11 opponents. How should he arrange the matches? He is concerned that if not well-planned, some players might find the opponents they needed unavailable in the final days, forcing them to wait and requiring extra days for the competition.
--p.79

Because of the fact that there exist angles (such as 60° angle) that cannot be trisected (using compass and straightedge finitely many times), mathematicians tended to shy away from problems involving trisections of angles. Perhaps that is why the following dandy theorem was discovered only in 1904 by the British geometer Frank Morley (1860 - 1937). The idea in our proof was essentially the one shown to me by Professor John H. Conway of Princeton University, when I (as the director of the New Mexico
Mathematics Contest) invited him to give a talk to the Contest finalists in February 1999.
--p.157