# A Decade of the Berkeley Math Circle: The American Experience, Volume I

Share this page *Edited by *
*Zvezdelina Stankova; Tom Rike*

A co-publication of the AMS and the Mathematical Sciences Research Institute

Many mathematicians have been drawn to mathematics through their
experience with math circles: extracurricular programs
exposing teenage students to advanced mathematical topics and a myriad
of problem solving techniques and inspiring in them a lifelong love
for mathematics. Founded in 1998, the Berkeley Math Circle
(BMC) is a pioneering model of a U.S. math circle, aspiring
to prepare our best young minds for their future roles as mathematics
leaders. Over the last decade, 50 instructors—from university
professors to high school teachers to business tycoons—have
shared their passion for mathematics by delivering more than 320 BMC
sessions full of mathematical challenges and wonders.

Based on a dozen of these sessions, this book encompasses a wide
variety of enticing mathematical topics: from inversion in the plane
to circle geometry; from combinatorics to Rubik's cube and abstract
algebra; from number theory to mass point theory; from complex numbers
to game theory via invariants and monovariants. The treatments of
these subjects encompass every significant method of proof and
emphasize ways of thinking and reasoning via 100 problem solving
techniques. Also featured are 300 problems, ranging from beginner to
intermediate level, with occasional peaks of advanced problems and
even some open questions.

The book presents possible paths to
studying mathematics and inevitably falling in love with it, via
teaching two important skills: thinking creatively while still
“obeying the rules,” and making connections between
problems, ideas, and theories. The book encourages you to apply the
newly acquired knowledge to problems and guides you along the way, but
rarely gives you ready answers. “Learning from our own
mistakes” often occurs through discussions of non-proofs and
common problem solving pitfalls. The reader has to commit to mastering
the new theories and techniques by “getting your hands
dirty” with the problems, going back and reviewing necessary
problem solving techniques and theory, and persistently moving forward
in the book. The mathematical world is huge: you'll never know
everything, but you'll learn where to find things, how to
connect and use them. The rewards will be substantial.

In the interest of fostering a greater awareness and appreciation of
mathematics and its connections to other disciplines and everyday life, MSRI
and the AMS are publishing books in the Mathematical Circles Library series as
a service to young people, their parents and teachers, and the mathematics
profession.

Titles in this series are co-published with the Mathematical Sciences Research Institute (MSRI).

#### Readership

High school teachers and students interested in mathematical problem solving; college professors and research mathematicians interested in the mathematical education of talented middle and high school students.

#### Reviews & Endorsements

. . . [F]rom the
authors' Introduction and the Foreword by Robert L. Bryant, Director of the
Mathematical Science Research Institute (MSRI), 'Math circles . . . all have
one thing in common: *to inspire in students an understanding of and
lifelong love for mathematics*.' The book, based on the notes from several
sessions of the Berkeley Math Circle (BMC), reaches that goal beautifully.
It is written with enthusiasm and flair; the book breathes love for mathematics
and the desire to convey and share it with the reader. . . . The editors
deserve praise for producing a coherent style that is followed throughout. . . . On the whole,
the book is thoughtfully organized and well written. . . . [A] welcome
beginning for the emerging tradition in the US math education.

-- Alex Bogomolny, MAA Reviews

#### Table of Contents

# Table of Contents

## A Decade of the Berkeley Math Circle: The American Experience, Volume I

- Cover Cover11
- Title page i2
- Contents iii4
- Foreword vii8
- Introduction ix10
- Inversion in the plane. Part I 120
- Combinatorics. Part I 2544
- Rubik’s cube. Part I 4766
- Number theory. Part I: Remainders, divisibility, congruences and more 6382
- A few words about proofs. Part I 87106
- Mathematical induction 103122
- Mass point geometry 127146
- More on proofs. Part II 155174
- Complex numbers. Part I 179198
- Stomp. Games with invariants 203222
- Favorite problems at BMC. Part I: Circle geometry 225244
- Monovariants. Part I: Mansion walks and frog migrations 249268
- Epilogue 285304
- Symbols and notation 299318
- Abbreviations 301320
- Biographical data 303322
- Bibliography 309328
- Credits 313332
- Index 315334
- Back Cover Back Cover1346