**MSRI Mathematical Circles Library**

Volume: 14;
2014;
376 pp;
Softcover

MSC: Primary 00;

**Print ISBN: 978-0-8218-4912-5
Product Code: MCL/14**

List Price: $25.00

Individual Price: $18.75

**Electronic ISBN: 978-1-4704-1959-2
Product Code: MCL/14.E**

List Price: $25.00

Individual Price: $18.75

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#### Supplemental Materials

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

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. The Berkeley
Math Circle (BMC) started in 1998 as one of the very first math
circles in the U.S. Over the last decade and a half, 100
instructors—university professors, business tycoons, high school
teachers, and more—have shared their passion for mathematics by
delivering over 800 BMC sessions on the UC Berkeley campus every week
during the school year.

This second volume of the book series is based on a dozen of these
sessions, encompassing a variety of enticing and stimulating
mathematical topics, some new and some continuing from Volume I:

- from dismantling Rubik's Cube and randomly putting it back together to solving it with the power of group theory;
- from raising knot-eating machines and letting Alexander the Great cut the Gordian Knot to breaking through knot theory via the Jones polynomial;
- from entering a seemingly hopeless infinite raffle to becoming friendly with multiplicative functions in the land of Dirichlet, Möbius, and Euler;
- from leading an army of jumping fleas in an old problem from the International Mathematical Olympiads to improving our own essay-writing strategies;
- from searching for optimal paths on a hot summer day to questioning whether Archimedes was on his way to discovering trigonometry 2000 years ago

Do some of these scenarios sound bizarre, having never before been associated with mathematics? Mathematicians love having fun while doing serious mathematics and that love is what this book intends to share with the reader. Whether at a beginner, an intermediate, or an advanced level, anyone can find a place here to be provoked to think deeply and to be inspired to create.

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 students and their parents and teachers and undergraduate students interested in math circles.

#### Table of Contents

# Table of Contents

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

- Front Cover Front Cover11
- Contents vii8
- Foreword xi12
- Introduction xiii14
- Session 1: Geometric Re-Constructions. Part I 126
- Session 2: Rubik’s Cube. Part II 2348
- Session 3: Knotty Mathematics 4974
- Session 4: Multiplicative Functions. Part I 79104
- Session 5: Introduction to Group Theory 103128
- Session 6: Monovariants. Part II 141166
- Session 7: Geometric Re-Constructions. Part II 171196
- Session 8: Complex Numbers. Part II 189214
- 1. Warning, “Teaser,” and Strategy 189214
- 2. Conventions from the Past 190215
- 3. Complex Division 190215
- 4. The Triangle Inequality: No “Respect” for Addition? 192217
- 5. Integer Powers in C 193218
- 6. Roots in C 196221
- 7. Roots of Unity and Regular Polygons 198223
- 8. Geometric Promise Fulfilled 200225
- 9. Venturing Everywhere in the Plane 202227
- 10. Which are the “Closest” Lines 205230
- 11. Hints and Solutions to Selected Problems 208233

- Session 9: Introduction to Inequalities. Part I 211236
- 1. The Language of Inequalities 211236
- 2. Arithmetic Mean – Geometric Mean Inequality 212237
- 3. Power Mean Inequality 216241
- 4. The Land of the Convex 218243
- 5. Applications of Convexity to Inequalities 220245
- 6. Geometry Leftovers and a Mean Summary 223248
- 7. Hints and Solutions to Selected Problems 225250

- Session 10: Multiplicative Functions. Part II 233258
- Session 11: Monovariants. Part III 263288
- Session 12: Geometric Re-Constructions. Part III 287312
- Epilogue 305330
- Symbols and Notation 321346
- Abbreviations 325350
- Biographical Data 327352
- Bibliography 331356
- Credits 335360
- Index 337362
- Back Cover Back Cover1372