A Decade of the Berkeley Math Circle – Volume II
Authors
Description
Many mathematicians were drawn to mathematics through their experience with mathematical circles. The Berkeley Mathematical Circle (CMB) began in 1998 as one of the first mathematical circles in the United States. Over the past decade and a half, 100 instructors—college professors, company bosses, high school professors, and others—have shared their passion for math, offering more than 800 CMB sessions on the campus of the University of California at Berkeley every week during the school year.
This second volume in this book series is based on some of these sessions, covering a variety of compelling and stimulating mathematical topics, some new and others continuing subjects from Volume I:
- from disassembling the Rubik’s Cube and assembling it, randomly, to solving it with the power of group theory;
- from creating knot-eating machines and letting Alexander the Great cut the Gordian knot, to unraveling the theory of knots through Jones’ polynomial;
- from entering an infinite, seemingly hopeless lottery, to becoming familiar with multiplicative functions in the lands of Dirichlet, Möbius, and Euler;
- from leading an army of jumping fleas on an age-old International Math Olympiad problem, to improving our own writing strategies;
- from searching for ideal paths on a hot summer’s day, to questioning whether Archimedes was on his way to discovering trigonometry 2,000 years ago.
Do some of these scenarios seem bizarre, never before associated with mathematics? Mathematicians love to have fun while doing serious math, and that passion is what this book aims to share with the reader. Whether at the beginner, intermediate or advanced level, anyone can find here a place to be provoked and think deeply and be inspired to create.
In the interest of promoting greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSR and AMS are publishing books from the Mathematical Circles Library series in service to young people, their parents and teachers, and the mathematical profession.
Target audience
Middle school
Name: A Decade of the Berkeley Math Circle – Volume II
Author(s): Zvezdelina Stankova e Thomas Rike
Pages: 403
Publication: IMPA, 2022
ISBN: 978-65-89124-06-1
Edition: 1
Introduction
1. Mathematical Circles of Excellence
2. What and for whom?
3. Notation and technical details
4. The art of being a mathematician and problem solving
5. Acknowledgments
Session 1. Geometric Reconstructions. Part I
1. Experimenting and Conjecturing
2. A workout in the triangle
3. Walking along an ideal path
4. Walking along an entire grid
5. To prove or accept?
6. Tips and solutions to selected problems
Session 2. Rubik’s cube. Part II
1. What is a group?
2. Permutation groups and group isomorphisms
3. Properties of groups and their subgroups
4. Odd and even worlds
5. How many cube positions can be reached?
6. Conclusions
7. Tips and solutions for selected problems
Session 3. Math with knots
1. To be or not to be a knot, that is the question
2. Reidemeister and machines eating from us
3. Three chalks defeat an army of knots
4. Jones’ polynomial
5. Is this the end?
6. Tips and solutions to selected problems
Session 4. Multiplicative functions. Part I
1. Raffle-∞: The Initial Setup
2. What are multiplicative functions?
3. Sum-functions
4. Tips and solutions to selected problems
Session 5. Introduction to group theory
1. Racking your brain
2. A polynomial prelude
3. Action groups
4. General groups
5. Some more examples of groups
6. Permutation groups or symmetric groups
7. The Game of 15
8. Tips and solutions for selected problems
Session 6. Monovariants. Part II
1. Numerical monovariants
2. Constructive activities
3. Not getting there
4. Conway’s Ladies
5. Tips and solutions to selected problems
Session 7. Geometric reconstructions. Part II
1. Optimal and endless challenges
2. A Pythagorean path to the intermediate
3. Physics and Mathematics combine forces
4. Ptolemy’s leadership in trigonometry
5. Tips and solutions to selected problems
Session 8. Complex numbers. Part II
1. Warning, “provocation” and strategy
2. Conventions of the past
3. Complex division
4. Triangular Inequality: no “respect” for addition?
5. Integer powers in C
6. C-roots
7. Roots of unity and regular polygons
8. Geometric promise fulfilled
9. Venturing Everywhere on the Plane
10. What are the “closest” lines
11. Tips and solutions for selected problems
Session 9. Introduction to inequalities. Part I
1. The language of inequalities
2. Inequality of the arithmetic mean-geometric mean
3. Middle-power inequality
4. The land of the convex
5. Applications of convexity to inequalities
6. Geometric Leftovers and Summarized Averages
7. Tips and solutions for selected problems
Session 10. Multiplicative functions. Part II
5. Dirichlet Product
6. Möbius inversion formula
7. Euler’s function φ(n)
8. Taming φ
9. Tips and solutions to selected problems
Session 11. Monovariants. Part III
1. The Balkan challenge
2. Softening and accentuating
3. Rearranging the Terms
4. Convexity and smoothing
5. Random Fun with Smoothing
6. Appendix on Limits and Infinite Smoothing
7. Tips and solutions for selected problems
Session 12. Geometric reconstructions. Part III
1. The farmer and the cow via inequalities and Calculation
2. Optimal bridge located!
3. Infinite angles and infinite series
4. Historical deviation: since today back to Archimedes?
5. Tips and solutions to selected problems
Epilogue
1. What comes from within
2. The culture of Circles
3. Eastern European vs. U.S. Mathematical Circles
4. History and power
5. Does the U.S. need CM excellence?
Symbols and notations
Abbreviations
Biographical data
Bibliography
Credits
Index