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Softcover ISBN:  9781470437183 
Product Code:  MCL/20 
List Price:  $55.00 
MAA Member Price:  $49.50 
AMS Member Price:  $44.00 
Sale Price:  $33.00 
eBook ISBN:  9781470448097 
Product Code:  MCL/20.E 
List Price:  $50.00 
MAA Member Price:  $45.00 
AMS Member Price:  $40.00 
Sale Price:  $30.00 
Softcover ISBN:  9781470437183 
eBook ISBN:  9781470448097 
Product Code:  MCL/20.B 
List Price:  $105.00 $80.00 
MAA Member Price:  $94.50 $72.00 
AMS Member Price:  $84.00 $64.00 
Sale Price:  $63.00 $48.00 

Book DetailsMSRI Mathematical Circles LibraryVolume: 20; 2018; 362 ppMSC: Primary 97; 00
Mathematical circles, with their questiondriven approach and emphasis on problem solving, expose students to the type of mathematics that stimulates the development of logical thinking, creativity, analytical abilities, and mathematical reasoning. These skills, while scarcely introduced at school, are in high demand in the modern world.
This book, a sequel to Mathematical Circle Diaries, Year 1, teaches how to think and solve problems in mathematics. The material, distributed among twentynine weekly lessons, includes detailed lectures and discussions, sets of problems with solutions, and contests and games. In addition, the book shares some of the knowhow of running a mathematical circle. The book covers a broad range of problemsolving strategies and proofing techniques, as well as some more advanced topics that go beyond the limits of a school curriculum. The topics include invariants, proofs by contradiction, the Pigeonhole principle, proofs by coloring, double counting, combinatorics, binary numbers, graph theory, divisibility and remainders, logic, and many others. When students take science and computing classes in high school and college, they will be better prepared for both the foundations and advanced material. The book contains everything that is needed to run a successful mathematical circle for a full year.
This book, written by an author actively involved in teaching mathematical circles for fifteen years, is intended for teachers, math coaches, parents, and math enthusiasts who are interested in teaching math that promotes critical thinking. Motivated students can work through this book on their own.
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 copublished with the Mathematical Sciences Research Institute (MSRI).
ReadershipTeachers, parents, research mathematicians, and undergraduate and graduate students interested in teaching extracurricular math to middle and high school students.

Table of Contents

Cover

Title page

Contents

Acknowledgments

Preliminaries

Mathematical Circles

A Few Words about This Book

Potential Students

Curriculum

Part 1 . Session Plans

Introduction

Lessons and Problem Sets

Chapter 1. Session 1: Checkerboard Problems

1.1. Introduction

1.2. Math Warmup

1.3. Discussion of the Day: Checkerboard Problems

1.4. InClass Problem Set

1.5. A Few Words about Problem Sets

1.6. TakeHome Problem Set

1.7. Additional “Checkerboard” Problems

Chapter 2. Session 2: Review: Math Logic and Other ProblemSolving Strategies

2.1. Math Warmup

2.2. Discussion of the Day: ProblemSolving Strategies

2.3. TakeHome Problem Set

Chapter 3. Session 3: Invariants

3.1. Warmup Discussion. Are Proofs Really Necessary?

3.2. Discussion of the Day: Invariants

3.3. TakeHome Problem Set

Chapter 4. Session 4: Proof by Contradiction

4.1. Math Warmup

4.2. Discussion of the Day: Proof by Contradiction

4.3. TakeHome Problem Set

Chapter 5. Session 5: Decimal Number System and Problems on Digits

5.1. Warmup Discussion. Egyptian Number System

5.2. Discussion of the Day: Problems on Digits

5.3. InClass Problem Set

5.4. TakeHome Problem Set

5.5. Additional Problems

Chapter 6. Session 6: Binary Numbers I

6.1. Math Warmup

6.2. Discussion of the Day: Binary Land—an Informal Introduction to Binaries

6.3. Binary Number System

6.4. Binary Notation

6.5. Computers and Binary Numbers

6.6. TakeHome Problem Set

Chapter 7. Session 7: Binary Numbers II

7.1. Math Warmup

7.2. Discussion of the Day: Binary Arithmetic

7.3. How to Convert Decimals to Binary

7.4. TakeHome Problem Set

Chapter 8. Session 8: Mathematical Dominoes Tournament

8.1. Math Warmup

8.2. Rules of Mathematical Dominoes

8.3. Mathematical Dominoes Problems

8.4. TakeHome Problem Set

Chapter 9. Session 9: Pigeonhole Principle

9.1. Math Warmup

9.2. Discussion of the Day: Pigeonhole Principle

9.3. TakeHome Problem Set

9.4. Additional Problems

Chapter 10. Session 10: Geometric Pigeonhole Principle

10.1. Math Warmup

10.2. Discussion of the Day: Geometric Pigeonhole

10.3. TakeHome Problem Set

10.4. Additional Problems

Chapter 11. Session 11: Mathematical Olympiad I

11.1. Event of the Day: Mathematical Olympiad

11.2. Mathematical Olympiad I. First Set of Problems

11.3. Mathematical Olympiad I. Second Set of Problems

11.4. Mathematical Olympiad I. Additional Problems

Chapter 12. Session 12: Combinatorics I. Review

12.1. Math Warmup

12.2. Discussion of the Day: Review of Combinatorics Techniques

12.3. InClass Problem Set

12.4. TakeHome Problem Set

12.5. Additional Problems

Chapter 13. Session 13: Combinatorics II. Combinations

13.1. Math Warmup

13.2. Discussion of the Day: Combinations

13.3. TakeHome Problem Set

Chapter 14. Session 14: Mathematical Auction

14.1. Math Warmup

14.2. Event of the Day: Mathematical Auction Game

14.3. Mathematical Auction Problems

14.4. TakeHome Problem Set

Chapter 15. Session 15: Combinatorics III. Complements. Snake Pit Game

15.1. Math Warmup

15.2. Discussion of the Day: Complements

15.3. Activity of the Day: Snake Pit on Combinatorics

15.4. TakeHome Problem Set

Chapter 16. Session 16: Combinatorics IV. Combinatorial Conundrum

16.1. Math Warmup

16.2. Discussion of the Day: Combinatorial Craftiness

16.3. TakeHome Problem Set

16.4. Additional Problems

Chapter 17. Session 17: Magic Squares and Related Problems

17.1. Math Warmup

17.2. Discussion of the Day: Magic Squares from 1 to 9

17.3. More on 3×3 Magic Squares

17.4. Magic Squares Extended

17.5. TakeHome Problem Set

Chapter 18. Session 18: Double Counting, or There Is More than One Way to Cut a Cake

18.1. Math Warmup

18.2. Discussion of the Day: Double Counting

18.3. TakeHome Problem Set

18.4. Additional Problems

Chapter 19. Session 19: Mathematical Olympiad II

19.1. Event of the Day: Mathematical Olympiad

19.2. Mathematical Olympiad II. First Set of Problems

19.3. Mathematical Olympiad II. Second Set of Problems

19.4. Mathematical Olympiad II. Additional Problems

Chapter 20. Session 20: Divisibility I. Review

20.1. Math Warmup

20.2. Discussion of the Day: Divisibility

20.3. Prime Factorization Practice. Set 1

20.4. Prime Factorization Practice. Set 2

20.5. TakeHome Problem Set

20.6. Additional Problems

Chapter 21. Session 21: Divisibility II. Relatively Prime Numbers; GCF and LCM

21.1. Math Warmup: Mysteries of Prime Numbers

21.2. Discussion of the Day: Relatively Prime Numbers

21.3. Greatest Common Factor (GCF)

21.4. Least Common Multiple (LCM)

21.5. How GCF and LCM Are Related

21.6. GCF and LCM. InClass Practice Problems

21.7. TakeHome Problem Set

21.8. Additional Problems

Chapter 22. Session 22: Divisibility III. Mathematical Race Game

22.1. Math Warmup

22.2. Event of the Day: Mathematical Race

22.3. TakeHome Problem Set

Chapter 23. Session 23: Mathematical Auction

23.1. Event of the Day: Mathematical Auction Game

23.2. Mathematical Auction Problems

23.3. TakeHome Problem Set

Chapter 24. Session 24: Divisibility IV. Divisibility by 3 and Remainders

24.1. Math Warmup

24.2. Discussion of the Day: Remainders When Divided by 3

24.3. Arithmetic of Remainders

24.4. TakeHome Problem Set

24.5. Additional Problems

Chapter 25. Session 25: Divisibility V. Divisibility and Remainders

25.1. Math Warmup

25.2. Discussion of the Day: Divisibility and Remainders

25.3. Divisibility and Remainders Practice

25.4. TakeHome Problem Set

25.5. Additional Problems

Chapter 26. Session 26: Graph Theory I. Graphs and Their Applications

26.1. Math Warmup

26.2. Discussion of the Day: Why Graphs Are Important

26.3. How to Calculate the Number of Edges in a Graph

26.4. TakeHome Problem Set

Chapter 27. Session 27: Graph Theory II. Handshaking Theorem

27.1. Math Warmup

27.2. Discussion of the Day: Odd Vertices Theorem

27.3. InClass Problem Set

27.4. TakeHome Problem Set

27.5. Additional Problems

Chapter 28. Session 28: Graph Theory II. Solving Problems with Graphs

28.1. Math Warmup

28.2. Discussion of the Day: Graphs Potpourri

28.3. TakeHome Problem Set

Chapter 29. Session 29: Mathematical Olympiad III

29.1. Event of the Day: Mathematical Olympiad

29.2. Mathematical Olympiad III. First Set of Problems

29.3. Mathematical Olympiad III. Second Set of Problems

Part 2 . Mathematical Contests and Competitions

Mathematical Contests

Mathematical Auction

What Is Special about Mathematical Auctions?

Rules of Mathematical Auction

A Sample Round

Team Work

Advice for a Teacher

Examples of Mathematical Auction Problems

Mathematical Dominoes

Rules of Mathematical Dominoes

Why Students Like Mathematical Dominoes

Why Teachers Like Mathematical Dominoes

Useful Details

Scorecards

Dominoes Cards: How to Make Them

Odds and Ends

Mathematical Snake Pit

Rules of Snake Pit Game

Useful Details

Score Table

Mathematical Race

Rules of Mathematical Race

Useful Details

Score Table

Mathematical Olympiad

Planning for an Oral Olympiad

Running an Olympiad

Olympiads in This Book

Awards and Prizes

Short Entertaining Math Games

Giotto and Math Giotto

Nim

Black Box

Part 3 . More Teaching Advice

How to Be a Great Math Circle Teacher

Teaching Style

Your Target Group

What Comes Next?

The Farewell

Part 4 . Solutions

Session 1. Checkerboard Problems

Session 2. Review: Math Logic and Other ProblemSolving Strategies

Session 3. Invariants

Session 4. Proof by Contradiction

Session 5. Decimal Number System and Problems on Digits

Session 6. Binary Numbers I

Session 7. Binary Numbers II

Session 8. Mathematical Dominoes Tournament

Session 9. Pigeonhole Principle

Session 10. Geometric Pigeonhole Principle

Session 11. Mathematical Olympiad I

Session 12. Combinatorics I. Review

Session 13. Combinatorics II. Combinations

Session 14. Mathematical Auction

Session 15. Combinatorics III. Complements. Snake Pit Game

Session 16. Combinatorics IV. Combinatorial Conundrum

Session 17. Magic Squares and Related Problems

Session 18. Double Counting, or There Is More than One Way to Cut a Cake

Session 19. Mathematical Olympiad II

Session 20. Divisibility I. Review

Session 21. Divisibility II. Relatively Prime Numbers; GCF and LCM

Session 22. Divisibility III. Mathematical Race Game

Session 23. Mathematical Auction

Session 24. Divisibility IV. Divisibility by 3 and Remainders

Session 25. Divisibility V. Divisibility and Remainders

Session 26. Graph Theory I. Graphs and Their Applications

Session 27. Graph Theory II. Handshaking Theorem

Session 28. Graph Theory III. Solving Problems with Graphs

Session 29. Mathematical Olympiad III

Appendix to Session 6

“Convert Decimal to Binary” Blank Table

Bibliography

Back Cover


Additional Material

RequestsReview Copy – for publishers of book reviewsAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Requests
Mathematical circles, with their questiondriven approach and emphasis on problem solving, expose students to the type of mathematics that stimulates the development of logical thinking, creativity, analytical abilities, and mathematical reasoning. These skills, while scarcely introduced at school, are in high demand in the modern world.
This book, a sequel to Mathematical Circle Diaries, Year 1, teaches how to think and solve problems in mathematics. The material, distributed among twentynine weekly lessons, includes detailed lectures and discussions, sets of problems with solutions, and contests and games. In addition, the book shares some of the knowhow of running a mathematical circle. The book covers a broad range of problemsolving strategies and proofing techniques, as well as some more advanced topics that go beyond the limits of a school curriculum. The topics include invariants, proofs by contradiction, the Pigeonhole principle, proofs by coloring, double counting, combinatorics, binary numbers, graph theory, divisibility and remainders, logic, and many others. When students take science and computing classes in high school and college, they will be better prepared for both the foundations and advanced material. The book contains everything that is needed to run a successful mathematical circle for a full year.
This book, written by an author actively involved in teaching mathematical circles for fifteen years, is intended for teachers, math coaches, parents, and math enthusiasts who are interested in teaching math that promotes critical thinking. Motivated students can work through this book on their own.
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 copublished with the Mathematical Sciences Research Institute (MSRI).
Teachers, parents, research mathematicians, and undergraduate and graduate students interested in teaching extracurricular math to middle and high school students.

Cover

Title page

Contents

Acknowledgments

Preliminaries

Mathematical Circles

A Few Words about This Book

Potential Students

Curriculum

Part 1 . Session Plans

Introduction

Lessons and Problem Sets

Chapter 1. Session 1: Checkerboard Problems

1.1. Introduction

1.2. Math Warmup

1.3. Discussion of the Day: Checkerboard Problems

1.4. InClass Problem Set

1.5. A Few Words about Problem Sets

1.6. TakeHome Problem Set

1.7. Additional “Checkerboard” Problems

Chapter 2. Session 2: Review: Math Logic and Other ProblemSolving Strategies

2.1. Math Warmup

2.2. Discussion of the Day: ProblemSolving Strategies

2.3. TakeHome Problem Set

Chapter 3. Session 3: Invariants

3.1. Warmup Discussion. Are Proofs Really Necessary?

3.2. Discussion of the Day: Invariants

3.3. TakeHome Problem Set

Chapter 4. Session 4: Proof by Contradiction

4.1. Math Warmup

4.2. Discussion of the Day: Proof by Contradiction

4.3. TakeHome Problem Set

Chapter 5. Session 5: Decimal Number System and Problems on Digits

5.1. Warmup Discussion. Egyptian Number System

5.2. Discussion of the Day: Problems on Digits

5.3. InClass Problem Set

5.4. TakeHome Problem Set

5.5. Additional Problems

Chapter 6. Session 6: Binary Numbers I

6.1. Math Warmup

6.2. Discussion of the Day: Binary Land—an Informal Introduction to Binaries

6.3. Binary Number System

6.4. Binary Notation

6.5. Computers and Binary Numbers

6.6. TakeHome Problem Set

Chapter 7. Session 7: Binary Numbers II

7.1. Math Warmup

7.2. Discussion of the Day: Binary Arithmetic

7.3. How to Convert Decimals to Binary

7.4. TakeHome Problem Set

Chapter 8. Session 8: Mathematical Dominoes Tournament

8.1. Math Warmup

8.2. Rules of Mathematical Dominoes

8.3. Mathematical Dominoes Problems

8.4. TakeHome Problem Set

Chapter 9. Session 9: Pigeonhole Principle

9.1. Math Warmup

9.2. Discussion of the Day: Pigeonhole Principle

9.3. TakeHome Problem Set

9.4. Additional Problems

Chapter 10. Session 10: Geometric Pigeonhole Principle

10.1. Math Warmup

10.2. Discussion of the Day: Geometric Pigeonhole

10.3. TakeHome Problem Set

10.4. Additional Problems

Chapter 11. Session 11: Mathematical Olympiad I

11.1. Event of the Day: Mathematical Olympiad

11.2. Mathematical Olympiad I. First Set of Problems

11.3. Mathematical Olympiad I. Second Set of Problems

11.4. Mathematical Olympiad I. Additional Problems

Chapter 12. Session 12: Combinatorics I. Review

12.1. Math Warmup

12.2. Discussion of the Day: Review of Combinatorics Techniques

12.3. InClass Problem Set

12.4. TakeHome Problem Set

12.5. Additional Problems

Chapter 13. Session 13: Combinatorics II. Combinations

13.1. Math Warmup

13.2. Discussion of the Day: Combinations

13.3. TakeHome Problem Set

Chapter 14. Session 14: Mathematical Auction

14.1. Math Warmup

14.2. Event of the Day: Mathematical Auction Game

14.3. Mathematical Auction Problems

14.4. TakeHome Problem Set

Chapter 15. Session 15: Combinatorics III. Complements. Snake Pit Game

15.1. Math Warmup

15.2. Discussion of the Day: Complements

15.3. Activity of the Day: Snake Pit on Combinatorics

15.4. TakeHome Problem Set

Chapter 16. Session 16: Combinatorics IV. Combinatorial Conundrum

16.1. Math Warmup

16.2. Discussion of the Day: Combinatorial Craftiness

16.3. TakeHome Problem Set

16.4. Additional Problems

Chapter 17. Session 17: Magic Squares and Related Problems

17.1. Math Warmup

17.2. Discussion of the Day: Magic Squares from 1 to 9

17.3. More on 3×3 Magic Squares

17.4. Magic Squares Extended

17.5. TakeHome Problem Set

Chapter 18. Session 18: Double Counting, or There Is More than One Way to Cut a Cake

18.1. Math Warmup

18.2. Discussion of the Day: Double Counting

18.3. TakeHome Problem Set

18.4. Additional Problems

Chapter 19. Session 19: Mathematical Olympiad II

19.1. Event of the Day: Mathematical Olympiad

19.2. Mathematical Olympiad II. First Set of Problems

19.3. Mathematical Olympiad II. Second Set of Problems

19.4. Mathematical Olympiad II. Additional Problems

Chapter 20. Session 20: Divisibility I. Review

20.1. Math Warmup

20.2. Discussion of the Day: Divisibility

20.3. Prime Factorization Practice. Set 1

20.4. Prime Factorization Practice. Set 2

20.5. TakeHome Problem Set

20.6. Additional Problems

Chapter 21. Session 21: Divisibility II. Relatively Prime Numbers; GCF and LCM

21.1. Math Warmup: Mysteries of Prime Numbers

21.2. Discussion of the Day: Relatively Prime Numbers

21.3. Greatest Common Factor (GCF)

21.4. Least Common Multiple (LCM)

21.5. How GCF and LCM Are Related

21.6. GCF and LCM. InClass Practice Problems

21.7. TakeHome Problem Set

21.8. Additional Problems

Chapter 22. Session 22: Divisibility III. Mathematical Race Game

22.1. Math Warmup

22.2. Event of the Day: Mathematical Race

22.3. TakeHome Problem Set

Chapter 23. Session 23: Mathematical Auction

23.1. Event of the Day: Mathematical Auction Game

23.2. Mathematical Auction Problems

23.3. TakeHome Problem Set

Chapter 24. Session 24: Divisibility IV. Divisibility by 3 and Remainders

24.1. Math Warmup

24.2. Discussion of the Day: Remainders When Divided by 3

24.3. Arithmetic of Remainders

24.4. TakeHome Problem Set

24.5. Additional Problems

Chapter 25. Session 25: Divisibility V. Divisibility and Remainders

25.1. Math Warmup

25.2. Discussion of the Day: Divisibility and Remainders

25.3. Divisibility and Remainders Practice

25.4. TakeHome Problem Set

25.5. Additional Problems

Chapter 26. Session 26: Graph Theory I. Graphs and Their Applications

26.1. Math Warmup

26.2. Discussion of the Day: Why Graphs Are Important

26.3. How to Calculate the Number of Edges in a Graph

26.4. TakeHome Problem Set

Chapter 27. Session 27: Graph Theory II. Handshaking Theorem

27.1. Math Warmup

27.2. Discussion of the Day: Odd Vertices Theorem

27.3. InClass Problem Set

27.4. TakeHome Problem Set

27.5. Additional Problems

Chapter 28. Session 28: Graph Theory II. Solving Problems with Graphs

28.1. Math Warmup

28.2. Discussion of the Day: Graphs Potpourri

28.3. TakeHome Problem Set

Chapter 29. Session 29: Mathematical Olympiad III

29.1. Event of the Day: Mathematical Olympiad

29.2. Mathematical Olympiad III. First Set of Problems

29.3. Mathematical Olympiad III. Second Set of Problems

Part 2 . Mathematical Contests and Competitions

Mathematical Contests

Mathematical Auction

What Is Special about Mathematical Auctions?

Rules of Mathematical Auction

A Sample Round

Team Work

Advice for a Teacher

Examples of Mathematical Auction Problems

Mathematical Dominoes

Rules of Mathematical Dominoes

Why Students Like Mathematical Dominoes

Why Teachers Like Mathematical Dominoes

Useful Details

Scorecards

Dominoes Cards: How to Make Them

Odds and Ends

Mathematical Snake Pit

Rules of Snake Pit Game

Useful Details

Score Table

Mathematical Race

Rules of Mathematical Race

Useful Details

Score Table

Mathematical Olympiad

Planning for an Oral Olympiad

Running an Olympiad

Olympiads in This Book

Awards and Prizes

Short Entertaining Math Games

Giotto and Math Giotto

Nim

Black Box

Part 3 . More Teaching Advice

How to Be a Great Math Circle Teacher

Teaching Style

Your Target Group

What Comes Next?

The Farewell

Part 4 . Solutions

Session 1. Checkerboard Problems

Session 2. Review: Math Logic and Other ProblemSolving Strategies

Session 3. Invariants

Session 4. Proof by Contradiction

Session 5. Decimal Number System and Problems on Digits

Session 6. Binary Numbers I

Session 7. Binary Numbers II

Session 8. Mathematical Dominoes Tournament

Session 9. Pigeonhole Principle

Session 10. Geometric Pigeonhole Principle

Session 11. Mathematical Olympiad I

Session 12. Combinatorics I. Review

Session 13. Combinatorics II. Combinations

Session 14. Mathematical Auction

Session 15. Combinatorics III. Complements. Snake Pit Game

Session 16. Combinatorics IV. Combinatorial Conundrum

Session 17. Magic Squares and Related Problems

Session 18. Double Counting, or There Is More than One Way to Cut a Cake

Session 19. Mathematical Olympiad II

Session 20. Divisibility I. Review

Session 21. Divisibility II. Relatively Prime Numbers; GCF and LCM

Session 22. Divisibility III. Mathematical Race Game

Session 23. Mathematical Auction

Session 24. Divisibility IV. Divisibility by 3 and Remainders

Session 25. Divisibility V. Divisibility and Remainders

Session 26. Graph Theory I. Graphs and Their Applications

Session 27. Graph Theory II. Handshaking Theorem

Session 28. Graph Theory III. Solving Problems with Graphs

Session 29. Mathematical Olympiad III

Appendix to Session 6

“Convert Decimal to Binary” Blank Table

Bibliography

Back Cover