Part 1 . Session Plans
Introduction
Lessons and Problem Sets
Chapter 1. Session 1: Checkerboard Problems
1.1. Introduction
1.2. Math Warm-up
1.3. Discussion of the Day: Checkerboard Problems
1.4. In-Class Problem Set
1.5. A Few Words about Problem Sets
1.6. Take-Home Problem Set
1.7. Additional “Checkerboard” Problems
Chapter 2. Session 2: Review: Math Logic and Other Problem-Solving Strategies
2.1. Math Warm-up
2.2. Discussion of the Day: Problem-Solving Strategies
2.3. Take-Home Problem Set
Chapter 3. Session 3: Invariants
3.1. Warm-up Discussion. Are Proofs Really Necessary?
3.2. Discussion of the Day: Invariants
3.3. Take-Home Problem Set
Chapter 4. Session 4: Proof by Contradiction
4.1. Math Warm-up
4.2. Discussion of the Day: Proof by Contradiction
4.3. Take-Home Problem Set
Chapter 5. Session 5: Decimal Number System and Problems on Digits
5.1. Warm-up Discussion. Egyptian Number System
5.2. Discussion of the Day: Problems on Digits
5.3. In-Class Problem Set
5.4. Take-Home Problem Set
5.5. Additional Problems
Chapter 6. Session 6: Binary Numbers I
6.1. Math Warm-up
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. Take-Home Problem Set
Chapter 7. Session 7: Binary Numbers II
7.1. Math Warm-up
7.2. Discussion of the Day: Binary Arithmetic
7.3. How to Convert Decimals to Binary
7.4. Take-Home Problem Set
Chapter 8. Session 8: Mathematical Dominoes Tournament
8.1. Math Warm-up
8.2. Rules of Mathematical Dominoes
8.3. Mathematical Dominoes Problems
8.4. Take-Home Problem Set
Chapter 9. Session 9: Pigeonhole Principle
9.1. Math Warm-up
9.2. Discussion of the Day: Pigeonhole Principle
9.3. Take-Home Problem Set
9.4. Additional Problems
Chapter 10. Session 10: Geometric Pigeonhole Principle
10.1. Math Warm-up
10.2. Discussion of the Day: Geometric Pigeonhole
10.3. Take-Home 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 Warm-up
12.2. Discussion of the Day: Review of Combinatorics Techniques
12.3. In-Class Problem Set
12.4. Take-Home Problem Set
12.5. Additional Problems
Chapter 13. Session 13: Combinatorics II. Combinations
13.1. Math Warm-up
13.2. Discussion of the Day: Combinations
13.3. Take-Home Problem Set
Chapter 14. Session 14: Mathematical Auction
14.1. Math Warm-up
14.2. Event of the Day: Mathematical Auction Game
14.3. Mathematical Auction Problems
14.4. Take-Home Problem Set
Chapter 15. Session 15: Combinatorics III. Complements. Snake Pit Game
15.1. Math Warm-up
15.2. Discussion of the Day: Complements
15.3. Activity of the Day: Snake Pit on Combinatorics
15.4. Take-Home Problem Set
Chapter 16. Session 16: Combinatorics IV. Combinatorial Conundrum
16.1. Math Warm-up
16.2. Discussion of the Day: Combinatorial Craftiness
16.3. Take-Home Problem Set
16.4. Additional Problems
Chapter 17. Session 17: Magic Squares and Related Problems
17.1. Math Warm-up
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. Take-Home Problem Set
Chapter 18. Session 18: Double Counting, or There Is More than One Way to Cut a Cake
18.1. Math Warm-up
18.2. Discussion of the Day: Double Counting
18.3. Take-Home 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 Warm-up
20.2. Discussion of the Day: Divisibility
20.3. Prime Factorization Practice. Set 1
20.4. Prime Factorization Practice. Set 2
20.5. Take-Home Problem Set
20.6. Additional Problems
Chapter 21. Session 21: Divisibility II. Relatively Prime Numbers; GCF and LCM
21.1. Math Warm-up: 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. In-Class Practice Problems
21.7. Take-Home Problem Set
21.8. Additional Problems
Chapter 22. Session 22: Divisibility III. Mathematical Race Game
22.1. Math Warm-up
22.2. Event of the Day: Mathematical Race
22.3. Take-Home Problem Set
Chapter 23. Session 23: Mathematical Auction
23.1. Event of the Day: Mathematical Auction Game
23.2. Mathematical Auction Problems
23.3. Take-Home Problem Set
Chapter 24. Session 24: Divisibility IV. Divisibility by 3 and Remainders
24.1. Math Warm-up
24.2. Discussion of the Day: Remainders When Divided by 3
24.3. Arithmetic of Remainders
24.4. Take-Home Problem Set
24.5. Additional Problems
Chapter 25. Session 25: Divisibility V. Divisibility and Remainders
25.1. Math Warm-up
25.2. Discussion of the Day: Divisibility and Remainders
25.3. Divisibility and Remainders Practice
25.4. Take-Home Problem Set
25.5. Additional Problems
Chapter 26. Session 26: Graph Theory I. Graphs and Their Applications
26.1. Math Warm-up
26.2. Discussion of the Day: Why Graphs Are Important
26.3. How to Calculate the Number of Edges in a Graph
26.4. Take-Home Problem Set
Chapter 27. Session 27: Graph Theory II. Handshaking Theorem
27.1. Math Warm-up
27.2. Discussion of the Day: Odd Vertices Theorem
27.3. In-Class Problem Set
27.4. Take-Home Problem Set
27.5. Additional Problems
Chapter 28. Session 28: Graph Theory II. Solving Problems with Graphs
28.1. Math Warm-up
28.2. Discussion of the Day: Graphs Potpourri
28.3. Take-Home 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 Problem-Solving 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

This book, a sequel to Mathematical Circle Diaries, Year 1, teaches how to think and solve problems in mathematics. The material, distributed among twenty-nine weekly lessons, includes detailed lectures and discussions, sets of problems with solutions, and contests and games. In addition, the book shares some of the know-how of running a mathematical circle. The book covers a broad range of problem-solving 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.