Mathematical Circles Library Mathematical Circle Diaries, Year 2 Complete Curriculum for Grades 6 to 8 Anna Burago Mathematical Circle Diaries, Year 2 Complete Curriculum for Grades 6 to 8 Mathematical Circles Library Mathematical Circle Diaries, Year 2 Complete Curriculum for Grades 6 to 8 Anna Burago Berkeley, California Advisory Board for the MSRI/Mathematical Circles Library Titu Andreescu Tatiana Shubin (Chair) David Auckly Zvezdelina Stankova H´el`ene Barcelo James Tanton Zuming Feng Ravi Vakil Tony Gardiner Diana White Nikolaj N. Konstantinov Ivan Yashchenko Andy Liu Paul Zeitz Alexander Shen Joshua Zucker Series Editor: Maia Averett, Mills College. Edited by Nelli Tkach and Maia Averett Illustrations by Susanna Hakobyan This volume is published with the generous support of the Simons Foundation and Tom Leighton and Bonnie Berger Leighton. 2010 Mathematics Subject Classification. Primary 97A20, 97A80, 00A07, 00A08, 00A09, 97D50. For additional information and updates on this book, visit www.ams.org/bookpages/mcl-20 Library of Congress Cataloging-in-Publication Data Names: Burago, Anna, 1967- author. Title:Mathematicalcirclediaries,year2: completecurriculumforgrades6to8/AnnaBurago. Description: Berkeley, California : MSRI Mathematical Sciences Research Institute ; Providence, Rhode Island : American Mathematical Society, c2018. | Series: MSRI mathematical circles library ; 20 | Includes bibliographical references. Identifiers: LCCN 2017058792 | ISBN 9781470437183 (alk. paper) Subjects: LCSH: Games in mathematics education. | Mathematics–Study and teaching (Middle school)–Activity programs. | AMS: Mathematics education – General, mathematics and educa- tion – Recreational mathematics, games. msc | Mathematics education – General, mathematics and education – Popularization of mathematics. msc | General – General and miscellaneous specific topics – Problem books. msc | General – General and miscellaneous specific topics – Recreational mathematics. msc | General – General and miscellaneous specific topics – Popularization of mathematics. msc | Mathematics education – Education and instruction in mathematics – Teaching problem solving and heuristic strategies. msc Classification: LCC QA20.G35 B8725 2018 | DDC 510.71/2–dc23 LC record available at https://lccn.loc.gov/2017058792 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2018 by Anna Burago. All rights reserved. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ Visit the MSRI home page at htpp://www.msri.org/ 10987654321 232221201918 Contents Acknowledgments xiii Preliminaries 1 Mathematical Circles 1 A Few Words about This Book 2 Potential Students 3 Curriculum 3 Part 1. Session Plans 5 Introduction 7 Lessons and Problem Sets 7 Session 1: Checkerboard Problems 9 1.1. Introduction 9 1.2. Math Warm-up 10 1.3. Discussion of the Day: Checkerboard Problems 10 1.4. In-Class Problem Set 14 1.5. A Few Words about Problem Sets 15 1.6. Take-Home Problem Set 15 1.7. Additional “Checkerboard” Problems 17 Session 2: Review: Math Logic and Other Problem-Solving Strategies 19 2.1. Math Warm-up 19 2.2. Discussion of the Day: Problem-Solving Strategies 20 2.3. Take-Home Problem Set 23 Session 3: Invariants 25 3.1. Warm-up Discussion. Are Proofs Really Necessary? 25 3.2. Discussion of the Day: Invariants 28 3.3. Take-Home Problem Set 32 v vi Contents Session 4: Proof by Contradiction 33 4.1. Math Warm-up 33 4.2. Discussion of the Day: Proof by Contradiction 33 4.3. Take-Home Problem Set 37 Session 5: Decimal Number System and Problems on Digits 39 5.1. Warm-up Discussion. Egyptian Number System 39 5.2. Discussion of the Day: Problems on Digits 41 5.3. In-Class Problem Set 45 5.4. Take-Home Problem Set 45 5.5. Additional Problems 46 Session 6: Binary Numbers I 47 6.1. Math Warm-up 47 6.2. Discussion of the Day: Binary Land—an Informal Introduction to Binaries 48 6.3. Binary Number System 51 6.4. Binary Notation 53 6.5. Computers and Binary Numbers 53 6.6. Take-Home Problem Set 55 Session 7: Binary Numbers II 59 7.1. Math Warm-up 59 7.2. Discussion of the Day: Binary Arithmetic 60 7.3. How to Convert Decimals to Binary 61 7.4. Take-Home Problem Set 65 Session 8: Mathematical Dominoes Tournament 67 8.1. Math Warm-up 68 8.2. Rules of Mathematical Dominoes 68 8.3. Mathematical Dominoes Problems 69 8.4. Take-Home Problem Set 78 Session 9: Pigeonhole Principle 81 9.1. Math Warm-up 81 9.2. Discussion of the Day: Pigeonhole Principle 81 9.3. Take-Home Problem Set 85 9.4. Additional Problems 87 Session 10: Geometric Pigeonhole Principle 89 10.1. Math Warm-up 89 10.2. Discussion of the Day: Geometric Pigeonhole 89 10.3. Take-Home Problem Set 92 10.4. Additional Problems 93 Contents vii Session 11: Mathematical Olympiad I 95 11.1. Event of the Day: Mathematical Olympiad 95 11.2. Mathematical Olympiad I. First Set of Problems 96 11.3. Mathematical Olympiad I. Second Set of Problems 97 11.4. Mathematical Olympiad I. Additional Problems 97 Session 12: Combinatorics I. Review 99 12.1. Math Warm-up 99 12.2. Discussion of the Day: Review of Combinatorics Techniques 100 12.3. In-Class Problem Set 105 12.4. Take-Home Problem Set 106 12.5. Additional Problems 107 Session 13: Combinatorics II. Combinations 109 13.1. Math Warm-up 109 13.2. Discussion of the Day: Combinations 110 13.3. Take-Home Problem Set 114 Session 14: Mathematical Auction 117 14.1. Math Warm-up 118 14.2. Event of the Day: Mathematical Auction Game 118 14.3. Mathematical Auction Problems 119 14.4. Take-Home Problem Set 120 Session 15: Combinatorics III. Complements. Snake Pit Game 121 15.1. Math Warm-up 121 15.2. Discussion of the Day: Complements 122 15.3. Activity of the Day: Snake Pit on Combinatorics 124 15.4. Take-Home Problem Set 126 Session 16: Combinatorics IV. Combinatorial Conundrum 129 16.1. Math Warm-up 129 16.2. Discussion of the Day: Combinatorial Craftiness 130 16.3. Take-Home Problem Set 135 16.4. Additional Problems 136 Session 17: Magic Squares and Related Problems 139 17.1. Math Warm-up 139 17.2. Discussion of the Day: Magic Squares from 1 to 9 140 17.3. More on 3 × 3 Magic Squares 143 17.4. Magic Squares Extended 144 17.5. Take-Home Problem Set 144 viii Contents Session 18: Double Counting, or There Is More than One Way to Cut a Cake 147 18.1. Math Warm-up 147 18.2. Discussion of the Day: Double Counting 148 18.3. Take-Home Problem Set 152 18.4. Additional Problems 153 Session 19: Mathematical Olympiad II 157 19.1. Event of the Day: Mathematical Olympiad 157 19.2. Mathematical Olympiad II. First Set of Problems 157 19.3. Mathematical Olympiad II. Second Set of Problems 158 19.4. Mathematical Olympiad II. Additional Problems 159 Session 20: Divisibility I. Review 161 20.1. Math Warm-up 161 20.2. Discussion of the Day: Divisibility 162 20.3. Prime Factorization Practice. Set 1 168 20.4. Prime Factorization Practice. Set 2 168 20.5. Take-Home Problem Set 169 20.6. Additional Problems 170 Session 21: Divisibility II. Relatively Prime Numbers; GCF and LCM 171 21.1. Math Warm-up: Mysteries of Prime Numbers 171 21.2. Discussion of the Day: Relatively Prime Numbers 173 21.3. Greatest Common Factor (GCF) 174 21.4. Least Common Multiple (LCM) 175 21.5. How GCF and LCM Are Related 177 21.6. GCF and LCM. In-Class Practice Problems 177 21.7. Take-Home Problem Set 179 21.8. Additional Problems 180 Session 22: Divisibility III. Mathematical Race Game 181 22.1. Math Warm-up 182 22.2. Event of the Day: Mathematical Race 182 22.3. Take-Home Problem Set 183 Session 23: Mathematical Auction 185 23.1. Event of the Day: Mathematical Auction Game 185 23.2. Mathematical Auction Problems 186 23.3. Take-Home Problem Set 187 Session 24: Divisibility IV. Divisibility by 3 and Remainders 189 24.1. Math Warm-up 189 24.2. Discussion of the Day: Remainders When Divided by 3 189 Contents ix 24.3. Arithmetic of Remainders 190 24.4. Take-Home Problem Set 196 24.5. Additional Problems 197 Session 25: Divisibility V. Divisibility and Remainders 199 25.1. Math Warm-up 199 25.2. Discussion of the Day: Divisibility and Remainders 199 25.3. Divisibility and Remainders Practice 204 25.4. Take-Home Problem Set 205 25.5. Additional Problems 205 Session 26: Graph Theory I. Graphs and Their Applications 207 26.1. Math Warm-up 207 26.2. Discussion of the Day: Why Graphs Are Important 208 26.3. How to Calculate the Number of Edges in a Graph 210 26.4. Take-Home Problem Set 211 Session 27: Graph Theory II. Handshaking Theorem 213 27.1. Math Warm-up 213 27.2. Discussion of the Day: Odd Vertices Theorem 214 27.3. In-Class Problem Set 217 27.4.