A Festival of Mathematics

232 Pages·2022·30.131 MB·English

by  Alice Peters, Mark Saul

M Mathematical Circles Library
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This book, inspired by the Julia Rob-
inson Mathematics Festival, aims to
engage students in mathematical dis-
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A Festival of
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requiring no more than simple arithmetic or intuitive concepts of symmetry.
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Follow-up problems and activities require intuitive logic and reveal more
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sophisticated notions of strategy and algorithms. Projects are designed so that l Mathematics

progress is more important than any end goal, ensuring that students will learn o
something significant no matter how far they get. The process of understanding f

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the questions and how they build on one another becomes an exhilarating
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ride, revealing serious mathematics before the reader is aware of the transition. a
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This book can be used in classrooms, math clubs, after school activities, home- h
schooling, and parent/student gatherings and is appropriate for students of age e
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8 to 18, as well as for teachers wanting to hone their skills.
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In the interest of fostering a greater awareness and appreciation of mathematics
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and its connections to other disciplines and everyday life, MSRI and the AMS ic
are publishing books in the Mathematical Circles Library series as a service to s

young people, their parents and teachers, and the mathematics profession.

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Alice Peters
Mark Saul
For additional information
and updates on this book, visit
www.ams.org/bookpages/mcl-28
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MCL/28 •
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230 pages • Spine: 1/2" • 50lb paper • Softcover
A Festival of
Mathematics
A Sourcebook

Mathematical Circles Library
A Festival of
Mathematics
A Sourcebook
Alice Peters
Mark Saul
Berkeley, California
Advisory Board for the MSRI/Mathematical Circles Library
Titu Andreescu Zvezdelina Stankova
David Auckly James Tanton
H´el`ene Barcelo Ravi Vakil
Zuming Feng Diana White
Tony Gardiner Ivan Yashchenko
Andy Liu Paul Zeitz
Alexander Shen Joshua Zucker
Tatiana Shubin (Chair)
Scientific Editor: David Scott
This volume is published with the generous support of the Simons Foundation
and Tom Leighton and Bonnie Berger Leighton.
2020 Mathematics Subject Classification. Primary 97-01, 97D50, 97E30,97U30, 97U40.
For additional informationand updates on this book, visit
www.ams.org/bookpages/mcl-28
Library of Congress Cataloging-in-Publication Data
Names: Peters,Alice,1946-author. |Saul,MarkE.,author.
Title: Afestivalofmathematics: asourcebook/AlicePeters,MarkSaul.
Description: Berkeley,California: MSRIMathematicalSciencesResearchInstitute;Providence,
Rhode Island : American Mathematical Society, [2022] | Series: MSRI mathematical circles
library,1944-8074;28|
Identifiers: LCCN2021052340|ISBN9781470453381(paperback)|9781470469559(ebook)
Subjects: LCSH:Mathematics–Studyandteaching. |Mathematicsteachers–Trainingof. |AMS:
Mathematicseducation–Introductoryexposition(textbooks,tutorialpapers,etc.) pertaining
tomathematicseducation. |Mathematicseducation–Educationandinstructioninmathemat-
ics–Teachingproblem solvingandheuristic strategies. |Mathematicseducation–Education
offoundations ofmathematics–Logic (educationalaspects). |Mathematicseducation– Ed-
ucational material and media, educational technology in mathematics education – Teachers’
manuals and planning aids (aspects of mathematics education). | Mathematics education –
Educationalmaterialandmedia,educationaltechnologyinmathematicseducation–Problem
books,competitions,examinations(aspectsofmathematicseducation).
Classification: LCCQA11.2.P5552022|DDC510.71/2–dc23/eng/20211208
LCrecordavailableathttps://lccn.loc.gov/2021052340
DOI:https://doi.org/10.1090/saul2/28
Copying and reprinting. Individual readersofthispublication,andnonprofit librariesacting
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,providedthecustomaryacknowledgmentofthesourceisgiven.
Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication
ispermittedonlyunderlicensefromtheAmericanMathematicalSociety. Requestsforpermission
toreuseportionsofAMSpublicationcontentarehandledbytheCopyrightClearanceCenter. For
moreinformation,pleasevisitwww.ams.org/publications/pubpermissions.
Sendreq[e-mail].
(cid:2)c 2022bytheauthors. Allrightsreserved.
PrintedintheUnitedStatesofAmerica.
(cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
establishedtoensurepermanenceanddurability.
VisittheAMShomepageathttps://www.ams.org/
VisittheMSRIhomepageathttp://www.msri.org/
10987654321 272625242322
To the memory of Nelson Blachman,
who inspired his daughter Nancy in her study of mathematics

Contents
Foreword xi
Preface xv
Acknowledgements xix
Part 1. Activity Guides 1
Chapter 1. Color Triangle Challenge 3
Some Initial Explorations 4
Toward a Generalization 9
Further Generalizations 13
Applying Arithmetic 16
Connection to the Binomial Coefficients 18
Historical Notes 21
Chapter 2. Magic Squares and Algebra 23
Constructing a Magic Square 23
The Geometry of Magic Squares 31
Some Facts about Groups 34
Uniqueness of the Magic Square: Some Combinatoric Results 36
New Magic Squares from Old 37
The Vector Space of a Magic Square 41
Magic Squares and Tic-tac-toe 49
Historical Notes 50
Chapter 3. Nim 53
One-Row Nim 53
Nim Variants 61
Two-Row Nim 63
Historical Notes 64
Chapter 4. Palindrome Grab! 67
The Basic Game 67
The Greedy Game 69
vii
viii CONTENTS
The Patient Game 71
Historical Notes 74
Chapter 5. To Twos, Too! Two Twos? More? 77
SDP2 Representations 77
SP2 Representations 81
S2P2 Representations 85
Some Extensions 89
Historical Notes 90
Chapter 6. Prisoner Puzzle 93
Last Man Sitting 93
Lucky 7? 96
Changing of the Guard 99
Historical Notes 101
Chapter 7. Broken Calculators 103
Calculator 1 103
Calculator 2 105
Calculator 3 108
Calculator 4 109
Calculator 5 110
Historical Notes 112
Chapter 8. Dominoes and Checkerboards 113
Constructing Tilings 114
Counting Tilings 121
Historical Notes 129
Chapter 9. Fair Division 131
Rectangles, Triangles, Squares 131
Quadrilaterals and Squares 136
Historical Notes 151
Chapter 10. Jumping Julia 153
Mazes and Graph Theory 153
Make Your Own Maze 161
Historical Notes 164
Part 2. Activity Handouts 167
Chapter 1. Color Triangle Challenge 169
Some Initial Explorations 170
Toward a Generalization 171
Applying Arithmetic 171
Chapter 2. Magic Squares and Algebra 175
Constructing a Magic Square 175
CONTENTS ix
The Geometry of Magic Squares 177
Uniqueness of the Magic Square: Some Combinatoric Results 178
New Magic Squares from Old 179
Chapter 3. Nim 181
One-Row Nim 181
Nim Variants 182
Two-Row Nim 182
Chapter 4. Palindrome Grab! 183
The Basic Game 183
The Greedy Game 184
The Patient Game 184
Chapter 5. To Twos, Too! Two Twos? More? 185
SDP2 Representations 185
SP2 Representations 185
S2P2 Representations 186
Some Extensions 187
Chapter 6. Prisoner Puzzle 189
Last Man Sitting 189
Lucky 7? 189
Changing of the Guard 190
Chapter 7. Broken Calculators 191
Calculator 1 191
Calculator 2 191
Calculator 3 191
Calculator 4 192
Calculator 5 192
Chapter 8. Dominoes and Checkerboards 193
Constructing Tilings 193
Counting Tilings 195
Chapter 9. Fair Division 197
Rectangles, Triangles, Squares 197
Quadrilaterals and Squares 198
Chapter 10. Jumping Julia 203
Mazes and Graph Theory 203
Make Your Own Maze 207