AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL 36 The Unity of Combinatorics Ezra Brown Richard K. Guy 10.1090/car/036 The Unity of Combinatorics AMS/MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL 36 The Unity of Combinatorics Ezra Brown Richard K. Guy 2018–2019 Editorial Committee Bruce P. Palka, Editor Francis Bonahon Annalisa Crannel Alex Iosevich Kristin Estella Lauter Steven J. Miller Henry Sagerman 2010 Mathematics Subject Classification. Primary 05B05, 05B10, 05B35, 05C15, 11Bxx, 20B25, 51E10, 52C20, 91A46, 94Bxx. Photo credits: The photograph of Richard K. Guy on the back cover is from Wikimedia. This file is licensed under the Creative Commons Attribution 2.0 Generic (https://creativecommons.org/licenses/by/2.0/deed.en) li- cense. For additional information and updates on this book, visit www.ams.org/bookpages/car-36 Page 353 constitutes an extension of this copyright page. © 2020 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ⃝1 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 https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 25 24 23 22 21 20 Contents Preface ix Credits and Permissions xiii Introduction 1 Chapter 1 Blocks, sequences, bow ties, and worms 5 1.1 Langford sequences 5 1.2 Partitioning sets of integers 7 1.3 Penrose tilings 12 Chapter 2 Combinatorial games 21 2.1 Wythoff’s game 21 2.2 Combinatorial games: rules and examples 22 2.3 Developing strategies: P-positions and N-positions 25 2.4 Nim, nimbers, and the Sprague–Grundy Theorem 27 2.5 Nim arithmetic and Nim algebra 35 Chapter 3 Fibonacci, Pascal, and Catalan 39 3.1 Fibonacci numbers 39 3.2 The triangle of Pingala, Al Karaji, Yang Hui, and Pascal 44 3.3 The Catalan numbers and the central column of Pascal’s triangle 54 Chapter 4 Catwalks, Sandsteps, and Pascal pyramids 61 Chapter 5 Unique rook circuits 77 Appendix 86 v vi Contents Chapter 6 Sums, colorings, squared squares, and packings 89 6.1 Triples satisfying 푥 + 푦 = 푧 89 6.2 Coil diagrams and the Ringel–Youngs Theorem 91 6.3 Squaring the square 94 6.4 Euler’s polyhedral formula 97 6.5 Packings and coverings of the complete graph 99 6.6 Steiner triple systems 103 Chapter 7 Difference sets and combinatorial designs 107 7.1 Difference sets 107 7.2 Multipliers 113 7.3 Difference sets, de Bruijn cycles, and de Bruijn graphs 115 7.4 Block designs 117 Chapter 8 Geometric connections 123 8.1 A quick tour of projective geometry 123 8.2 Finite projective geometries and Singer designs 132 8.3 Examples: 푛 = 2, 푞 = 2 and 3 137 8.4 Affine planes and magic squares 140 8.5 Heawood’s map on the torus revisited 142 8.6 (7, 3, 1) and Nim 144 8.7 The automorphism group of the Fano plane 148 Chapter 9 The groups 푃푆퐿(2, 7) and 퐺퐿(3, 2) and why they are isomorphic 155 9.1 The group 퐺퐿(3, 2) 157 9.2 The group 푃푆퐿(2, 7) 159 9.3 Constructing an isomorphism of 푃푆퐿(2, 7) onto 퐺퐿(3, 2) 161 Chapter 10 Incidence matrices, codes, and sphere packings 165 10.1 Introducing incidence matrices 165 10.2 Error-correcting codes 167 10.3 Sphere packing 174 10.4 Hadamard matrices and Hadamard difference sets 181 10.5 Hadamard matrices and projective geometries 183 Chapter 11 Kirkman’s schoolgirls, fields, spreads, and hats 187 11.1 Kirkman’s Schoolgirls Problem 187 Contents vii 11.2 Fifteen young ladies at school 188 11.3 Resolvable block designs and Kirkman triple systems 189 11.4 Kirkman’s schoolgirls and difference sets 191 11.5 퐾 = ℚ (√2, √3, √5, √7) and the designs it contains 194 11.6 Spreads in 푃퐺(3, 픽2) and the geometry of Kirkman 199 11.7 Fifteen schoolgirls, fifteen hats, and coding theory 202 11.8 Questions 205 Chapter 12 (7, 3, 1) and combinatorics 209 12.1 (7, 3, 1) and the Heawood graph 210 12.2 (7, 3, 1) and Latin squares 211 12.3 (7, 3, 1) and round-robin tournaments 212 Chapter 13 (7, 3, 1) and normed algebras 217 13.1 Sums of squares 217 13.2 The quaternions and the octonions 220 13.3 Beyond the octonions 225 Chapter 14 (7, 3, 1) and matroids 229 14.1 Why matroids? 230 14.2 Declaration of (in)dependence 230 14.3 Thus, matroids 234 14.4 Matroids and greed 239 Chapter 15 Coin-turning games and Mock Turtles 243 15.1 A review of some combinatorial game theory 243 15.2 Turning Turtles 245 15.3 Turning Corners: coins on a grid 247 15.4 Mock Turtles: more turtles in a line 250 15.5 More about turtle-turning games 252 Chapter 16 The (11, 5, 2) biplane, codes, designs, and groups 257 16.1 “How do you make math exciting for students?” 257 16.2 Difference sets, block designs, and biplanes 259 16.3 The automorphism group of the biplane 261 16.4 Incidence matrices, revisited 265 16.5 Error-correcting codes 266 16.6 Steiner systems 270 viii Contents 16.7 Automorphisms, transitivity, simplicity, and the Mathieu groups 274 Chapter 17 Rick’s Tricky Six Puzzle: More than meets the eye 279 17.1 Sliding-block puzzles 279 17.2 What is the exception? 282 17.3 Not much of a puzzle? 283 17.4 What is the automorphism group of the Tricky Six Puzzle? 286 17.5 Two different group actions 287 17.6 The projective plane of order 4 294 17.7 Buy one, get several free! 299 17.8 The Hoffman–Singleton graph 303 17.9 The Steiner system 푆(5, 6, 12) 305 17.10 A (12,132,4) binary code and Golay’s ternary code 풢12 307 17.11 Conclusions 309 Chapter 18 푆(5, 8, 24) 311 Chapter 19 The Miracle Octad Generator 317 19.1 The Miracle Octad Generator (MOG) 318 19.2 An elementary approach 319 19.3 A more mathematical approach 324 Bibliography 329 Index 338 Preface The Unity of Combinatorics, or TUoC, began as Richard K. Guy’s ple- nary lecture at the combinatorics session of an international conference in 1994. Richard wrote up the lecture as a 30-page paper that appeared in the 1995 session proceedings “Combinatorics Advances”. TUoC was framed as an outline for a series of expository lectures on the various top- ics in the paper. It was a quick but fascinating survey of many different combinatorial problems and their relations to one another. Some years later, Don Albers, Publications Director of the Mathe- matical Association of America (the MAA) approached Richard with a proposal that the MAA publish an expanded version of TUoC in their Carus Monograph Series. Some time later, Don and Richard—now in his 90s—decided to take on a younger coauthor to help get the project moving to completion. That younger author turned out to be the 70ish Ezra (Bud) Brown, a number theorist, combinatorialist, and expository writer of mathematics. Their combined efforts resulted in the book you are now reading. We have divided Richard’s original paper into chapters, expanded the exposition, added explanations, examples, and references, built up back- ground material as needed, and augmented the original paper with ad- ditional connections as seemed appropriate. Eleven chapters consist of previously published articles or excerpts of articles relevant to the topics outlined in TUoC; here is a list of their titles and (authors). • Chapter 4: “Catwalks, Sandsteps, and Pascal pyramids” (Richard K. Guy) • Chapter 5: “Unique rook circuits” (Richard K. Guy and Mark Paul- hus) ix x Preface • Chapter 9: “The groups 푃푆퐿(2, 7) and 퐺퐿(3, 2) and why they are iso- morphic” (Ezra Brown and Nicholas Loehr) • Chapter 11: “Kirkman’s schoolgirls, fields, spreads, and hats” (Ezra Brown and Keith Mellinger) • Chapter 12:“(7, 3, 1) and combinatorics” (Ezra Brown) • Chapter 13:“(7, 3, 1) and normed algebras” (Ezra Brown) • Chapter 14:“(7, 3, 1) and matroids” (David Neel and Nancy Neu- dauer) • Chapter 16: “The (11, 5, 2) biplane, codes, designs, and groups” (Ezra Brown) • Chapter 17: “Rick’s Tricky Six Puzzle: more than meets the eye” (Alex Fink and Richard K. Guy) • Chapter 18:“푆(5, 8, 24)” (Ian Anderson) • Chapter 19: “The Miracle Octad Generator” (Robert Curtis) Chapters 1, 2, 3, 6, 7, 8, 10, and 15 are Guy–Brown collaborations. Thus, we have modeled the structure on John H. Conway and Neil J. A. Sloane’s classic reference work, “Sphere Packings, Lattices and Groups” (see [46]). We begin with sequences, including Langford sequences, Beatty se- quences, the Fibonacci numbers, Pascal’s triangle, and the Catalan num- bers. Topics from graph theory include colorings, packings, embeddings, Kirchoff’s current graphs, and perfect squared squares. Topics from com- binatorial designs include finite projective planes, block designs, Steiner systems, Kirkman’s resolvable designs, difference sets, and symmetries of combinatorial designs. Other topics include Penrose tilings, combi- natorial game theory, matroids, error-correcting codes, matrices with interesting combinatorial properties, and finite geometries. Along the way, we identify and describe some of the many connections between and among these topics. Chapter 15, which treats such curious combinatorial games as Mock Turtles, Turning Corners, Moebius, and Mogul is a bridge between the Preface xi topics treated in Richard’s original paper and the more advanced Chap- ter 17 on Rick’s Tricky Six Puzzle by Alex Fink and Richard.