The Unity of Combinatorics

Home , Fano plane, Steiner system, Zarankiewicz problem

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) license.

For additional information and updates on this book, visit www.ams.org/bookpages/car-36

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© 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. ⃝∞ 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 topics 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 background material as needed, and augmented the original paper with additional 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 isomorphic” (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 sequences, the Fibonacci numbers, Pascal’s triangle, and the Catalan numbers. Topics from graph theory include colorings, packings, embeddings, Kirchoff’s current graphs, and perfect squared squares. Topics from combinatorial 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, combinatorial 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. In Chapter 15, we make mention of binary Golay codes, Mathieu groups, and the Leech lattice of dimension 24. Chapter 16 on the (11, 5, 2) biplane is another way to connect all of the above three objects, as well as the ternary Golay codes and the Steiner systems 푆(4, 5, 11) and 푆(5, 6, 12). We conclude with two chapters on that most unusual of all combinatorial designs, the Steiner system 푆(5, 8, 24). This is a collection of octads (eight-element subsets of a 24-element set Ω) such that every five- element subset of Ω is contained in exactly one octad. Chapter 18 be- gins with a question: based only on this definition, what can we learn about the internal structure of such an 푆(5, 8, 24)? (Quite a lot, as it turns out.) Chapter 19 is Rob Curtis’s beautiful exposition of the Miracle Oc- tad Generator, an object constructed from an 푆(5, 8, 24) that will take a five-element subset of the 24-element set Ω and find the unique octad containing that five-element subset. *** We want to acknowledge and thank the many people who helped make this book a reality. Don Albers invited Richard to expand his 30- page talk into a book for the Carus Monograph Series (CMS) and sug- gested that Richard take on Bud as a coauthor. CMS editor Fernando Gouvêa gave the entire manuscript—and its authors—the benefit of years of experience as both an editor and a published author: Fernando, we can’t thank you enough. Steve Kennedy of the MAA gave us shout- outs of encouragement as well as gentle “noodges” when we needed them. Woody Dudley gave Bud a start on his career as an expository writer, and gave us editorial help early on. Rob Curtis, Ian Anderson, David Neel, and Nancy Neudauer generously contributed their works that became chapters or sections in the book. The CMS editorial board gave individual chapters plenty of scrutiny, and board member Steve Miller went the extra mile on that score. Hendrik Lenstra’s writings on Nim multiplication and other game-theoretic works were an inspi- ration. David Roselle’s graduate class introduced Bud to combinatorics in general and the Fano plane in particular, for which Bud is eternally grateful. Beverly Ruedi of the MAA gave us encouraging shout-outs as well as plenty of practical advice. Christine Thivierge of the AMS helped xii Preface

Bud deal with, and emerge unscathed from, the world of permissions. Finally, we thank our coauthors of those works that became chapters in the book, namely Alex Fink, Nicholas Loehr, Keith A. Mellinger, and Mark M. Paulhus; and coauthors of ours in the list of references, namely Arthur Benjamin, Elwyn R. Berlekamp, John H. Conway, Hallard T. Croft, Kenneth J. Falconer, Christian Krattenthaler, Richard Nowakowski, K. Brooks Reid, Adrian Rice, Gerhard Ringel, Bruce E. Sagan, John Selfridge, Cedric A. B. Smith, and J. W. T. Youngs. Credits and Permissions

• Reprinted/adapted by permission from Springer Nature: Kluwer Academic Publishers, “The Unity of Combinatorics” by Richard K. Guy, in C. J. Colbourn and E. S. Mahmoodian (eds.), Combinatorics Advances, 129–159 © 1995. • Richard K. Guy, “Catwalks, Sandsteps and Pascal Pyramids,” Journal of Integer Sequences 3 (2000), Article 00.1.6. (https://cs.uwaterloo.ca/journals/JIS/) Copyright Statement: Authors agree that by publishing in the Journal of Integer Sequences, they have created an original paper which shall not be published in the same, or substantially the same, form in any other journal without acknowledging prior publication in the Journal. By publishing a paper in the Journal, authors grant the Journal a perpetual, royalty-free license to publish this paper in any collection of Journal papers in any form. Authors retain the copyright of their submitted papers. Authors may, of course, submit their paper to the arXiv or any other preprint archive.

xiii xiv Credits and Permissions

• Ezra Brown and Keith A. Mellinger, “Kirkman’s schoolgirls wearing hats and walking through fields of numbers,” Math. Magazine, 82 (2009), 3–15. © Mathematical Association of America, 2019. All rights reserved. • Ezra Brown, “The many names of (7, 3, 1),” Math. Magazine 75 (2002), 83–94. © Mathematical Association of America, 2019. All rights reserved. • Ezra Brown, “Many more names of (7, 3, 1),” Math. Magazine 88 (2015), 103–120. © Mathematical Association of America, 2019. All rights reserved. • David L. Neel and Nancy Ann Neudauer, “Matroids you have known,” Math. Magazine 82 (2009), 26–41. © Mathematical Associ- ation of America, 2019. All rights reserved.

• Figures 1.6, 1.9, 1.10, 1.11, 1.12, and 1.13, permission given by Fer- nando Q. Gouvêa. • Robert T. Curtis, “The Miracle Octad Generator,” unpublished, permission given by Robert T. Curtis. • Ian Anderson, A first course in combinatorial mathematics (1st, ed.) Clarendon Press Oxford © Oxford University Press 1974, Section 7.3. Reproduced with permission of Oxford Publishing Limited through PSClear.

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211 − 1 = 2047 = 23 ⋅ 89 (11, 5, 2) biplane, 257–273 and binary Golay codes, 268 and difference sets, 259 and perfect numbers, 267 and error-correcting codes, 266– ? 211 − 1 = 2047 = 23 ⋅ 89 270 and Hudalricus Regius, 267 and Galois, 265 and L. E. Dickson, 267 and Mathieu groups, 274–277 (7, 3, 1) and Steiner systems, 270–272 block design, 29, 105, 137, 165, affine planes, 140–142 209, 210, 214 퐴퐺(2, 픽 ), 141 complex and quaternion subal- 5 Alberti, Leon Battisti gebras of the octonions, 225 and perspective in art, 123 difference set, 107, 182, 214 Alekseev, V. E., 89 doubly regular tournament, 214 and partitioning problems Fano plane, 137, 165 푥 + 푦 = 푧, 89 Hamming 퐻(7, 4) code, 172 algebraic number fields, 192–198 Heawood graph, 210 quadratic (degree-2) fields, 194, Heawood’s map on the torus, 210 195 multiplication rule for octonion biquadratic (degree-4) fields, 194, units, 224 195 orthogonal Latin squares of or- octic (degree-8) fields, 194, 217 der 2, 212 Anderson, Ian, 311 P-positions for three-heap Nim, arithmetic progressions, 91 29 Aschbacher, Michael, 122 Rick’s Tricky Six Puzzle, 299–302 and symmetric (79, 13, 2) designs, skew-Hadamard matrix, 215 122 Steiner triple system, 105 associahedrons, 55 the ℱ matroid, 229 7 automorphism, 147 three-circle Venn diagram, 172 of a graph, 147 339 340 Index

outer automorphism, 279, 287 (11, 5, 2), 120, 259 automorphism group, 147, 152, 153 (13, 4, 1), 139 of a block design, 147 (15, 3, 1), 187, 190, 195, 196, 203, of a Steiner system, 274 204 of Rick’s Tricky Six Puzzle, 279, (15, 7, 3), 196, 198 286–293 (16, 6, 2), 111 of the (7, 3, 1) block design, 148– (25, 9, 3), 122 153, 265 (31, 15, 7), 139 on the octahedron, 150–152 Bruck–Ryser–Chowla Theorem, on the regular heptagon, 150 119 of the Fano plane, 147, 148 dual designs, 139

of the Golay code 풢12, 309 elementary relations, 119 of the Hamming code 퐻(7, 4), 253 resolvable designs, 187, 189, 190, of the Heawood graph, 147 196, 203, 204 of the Hoffman–Singleton graph, symmetric designs, 119–122, 133– 303 138, 147, 185, 194, 196–198

of the quadratic residue code 퐶17, Bose’s theorem 255 for projective planes, 135 of 푆(5, 8, 24), 311 on Singer designs, 133, 135 of the (11, 5, 2) biplane, 261–265 Bouton’s Theorem, 28, 34, 244 Bouton, Charles L., 28 Babylonic cuneiform, 51 bow ties, 13–19 Beatty sequences, 8–18, 21, 34, 41, bricks 78 and enumeration problems, 77 complementary, 9–11, 21, 22, 34 and the Fibonacci numbers, 41 Beatty’s Theorem, 9 and the MOG, 319–324 Beatty, Samuel, 10 Brooks, R. L., 94 Berlekamp, Elwyn R., 31 and perfect squared squares, 94 Binet J. P. M. Brown, Ezra, 155, 187, 209, 217, 257 and the Fibonacci numbers, 41 (7, 3, 1) and combinatorics, 209– binomial coefficients, 45–47 215 biplanes (7, 3, 1) and normed algebras, 217– symmetric (푣, 푘, 2) designs, 261 227 block designs, 117–123, 132–136 Kirkman’s Schoolgirls, fields, spreads, (7, 3, 1), 120, 138, 139, 142, 146– and hats, 187–207 152, 194, 209 Index 341

the (11, 5, 2) biplane, codes, de- Mogul, 255 signs, and groups, 257–273 Turning Corners, 247–249 the groups 푃푆퐿(2, 7) and 퐺퐿(3, 2) Turning Turtles, 245–247 and why they are isomorphic, combinations, 46 155–163 combinatorial games, 21–38, 243– Bruce, J. W., 49 255 and the Lucas–Lehmer primal- complete graphs, 90

ity test, 49 genus of 퐾푛, 92 Bruck–Ryser–Chowla Theorem, 119 퐾25, 103 Brunelleschi, Filippo 퐾5, 100, 101, 294 and perspective in art, 123 퐾푛, 90–92 complex numbers, 220 Catalan numbers, 39, 54–65, 69, 71, and the Two-Squares Identity, 220 74 continued fraction, 43 André’s Reflection Principle, 57 Conway worms, 15 associahedrons, 56 Conway, John Horton, 12–19, 35 Dyck paths, 58–60 On Numbers And Games, 36 formula for, 57 and Nim multiplication, 247 grouping of products, 55 and Rick’s Tricky Six Puzzle, 283 in Catalan, 60 and the “alias-alibi” problem, 284 strings of 퐿’s and 푅’s, 57 and the hexacode ℋ, 326 triangulations of regular polygons, and the Hoffman–Singleton graph, 59 304 walks, 60 Conway worms, 15 Cayley, Arthur kites and darts, 15 and Kirkman’s Schoolgirls Prob- monads, duads, and synthemes, lem, 206 291 and partitioning polygons, 64 short and long bow ties, 15 and the Cayley–Dickson construc- the Ace, 15 tion, 223–226 current graphs, 93–97 and the octonions, 181, 223 codewords, 167–173, 203, 205 Dürer, Albrecht coil diagrams, 91 and perspective in art, 124 coin-turning games, 243–255 darts, 13–15 Mock Turtles, 250–254 Dawson’s Chess, 31 Moebius, 254–255 de Bruijn cycles, 115–116 342 Index de Bruijn graph, 116 dodecads, 314 de Moivre, Abraham and 푆(5, 8, 24), 314 and the Fibonacci numbers, 41 Dots and Boxes, 31 Denniston, R. H. F., 206 duads, 287–297 and Kirkman parades, 206 dual graphs, 96, 97, 144 Desargues, Girard Duijvestijn, A. J. W., see perfect squared and Desargues’ Theorem, 124 square, 94 and projective geometry, 124 Dyck paths, 58–60, 77, 78 Dickson, Leonard Eugene, 155 ? eigenvalues and eigenvectors and 211 −1 = 2047 = 23⋅89, 268 and the Fibonacci numbers, 41 and 푃푆퐿(2, 7), 155 Emmy Noether Boarding School, 188 and the Cayley–Dickson construc- error-correcting codes, 167–174, 266– tion, 223–226 270 difference sets, 105, 107–113, 115– Euclid 117, 120–122 and the Elements, 42 (7, 3, 1), 107, 110, 113–115, 117 and the GCD algorithm, 42 (11, 5, 2), 108, 110, 113, 114, 120, Euler, Leonard 260 Euler’s polyhedral formula, 97 (13, 4, 1), 110 Euler, Leonhard (15, 7, 3), 115 and Latin squares, 212 (16, 6, 2), 38, 111, 112, 183 Euler characteristic, 98 (19, 9, 4), 110 Euler circuit, 116 (21, 5, 1), 115, 294 Euler’s conjecture disproved, 212 (23, 11, 5), 110 Euler’s misattribution, 51 (31, 15, 7), 110 Euler’s polyhedral formula, 97, 99, (37, 9, 2), 115 99 (40, 13, 4), 115 for nonplanar graphs, 99 (43, 21, 10), 110 sketch of a proof, 98 Hadamard, 111, 183 evil numbers, 250–252 Paley, 109, 183, 214 planar, 108, 109 face-centered-cubic lattice, 176–179 Singer, 116, 135, 136 Fano plane, 132, 137, 138, 144–148, Whiteman, 112 152, 155, 156, 165, 172, 182, 183, Diophantus, 218 238, 300 Two-Squares Identity, 220 Fano, Gino, 132 directed Euler circuit, 116 Fermat, Pierre Index 343

and Fermat’s Little Theorem, 50 Galois, Évariste, 194, 217 and the Fermat powers 22푛 , 36, Fundamental Theorem of Galois 247 Theory, 207 and the Pell Equation, 51 Galois group, 207, 217 Fibonacci Galois theory, 194 Liber abaci, 39 generalized Fibonacci numbers, 48– numbers, 12, 39–44, 47 53 Beatty sequences, 41 Brahmagupta–Pell numbers, 50 counting problems, 40–41 Chebyshev polynomials, 52, 53 de Moivre–Binet Formula, 41 Jacobsthal numbers, 50 Euclidean algorithm, 42 Mersenne numbers, 49 Virahanka, 40 genus, 91 origin of the name, 40 of a graph, 91

rabbits, 39 of the complete graph 퐾푛, 92 field, 126, 127, 129, 131, 133, 145 Golay codes finite field, 129, 130, 133, 146 풢11, 170, 259, 266, 269, 270, 308 construction, 129 풢12, 170, 259, 266, 269, 270, 283, finite projective planes, 132–140, 144– 307, 308 152, 183, 211, 230, 238, 261, 283, and Rick’s Tricky Six Puzzle, 294 307 finite projective spaces, 127–152, 198– 풢23, 170, 255, 259, 266, 269, 270 201, 238, 239 풢24, 170, 255, 259, 266, 269, 270, 푃퐺(2, 픽2), 132 272, 325, 326 푃퐺(3, 픽2) sits inside 푃퐺(2, 픽4), 300 Golay, Marcel, 167 푃퐺(3, 픽2), 183, 185, 189, 199–201, see also Golay codes, 167 205 golden section 휙, 8, 12, 14, 43

푃퐺(푛, 픽푞), 137 graph coloring, 91–93 Fink, Alex, 279 proper, 91–93 Rick’s Tricky Six Puzzle, 279–309 graph covering, 90, 99–103 Fisher, R. A., 118 graph decomposition, 99–105

Fisher’s Inequality, 119, 139, 166 of 퐾25 into 100 triangles, 103 Four Color Theorem, 91 of 퐾5 into two 5-cycles, 101 fractals, 53 of 퐾7 into seven triangles, 105 and Pascal’s triangle, 53 graph packing, 90, 99–104 Fundamental Theorem of Matroids graphs, 56 and Greedy Algorithms, 241 associahedron graph, 55 344 Index

connected, 71, 97 Multiple Transitivity Theorem, 275 de Bruijn, 116 multiply transitive, 275 genus of a graph, 91 푃퐺퐿(2, 5), 279 Moore graphs, 303 푃푆퐿(2, 7), 147, 148, 153, 155, 156, paths, 58 159–161, 163, 275 planar, 56, 97–99 푃푆퐿(2, 11), 257 walks, 78 푃푆퐿(2, 17), 254 Graves, John T. 푃푆퐿(3, 4), 156

and the Eight-Squares Identity, 푆5, 279 223 푆5 sits specially in 푆6, 279 and the octonions, 181, 222 푆6, 279 greedy algorithms simple, 275, 276 Dijkstra and minimal connectors, 푆퐿퐹(7), 159–163 240 why 푃푆퐿(2, 7) and 퐺퐿(3, 2) are iso- Euler and the Königsberg Bridges, morphic, 152–163 240 Grundy number, 31, 33–35, 244, 245, Ford–Fulkerson and max-flow min- 247–252, 254, 255 cut, 240 Grundy’s Theorem (see Sprague–Grundy Hall and maximal matchings, 240 Theorem), 32 Hungarian Assignment algorithm,Grundy, Patrick M., 31 240 Guy, Richard K., 61, 77, 279 Kruskal and minimal spanning Catwalks, Sandsteps and Pascal trees, 240 Pyramids, 61–75 Menger’s theorem, 240 Rick’s Tricky Six Puzzle, 279–309 Prim and minimal spanning trees, unique rook circuits, 77–87 240 groups Hadamard difference sets, 111 Hadamard matrix, 111, 181–184, 215 퐴8, 156 퐺퐿(3, 2), 147, 148, 152, 153, 155– normalized, 182 158, 161, 163, 253 regular, 183 skew-Hadamard matrix, 215 퐺퐿(픽8), 157, 161, 163 푘-transitive groups, 275 Hadamard, Jacques, 182 Hamilton, William R. 푀11, 258, 275, 277 and the quaternions, 221 푀12, 258, 275, 277 could not multiply triples, 221 푀24, 255, 258, 275, 277 Mathieu groups, 258, 274–277 Hamming Index 345

distance, 168, 179, 307 of 푃퐺(3, 픽2), 183 sphere, 168, 169, 180 inner automorphism, 287 weight, 168, 180 Isaacs’s game, 32 Hamming codes, 170–174, 179, 201, Isaacs, Rufus, 21 203–205, 252, 253 isomorphic (3, 1) triplication code, 170 block designs, 139, 205 (7, 4) code, 170 finite fields, 129 (15, 11) code, 203 games, 21 extended Hamming code, 179, 252, graphs, 99 253 groups, 265, 275

Hamming, Richard, 167 푃퐺퐿(2, 5) and 푆5, 279 Hats Games, 202, 203 푃푆퐿(2, 7) and 퐺퐿(3, 2), 152

The Three Hats Game, 202 the alternating group 퐴8 and The Fifteen Hats Game, 203 the linear group 퐿4(2), 325 Heawood graph, 93, 99, 142, 143 Hadamard matrices, 182 Heawood’s conjecture, 91 matroids, 236 Heawood, Percy J., 91 subalgebras hexacode, 326–327 of the octonions, 225 hexads of the sedenions, 226 and 푆(5, 6, 12), 305, 307 isomorphism Hoffman–Singleton graph, 303–305 between 퐺퐿(3, 2) and Aut(7, 3, 1), homomorphisms, 152–163 153

Hudalricus Regius, 268 between 푃퐺퐿(2, 5) and 푆5, 279 ? and 211 −1 = 2047 = 23⋅89, 268 between 푃푆퐿(2, 7) and 퐺퐿(3, 2), hyperplanes, 133, 134 155–163 between graphs, 99 incidence graph, 210 between projective spaces and Kirk- incidence matrix, 265, 266, 270 man designs, 201

of (7, 3, 1), 166 exceptional: 퐴8 ≅ 퐿4(2), 324 of a block design, 166, 190 of combinatorial games, 22 of a finite projective geometry, 165 the Galois correspondence, 207 of the (15, 11) Hamming code, 205 of the Fano plane, 165, 173, 182, Kepler conjecture, 176, 178 183 proof by Hales et al., 178 of the Kirkman design, 190, 203, Kirchhoff’s current law, 93, 95 204 Kirchhoff, Gustav, 93 346 Index

Kirkman parades, 206 the groups 푃푆퐿(2, 7) and 퐺퐿(3, 2) Kirkman’s Schoolgirls Problem, 187– and why they are isomorphic, 206 155–163 and algebraic number fields, 194– Lucas numbers, 12 198 and difference sets, 191 푀11, 258, 275, 277 and Hamming codes, 203, 204 푀12, 258, 275, 277 and hats, 201–205 푀24, 255, 258, 275, 277 and Rick’s Tricky Six Puzzle, 299– magic squares, 140–142, 191 303 3 × 3, 140, 191 and spreads, 199, 201 5 × 5, 141 and the group (ℤ/2ℤ)4, 207 pandiagonal, 141 solutions, 188, 192, 196, 200, 203, map, 71 207 Hamiltonian, 71 Kirkman, Thomas P., 133, 185, 187– rooted, 71 191, 196, 200, 201, 205 matroids, 229–242 and Kirkman designs, 196, 200, and bases, 234 201, 205 and fields, 237 kites, 13–15 and greedy algorithms, 239 and independence, 236 Langford sequences, 6, 78, 90 and independent sets, 231, 232 Langford, Dudley, 5 and optimal spanning trees, 230 son playing with blocks, 5, 89, 327 and projective planes, 238 Laplace transform coefficients, 63 and spanning trees, 233 Latin squares, 141 and the Fano plane, 230, 238 and finite projective planes, 211 basis axioms, 234 Euler’s conjecture, 212 cycle matroid, 235 Euler’s conjecture disproved, 212 graph matroid, 235 of order 푛, 211 origin of the name, 235 orthogonal, 141, 211 uniform matroid, 235 Leech lattice vector matroid, 235 8 퐸8 in ℝ , 179, 180 Meeker, Darcy Λ in ℝ24, 179, 315, 327 and perfect squared squares, 94 Leech, John, 179 Mellinger, Keith A., 187 Leonardo of Pisa, see Fibonacci Kirkman’s Schoolgirls, fields, spreads, Loehr, Nicholas A., 155 and hats, 187–207 Index 347

Mersenne numbers, 49 N-positions, 25, 26, 28, 29, 31, 34, Lucas–Lehmer primality test, 49 244, 248, 249, 251 Mersenne primes, 49 Nim, 23, 24, 27, 28, 34, 37, 144–146, mex, 31, 32, 35, 247, 250 243–253 Miracle Octad Generator (MOG) addition, see Nim sum and parity, 319 multiplication, 35–38, 247, 249 Miracle Octad Generator (MOG) product, 247, 248 octads of 푆(5, 8, 24), 317 sum, 27, 34–36, 144–146, 201, 203, Miracle Octad Generator (MOG), 317– 204, 244, 246–248, 251, 252 327 nimbers, 27, 28, 31, 144–146, 203, a picture of 푆(5, 8, 24), 317, 319 244, 247–249

and the Mathieu group 푀24, 317 nonexistence of three-dimensional and 푆(3, 4, 16), 319 real normed algebras, 226 and apples in buckets, 319 nonisomorphic simple groups of the and parity, 324 same order, 156 and the exceptional isomorphism normed algebra, 219

퐴8 ≅ 퐿4(2), 324 and the hexacode, 326 octads, 312 and the Leech lattice in ℝ24, 327 and 푆(5, 8, 24), 312–315 bricks (eight-element subsets), 319– nonobvious properties, 312–315 324 octonions, 181, 222, 223 elementary construction, 319 and (7, 3, 1), 224 octads of 푆(5, 8, 24), 326 and sums of eight squares, 223 tetrads, 319–324 odious numbers, 250–255 the fundamental correspondence, OEIS, 61–63, 71 319 Online Encyclopedia of Integer Se- Mock Turtles, 250–254 quences, see OEIS Moebius transformations, 159, 254, Orbit-Stabilizer Theorem, 148–152, 265, 279, 285, 286, 289 254, 265 MOG, see Miracle Octad Generator orbits, 114, 115, 148–152 monads, 287–306 planetary, 176 Moore graphs, 303 P-positions, 25, 26, 28–33, 35, 244– multipliers, 113–115 255 The Multiplier Theorem, 114 packings, see graph packings, sphere Mystical Hexagram Theorem, 124, packings 125 in 푃퐺(3, 픽2), 189 348 Index partitioning problems, 63, 64, 89, on sliding block puzzles, 280 91, 99 on the (11, 5, 2) biplane, 261–265 partitioning polygons, 63, 64 on the (7, 3, 1) block design, 147– 푥 + 푦 = 푧, 90, 91, 99 153 푥 + 푦 = 2푧, 90 that preserve adjacency in a graph, 푥 + 푦 = 3푧, 91 147 Pascal’s triangle, 39, 44–54, 60 perspective art and the Catalan numbers, 55 and projective geometry, 125 mod 2 triangle, 53 perspective drawing, 123–125

origins, 47 푃퐺(2, 픽2), 132, 144 quarter-pyramid, 69, 70 푃퐺(2, 픽3), 139, 140 semi-pyramid, 66, 67, 74 푃퐺(2, 픽4), 294 semi-triangle, 63 푃퐺(2, 픽5), 141 Pascal, Blaise, 124 푃퐺(3, 픽2), 183, 185, 189, 199–201, and projective geometry, 124 205

and the Mystical Hexagram The- 푃퐺(3, 픽푞), 199, 200 orem, 124 푃퐺(푛, 픽), 126–129

Paulhus, Mark M., 77 푃퐺(푛, 픽푞), 130–137 unique rook circuits, 77–87 Piero della Francesca Penrose tiling, 13–15, 17, 18, 20 and perspective in art, 124 ace, 15, 17 Plücker, Julius aperiodic set, 13 and plane cubic curves, 117 bow ties, 12–17 and the (9, 12, 4, 3, 1) block design, Conway worms, 17, 18, 20 118, 120, 122 darts, 13–15 planar graph, 92, 97–99 kites, 13–15 point classes, 126, 129 rhombus, 14 Poncelet, Jean-Victor perfect squared squares, 94–96 and projective geometry, 125 permutations, 46 positions, see P-positions, N-positions, on an error-correcting code, 253 terminal positions and multipliers, 114–115 projective geometry, 131 on ℤ/푣ℤ, 113 projective geometry, 123 on a set, 46, 114 and perspective art, 125 on block designs, 139 projective planes, 86, 126, 132, 135, on Rick’s Tricky Six Puzzle, 282– 137–141, 147, 152, 153 285 axioms, 126 Index 349 projective spaces, 126–137 and the projective plane of order 4, definition, 126–127 294–300 dual subspaces, 128 the six equivalence classes, 284– 푃퐺(푛, 픽), 126–129 285

푃퐺(푛, 픽푞), 130 Ringel–Youngs Theorem, 92 Subspace Correspondence The- rook circuits, 77–87 orem, 127 4-corner, 80–82 projective transformation, 125 guideposts, 79, 81, 83, 84, 86 푃푆퐿(2, 5), 265 the 4-Corner Principle, 80, 83, 86 푃푆퐿(2, 7), 265 the Cul-de-sac Principle, 80 푃푆퐿(2, 11), 265 the Parity Principle, 79, 81 푃푆퐿(2, 푝) the Two Neighbor Principle, 79 and Galois, 265 rook tour, 78 quaternions, 181, 221, 222 푆(5, 8, 24), 259, 277, 311–315

and (7, 3, 1), 225 and 푀24, 311 and the Golay code 풢24, 272 Rank-Nullity Theorem, 128 internal structure, 311–315 resolvable block designs, see block Restriction Theorem, 312 designs Sands, Bill Restriction Theorem for Steiner Sys- Sands walks, 63 tems, 312 Sands’ Theorem, 61 Rick’s Tricky Six Puzzle, 279–309 walks of length 푛, 67 and 퐾5, 294 sedenions, 225 and 푆(5, 6, 12), 305 and a (15, 3, 1) block design, 226 and five-coloring the edges of 퐾6, Shannon, Claude, 167, 170 288 and the first Hamming code, 170 and Kirkman’s Schoolgirls Prob- Singer designs, 132, 133, 135, 139 lem, 299–303 Singer difference sets, 110, 136 and outer automorphisms, 287– Singer, James, 135–140 294 Singer’s two theorems on differ- and Rick Wilson’s theorem, 282 ence sets, 135 and the Fano plane, 299–302 Skolem sequences, 7, 8, 78, 89, 90 and the Golay code 풢12, 307 Skolem, Thoralf, 8 and the Hoffman–Singleton graph, and Skolem sequences, 7, 8, 78, 303–304 89, 90 350 Index

and Steiner Triple Systems, 89 with 푝 > 3, 271 Sloane, Neil A. J., 61 Steiner triple systems, 90, 103–105, and the OEIS, 61 107, 120, 187, 190, 206 Smith, C. A. B., 94 Stirling numbers of the second kind, and perfect squared squares, 94 64 sphere packing, 174 Stone, A. H., 94 density, 175 and perfect squared squares, 94 in ℝ2, 175 Subspace Correspondence Theorem, in ℝ3, 176, 178 127, 130, 134 in ℝ4, 179 sums of squares, 217–226 in ℝ8, 179–181 Two-Squares Identity, 218 in ℝ24, 179, 315 Four-Squares Identity, 218 kissing number, 178 Eight-Squares Identity, 219 lattice packing, 177 no Sixteen-Squares Identity, 226 Sprague, Roland P., 34, 94 Sylvester, J. J., 287 Sprague–Grundy Theorem, 34, 37, monads, duads, and synthemes, 244, 251 287 spreads, 189 symmetric designs, see block designs

in 푃퐺(3, 픽2), 198–200 symmetry, 13, 16, 33, 67, 70, 103, stabilizers, 149, 151, 152 146, 152, 176, 190, 257, 258, 261, Steiner systems 262, 274 Restriction Theorem, 271, 274, 312 synthemes, 297 푆(2, 3, 7), 270, 274 푆(3, 4, 8), 156, 270 Taxicab metric, 72 푆(4, 5, 11), 271, 273 terminal positions, 24–26, 32, 243, 245, 250 and 푀11, 273 푆(5, 6, 12), 271, 273, 305 three-dimensional real normed algebras, nonexistence of, 226 and 푀12, 273 and Rick’s Tricky Six Puzzle, tiling, 13, 15 305 aperiodic set, 14 푆(5, 7, 28), 271 nonperiodic, 13 푆(5, 8, 24), 259, 271, 272, 274, 277, periodic, 13 311–315 prototile, 13, 14 tile, 13 and 풢24, 272 tournaments, 213 and 푀24, 275 푆(푝, 푞, 푟), 270 doubly regular, 213–215 Index 351

regular, 213 transitive, 213 transitivity of groups, see groups Turning Corners, 243 Turning Turtles, 243, 245–249 Tutte, W. T., 94 and perfect squared squares, 94 vanishing point, 123–125 vector spaces, 126–131, 152, 229, 231, 232, 235, 236, 238, 241 Venn diagram, 172 Viazofska, Maryna and sphere-packing in ℝ8 and ℝ24, 315 Virahanka, 40 walks, 61–75 one-dimensional, 63, 69 two-dimensional, 74 three-dimensional, 74 four-dimensional, 74 on lattice points, 61–75 using the Taxicab metric, 72 Winning Ways, 31 Wyt Queens, 23 Wythoff’s game, 21–23, 32, 33, 35, 37 Wythoff, Willem Abraham, 21

Zarankiewicz’s problem, 166

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Library of Congress Cataloging-in-Publication Data Names: Brown, Ezra (Ezra A.), author. | Guy, Richard K., author. Title: The unity of combinatorics / Ezra Brown, Richard K. Guy. Description: Providence, Rhode Island : MAA Press, an imprint of the American Math- ematical Society, [2020] | Series: The Carus mathematical monographs, 0069-0813 ; volume 36 | Includes bibliographical references and index. Identifiers: LCCN 2019056911 | ISBN 9781470452797 (hardcover) | ISBN 9781470456672 (ebook) Subjects: LCSH: Combinatorial analysis. | Game theory. | Error-correcting codes (Infor- mation theory) | AMS: Combinatorics {For finite fields, see 11Txx} – Designs and configurations {For applications of design theory, see 94C30} – Block designs [See also 51E05, 62K10]. | Combinatorics {For finite fields, see 11Txx} – Designs and configurations {For applications of design theory, see 94C30} – Difference sets (number-theoretic, group-theoretic, etc.) [See also 11B13]. | Combinatorics {For finite fields, see 11Txx} – Designs and configurations {For applications of design theory, see 94C30} – Matroids, geometric lattices [See also 52B40, 90C27]. | Combinatorics {For finite fields, see 11Txx} – Graph theory {For applications of graphs, see 68R10, 81Q30, 81T15, 82B20, 82C20, 90C35, 92E10, 94C15} – Coloring of graphs and hypergraphs. | Number theory – Se- quences and sets. | Group theory and generalizations – Permutation groups – Finite automorphism groups of algebraic, geometric, or combinatorial structures [See also 05Bxx, 12F10, 20G40, 20H30, 51-XX]. | Geometry {For algebraic geometry, see 14-XX} – Finite geometry and special incidence structures – Steiner systems. | Convex and discrete geometry – Discrete geometry – Tilings in 2 dimensions [See also 05B45, 51M20]. | Game theory, economics, social and behavioral sciences – Game theory – Combina- torial games. | Information and communication, circuits – Theory of error-correcting codes and error-detecting codes. Classification: LCC QA164 .B758 2020 | DDC 511/.6–dc23 LC record available at https://lccn.loc.gov/2019056911

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Combinatorics, or the art and science of counting, is a vibrant and active area of pure mathematical research with many applications. The Unity of Combinatorics succeeds in showing that the many facets of combinatorics are not merely isolated instances of clever tricks but that they have numerous connections and threads weaving them together to form a beautifully patterned tapestry of ideas. Topics include combinatorial designs, combinatorial games, matroids, difference sets, Fibonacci numbers, fi nite geometries, Pascal’s triangle, Penrose tilings, error-correcting codes, and many others. Anyone with an interest in mathematics, profes- sional or recreational, will be sure to fi nd this book both enlightening and enjoyable. Few mathematicians have been as active in this area as Richard Guy, now in his eighth decade of mathematical productivity. Guy is the author of over 300 papers and twelve books in geometry, number theory, graph theory and combinatorics. In addition to being a life- long number-theorist and combinatorialist, Guy’s co-author, Ezra Brown, is a multi-award-winning expository writer. Together, Guy and Brown have produced a book that, in the spirit of the founding words of the Carus book series, is accessible “not only to mathematicians but to scientifi c workers and others with a modest mathematical background.”

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