Volume: 36; 2020; 353 pp; Hardcover
MSC: Primary 05; 11; 20; 51; 52; 91; 94;
Print ISBN: 978-1-4704-5279-7
Product Code: CAR/36
List Price: $65.00
AMS Member Price: $48.75
MAA Member Price: $48.75
Electronic ISBN: 978-1-4704-5667-2
Product Code: CAR/36.E
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The Unity of Combinatorics
Share this pageEzra Brown; Richard K. Guy
MAA Press: An Imprint of the American Mathematical Society
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, finite geometries, Pascal's triangle, Penrose
tilings, error-correcting codes, and many others. Anyone with an
interest in mathematics, professional or recreational, will be sure to
find 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
scientific workers and others with a modest mathematical
background.”
Table of Contents
Table of Contents
The Unity of Combinatorics
- Cover Cover11
- Title page iii5
- Copyright iv6
- Contents v7
- Preface ix11
- Credits and Permissions xiii15
- Introduction 117
- Chapter 1. Blocks, sequences, bow ties, and worms 521
- Chapter 2. Combinatorial games 2137
- Chapter 3. Fibonacci, Pascal, and Catalan 3955
- Chapter 4. Catwalks, Sandsteps, and Pascal pyramids 6177
- Chapter 5. Unique rook circuits 7793
- Chapter 6. Sums, colorings, squared squares, and packings 89105
- Chapter 7. Difference sets and combinatorial designs 107123
- Chapter 8. Geometric connections 123139
- 8.1. A quick tour of projective geometry 123139
- 8.2. Finite projective geometries * and Singer designs 132148
- 8.3. Examples: 𝑛=2, 𝑞=2 and 3 137153
- 8.4. Affine planes and magic squares 140156
- 8.5. Heawood’s map on the torus revisited 142158
- 8.6. Ù and Nim 144160
- 8.7. The automorphism group * of the Fano plane 148164
- Chapter 9. The groups 𝑃𝑆𝐿(2,7) and 𝐺𝐿(3,2) and why they are isomorphic 155171
- Chapter 10. Incidence matrices, codes, and sphere packings 165181
- Chapter 11. Kirkman’s schoolgirls, fields, spreads, and hats 187203
- 11.1. Kirkman’s Schoolgirls Problem 187203
- 11.2. Fifteen young ladies at school 188204
- 11.3. Resolvable block designs * and Kirkman triple systems 189205
- 11.4. Kirkman’s schoolgirls and difference sets 191207
- 11.5. 𝐾=Rats (√2,√3,√5,√7) * and the designs it contains 194210
- 11.6. Spreads in 𝑃𝐺(3,zF ₂) * and the geometry of Kirkman 199215
- 11.7. Fifteen schoolgirls, fifteen hats, * and coding theory 202218
- 11.8. Questions 205221
- Chapter 12. (7,3,1) and combinatorics 209225
- Chapter 13. (7,3,1) and normed algebras 217233
- Chapter 14. (7,3,1) and matroids 229245
- Chapter 15. Coin-turning games and Mock Turtles 243259
- Chapter 16. The (11,5,2) biplane, codes, designs, and groups 257273
- 16.1. “How do you make math * exciting for students?” 257273
- 16.2. Difference sets, block designs, * and biplanes 259275
- 16.3. The automorphism group of the biplane 261277
- 16.4. Incidence matrices, revisited 265281
- 16.5. Error-correcting codes 266282
- 16.6. Steiner systems 270286
- 16.7. Automorphisms, transitivity, * simplicity, and the Mathieu groups 274290
- Chapter 17. Rick’s Tricky Six Puzzle: More than meets the eye 279295
- 17.1. Sliding-block puzzles 279295
- 17.2. What is the exception? 282298
- 17.3. Not much of a puzzle? 283299
- 17.4. What is the automorphism group * of the Tricky Six Puzzle? 286302
- 17.5. Two different group actions 287303
- 17.6. The projective plane of order 4 294310
- 17.7. Buy one, get several free! 299315
- 17.8. The Hoffman–Singleton graph 303319
- 17.9. The Steiner system 𝑆(5,6,12) 305321
- 17.10. A (12,132,4) binary code * and Golay’s ternary code 𝒢₁₂ 307323
- 17.11. Conclusions 309325
- Chapter 18. 𝑆(5,8,24) 311327
- Chapter 19. The Miracle Octad Generator 317333
- Bibliography 329345
- Index 338354
- Copyright 353369
- Back Cover Back Cover1370