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# Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other…

## by David Joyner

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Indeholder "Preface", "Acknowledgements", "Where to begin",
"Chapter 1: Elementary my dear Watson", "1.1 You have a logical mind if...", "1.1.1 'You talking to me?'", "1.2 Elements, my dear Watson",
"Chapter 2: 'And you do addition?'", "2.2.1 History", "2.2.2 3 × 3 matrices", "2.2.3 m × n matrices", "2.2.4 Multiplication and inverses", "2.2.5 Determinants", "2.3 Relations", "2.4 Counting and mathematical induction",
"Chapter 3: Bell ringing and other permutations", "3.1 Definitions", "3.2 Inverses", "3.3 Cycle notation", "3.4 An algorithm to list all the permutations", "3.4.1 Why Steinhaus's algorithm works", "3.4.2 A side order of dessert: cake cutting", "3.5 Permutations and bell ringing",
"Chapter 4: A procession of permutation puzzles", "4.1 15 Puzzle", "4.2 The Hockeypuck puzzle", "4.3 Rainbow Masterball", "4.4 Pyraminx", "4.5 Rubik's Cubes", "4.5.1 2 × 2 × 2 Rubik's Cube", "4.5.2 3 × 3 × 3 Rubik's Cube", "4.5.3 Some two-player Rubik's Cube games", "4.6 Skewb", "4.7 Megaminx", "4.8 Other permutation puzzles",
"Chapter 5: What's commutative and purple?", "5.1 The unit quaternions", "5.2 Finite cyclic groups", "5.3 The dihedral group", "5.4 The symmetric group", "5.5 General definitions", "5.5.1 Cauchy's theorem", "5.5.2 The Gordon game", "5.6 Subgroups", "5.7 Puzzling examples", "5.7.1 2", "5.7.2 Example: The two squares group", "5.8 Commutators", "5.9 Conjugation", "5.10 Orbits and actions", "5.11 Cosets", "5.12 Campanology, revisited", "5.13 Dimino's algorithm",
"Chapter 6: Welcome to the machine", "6.1 Some history", "6.2 Merlin's Machine", "6.2.1 The machine", "6.2.2 The rectangular graph", "6.3 Variants", "6.3.1 Merlin's Magic and 3 × 3 Lights Out", "6.3.2 The Orbix", "6.3.3 Keychain Lights Out", "6.3.4 Lights Out", "6.3.5 Deluxe Lights Out", "6.3.6 Lights Out Cube", "6.3.7 Alien Tiles", "6.3.8 Theoretical generalizations and variants", "6.4 Finite-state machines", "6.5 The mathematics of the machine", "6.5.1 The square case", "6.5.2 Downshifting", "6.5.3 The rectangular case", "6.5.4 Alien Tiles again", "6.5.5 Orbix, revisited", "6.5.6 Return of the Keychain Lights Out",
"Chapter 7: 'God's algorithm' and graphs", "7.1 In the beginning...", "7.2 Cayley graphs", "7.3 God's algorithm", "7.4 The graph of the 15 Puzzle", "7.4.1 General definitions", "7.4.2 Remarks on applications",
"Chapter 8: Symmetry and the Platonic solids", "8.1 Descriptions", "8.2 Background on symmetries in 3-space", "8.3 Symmetries of the tetrahedron", "8.4 Symmetries of the cube", "8.5 Symmetries of the dodecahedron", "8.6 Some thoughts on the icosahedron", "8.7 901083404981813616 cubes",
"Chapter 9: The illegal cube group", "9.1 Functions between two groups", "9.2 Group actions", "9.3 When two groups are really the same", "9.3.1 Conjugation in Sn", "9.3.2 ... and a side order of automorphisms, please", "9.4 Kernels are normal, some subgroups are not", "9.4.1 Examples of non-normal subgroups", "9.4.2 The alternating group", "9.5 Quotient groups", "9.6 Dabbling in direct products", "9.6.1 First fundamental theorem of cube theory", "9.6.2 Example: cube twists and flips", "9.6.3 Example: the slice group of the cube", "9.6.4 Example: the slice group of the Megaminx", "9.7 A smorgasbord of semi-direct products", "9.8 A reification of wreath products", "9.8.1 The illegal Rubik's Cube group", "9.8.2 Elements of order d in Cm wr Sn",
"Chapter 10: Words which move", "10.1 Words in free groups", "10.1.1 Length", "10.1.2 Trees", "10.2 The word problem", "10.3 Presentations and Plutonian robots", "10.4 Generators, relations for groups of order ( )
bnielsen | Jan 8, 2013 |
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## Wikipedia in English (3)

 God's algorithmRubik's Cube group Wikipedia:Reference desk/Archives/Mathematics/2007 November 15
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#### Amazon.com Product Description (ISBN 0801869471, Paperback)

Group theory deals with symmetry, in the most abstract form possible. It is a core part of the undergraduate math curriculum, and forms part of the training of theoretical physicists and chemical crystallographers. Group theory has tended to be very dry -- until now. David Joyner uses mathematical toys (primarily the Rubik's Cube and its more modern cousins, the Megaminx, the Pyraminx, and so on) as well as other mathematical examples (e.g., bell ringing) to breathe new life into a time-honored subject.

"Why," asks the author, "should two such different topics, mechanical puzzles and abstract group theory, be related? This book takes the reader on an intellectual trip to answer this curiosity." Adventures in Group Theory will not only appeal to all math enthusiasts and interested general readers but will also find use in the classroom as a wonderful supplementary text in any abstract algebra or group theory course.

Group theory deals with symmetry, in the most abstract form possible. It is a core part of the undergraduate math curriculum, and forms part of the training of theoretical physicists and chemical crystallographers. Group theory has tended to be very dry--until now. David Joyner uses mathematical toys (primarily the Rubik's Cube and its more modern cousins, the Megaminx, the Pyraminx, and so on) as well as other mathematical examples (e.g., bell ringing) to breathe new life into a time-honored subject.… (more)

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