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Mathematics of the Rubik's Cube

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Mathematics of the Rubik's cube Associate Professor W. D. Joyner Spring Semester, 1996{7 2 \By and large it is uniformly true that in mathematics that there is a time lapse between a mathematical discovery and the moment it becomes useful; and that this lapse can be anything from 30 to 100 years, in some cases even more; and that the whole system seems to function without any direction, without any reference to usefulness, and without any desire to do things which are useful." John von Neumann COLLECTED WORKS, VI, p. 489 For more mathematical quotes, see the first page of each chapter below, [M], [S] or the www page at http://math.furman.edu/~mwoodard/mquot. html 3 \There are some things which cannot be learned quickly, and time, which is all we have, must be paid heavily for their acquiring. They are the very simplest things, and because it takes a man's life to know them the little new that each man gets from life is very costly and the only heritage he has to leave." Ernest Hemingway (From A. E. Hotchner, PAPA HEMMINGWAY, Random House, NY, 1966) 4 Contents 0 Introduction 13 1 Logic and sets 15 1.1 Logic................................ 15 1.1.1 Expressing an everyday sentence symbolically..... 18 1.2 Sets................................ 19 2 Functions, matrices, relations and counting 23 2.1 Functions............................. 23 2.2 Functions on vectors....................... 28 2.2.1 History........................... 28 2.2.2 3 × 3 matrices....................... 29 2.2.3 Matrix multiplication, inverses.............. 30 2.2.4 Muliplication and inverses................ 31 2.3 Relations.............................. 31 2.4 Counting.............................. 34 3 Permutations 37 3.1 Inverses.............................. 40 3.2 Cycle notation........................... 44 3.3 An algorithm to list all the permutations............ 49 4 Permutation Puzzles 53 4.1 15 puzzle.............................. 54 4.2 Devil's circles (or Hungarian rings)............... 56 4.3 Equator puzzle.......................... 57 4.4 Rainbow Masterball........................ 61 4.5 Rubik's cubes........................... 65 5 6 CONTENTS 4.5.1 2 × 2 Rubik's cube.................... 65 4.5.2 3 × 3 Rubik's cube.................... 67 4.5.3 4 × 4 Rubik's cube.................... 70 4.5.4 n × n Rubik's cube.................... 73 4.6 Skewb............................... 73 4.7 Pyraminx............................. 76 4.8 Megaminx............................. 79 4.9 Other permutation puzzles.................... 83 5 Groups, I 85 5.1 The symmetric group....................... 86 5.2 General definitions........................ 87 5.2.1 The Gordon game..................... 92 5.3 Subgroups............................. 93 5.4 Examples of groups........................ 95 5.4.1 The dihedral group.................... 95 5.4.2 Example: The two squares group............ 97 5.5 Commutators........................... 99 5.6 Conjugation............................ 100 5.7 Orbits and actions........................ 103 5.8 Cosets............................... 107 5.9 Dimino's Algorithm........................ 109 5.10 Permutations and campanology................. 111 6 Graphs and "God's algorithm" 117 6.1 Cayley graphs........................... 118 6.2 God's algorithm.......................... 121 6.2.1 The Icosian game..................... 123 6.3 The graph of the 15 puzzle.................... 123 6.3.1 General definitions.................... 125 6.4 Remarks on applications, NP-completeness........... 128 7 Symmetry groups of the Platonic solids 129 7.1 Descriptions............................ 129 7.2 Background on symmetries in 3-space.............. 131 7.3 Symmetries of the tetrahedron.................. 134 7.4 Symmetries of the cube...................... 135 7.5 Symmetries of the dodecahedron................. 137 CONTENTS 7 7.6 Appendix: Symmetries of the icosahedron and S6 ....... 139 8 Groups, II 143 8.1 Homomorphisms......................... 143 8.2 Homomorphisms arising from group actions.......... 146 8.3 Examples of isomorphisms.................... 147 8.3.1 Conjugation in Sn ..................... 149 8.3.2 Aside: Automorphisms of Sn ............... 150 8.4 Kernels and normal subgroups.................. 151 8.5 Quotient subgroups........................ 153 8.6 Direct products.......................... 155 8.7 Examples............................. 156 8.7.1 The twists and flips of the Rubik's cube........ 156 8.7.2 The slice group of the Rubik's cube........... 157 8.8 Semi-direct products....................... 162 8.9 Wreath products......................... 165 8.9.1 Application to order of elements in Cm wr Sn ...... 167 9 The Rubik's cube and the word problem 169 9.1 Background on free groups.................... 169 9.1.1 Length........................... 170 9.1.2 Trees............................ 171 9.2 The word problem........................ 172 9.3 Generators, relations, and Plutonian robots.......... 173 9.4 Generators, relations for groups of order < 26......... 174 9.5 The presentation problem.................... 180 n 9.5.1 A presentation for Cm >C Sn+1 ............. 181 9.5.2 Proof............................ 183 10 The 3 × 3 Rubik's cube group 185 10.1 Mathematical description of the 3 × 3 cube moves....... 185 10.1.1 Notation.......................... 185 10.1.2 Corner orientations.................... 187 10.1.3 Edge orientations..................... 188 10.1.4 The semi-direct product................. 189 10.2 Second fundamental theorem of cube theory.......... 190 10.2.1 Some consequences.................... 194 10.3 The homology group of the square 1 puzzle........... 195 8 CONTENTS 10.3.1 The main result...................... 196 10.3.2 Proof of the theorem................... 198 11 Other Rubik-like puzzle groups 201 11.1 On the group structure of the skewb.............. 201 11.2 Mathematical description of the 2 × 2 cube moves....... 205 11.3 On the group structure of the pyraminx............ 208 11.3.1 Orientations........................ 209 11.3.2 Center pieces....................... 212 11.3.3 The group structure................... 212 11.4 A uniform approach........................ 213 11.4.1 General remarks..................... 214 11.4.2 Parity conditions..................... 214 12 Interesting subgroups of the cube group 217 12.1 The squares subgroup....................... 218 12.2 P GL(2; F5) and two faces of the cube.............. 220 12.2.1 Finite fields........................ 220 12.2.2 M¨obiustransformations................. 224 12.2.3 The main isomorphism.................. 226 12.2.4 The labeling........................ 227 12.2.5 Proof of the second theorem............... 228 12.3 The cross groups......................... 229 12.3.1 P SL(2; F7) and crossing the cube............ 230 12.3.2 Klein's 4-group and crossing the pyraminx....... 233 13 Crossing the Rubicon 235 13.1 Doing the Mongean shuffle.................... 236 13.2 Background on P SL2 ....................... 236 13.3 Galois' last dream......................... 238 13.4 The M12 generation........................ 239 13.5 Coding the Golay way...................... 240 13.6 M12 is crossing the rubicon.................... 242 13.7 An aside: A pair of cute facts.................. 243 13.7.1 Hadamard matrices.................... 243 13.7.2 5-transitivity....................... 245 CONTENTS 9 14 Appendix: Some solution strategies 247 14.1 The subgroup method...................... 247 14.1.1 Example: the corner-edge method............ 248 14.1.2 Example: Thistlethwaite's method........... 249 14.2 3 × 3 Rubik's cube........................ 250 14.2.1 Strategy for solving the cube............... 250 14.2.2 Catalog of 3 × 3 Rubik's "supercube" moves...... 251 14.3 4 × 4 Rubik's cube........................ 251 14.4 Rainbow masterball........................ 253 14.4.1 A catalog of rainbow moves............... 254 14.5 Equator puzzle.......................... 255 14.6 The skewb............................. 258 14.6.1 Strategy.......................... 258 14.6.2 A catalog of skewb moves................ 258 14.7 The pyraminx........................... 259 14.8 The megaminx.......................... 260 14.8.1 Catalog of moves..................... 261 10 CONTENTS Illustrations in this text (jpg files): chapter 2: VENN1.JPG box2.JPG chapter 3: plotf1b.jpg plotf2b.jpg chapter 4: 15a1.jpg 15b1.jpg 15c1.jpg circles1.JPG equator1.JPG ball3.jpg ball4.jpg ball5.jpg box4.jpg box3.jpg skewb4.jpg tetra2.jpg tetra4.jpg penta1.jpg penta2.jpg chapter 5: square2.jpg cube2.jpg hexa1.jpg bells3.jpg chapter 6: cayley1.jpg cayley2.jpg icosian.jpg 15puzz1.jpg CONTENTS 11 15puzz3.jpg 15puzz9.jpg and 15puzz6.jpg 15puzz2.jpg 15puzz10.jpg and 15puzz11.jpg 15puzz4.jpg and 15puzz5.jpg chapter 7: tetra3.jpg octah2.jpg dodec2.jpg icosah2.jpg icosah3.jpg icosah4.jpg chapter 8: none chapter 9: groups1.JPG coxeter2.JPG chapter 10: rubike1.jpg, rubike2.jpg, rubike3.JPG rubiko1.jpg, rubiko2.jpg sqr1c.JPG sqr1a.JPG and sqr1b.jpg chapter 11: none chapter 12: cube1.jpg crossgp.jpg chapter 13, 14: none 12 CONTENTS Chapter 0 Introduction "The advantage is that mathematics is a field in which one's blunders tend to show very clearly and can be corrected or erased with a stroke of the pencil. It is a field which has often been compared with chess, but differs from the latter in that it is only one's best moments that count and not one's worst." Norbert Wiener EX-PRODIGY: MY CHILDHOOD AND YOUTH Groups measure symmetry. No where is this more evident than in the study of symmetry in 2- and 3-dimensional
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