Permutation Puzzles: A Mathematical Perspective 15 Puzzle, Oval Track, Rubik’s Cube and Other Mathematical Toys Lecture Notes Jamie Mulholland Department of Mathematics Simon Fraser University c Draft date June 30, 2016 Contents Contents i Preface vii Greek Alphabet ix 1 Permutation Puzzles 1 1.1 Introduction . .1 1.2 A Collection of Puzzles . .2 1.3 Which brings us to the Definition of a Permutation Puzzle . 10 1.4 Exercises . 12 2 A Bit of Set Theory 15 2.1 Introduction . 15 2.2 Sets and Subsets . 15 2.3 Laws of Set Theory . 16 2.4 Examples Using SageMath . 18 2.5 Exercises . 19 3 Permutations 21 3.1 Permutation: Preliminary Definition . 21 3.2 Permutation: Mathematical Definition . 23 3.3 Composing Permutations . 26 3.4 Associativity of Permutation Composition . 28 3.5 Inverses of Permutations . 29 3.6 The Symmetric Group Sn ........................................ 33 3.7 Rules for Exponents . 33 3.8 Order of a Permutation . 35 3.9 Exercises . 36 i ii CONTENTS 4 Permutations: Cycle Notation 39 4.1 Permutations: Cycle Notation . 39 4.2 Products of Permutations: Revisited . 41 4.3 Properties of Cycle Form . 42 4.4 Order of a Permutation: Revisited . 43 4.5 Inverse of a Permutation: Revisited . 44 4.6 Summary of Permutations . 46 4.7 Working with Permutations in SageMath . 46 4.8 Exercises . 47 5 From Puzzles To Permutations 51 5.1 Introduction . 51 5.2 Swap ................................................... 52 5.3 15-Puzzle . 54 5.4 Oval Track Puzzle . 55 5.5 Hungarian Rings . 58 5.6 Rubik’s Cube . 59 5.7 Exercises . 62 6 Permutations: Products of 2-Cycles 67 6.1 Introduction . 67 6.2 Product of 2-Cycles . 68 6.3 Solvability of Swap . 69 6.4 Exercises . 70 7 Permutations: The Parity Theorem 71 7.1 Introduction . 71 7.2 Variation of Swap . 73 7.3 Proof of the Parity Theorem . 74 7.4 Exercises . 79 8 Permutations: An and 3-Cycles 83 8.1 Swap Variation: A Challenge . 83 8.2 The Alternating Group An ....................................... 83 8.3 Products of 3-cycles . 85 8.4 Variations of Swap: Revisited . 86 8.5 Exercises . 88 CONTENTS iii 9 The 15-Puzzle 91 9.1 Solvability Criteria . 91 9.2 Proof of Solvability Criteria . 93 9.3 Strategy for Solution . 96 9.4 Exercises . 97 10 Groups 103 10.1 Group: Definition . 103 10.2 Some Everyday Examples of Groups . 106 10.3 Further Examples of Groups . 108 10.4 Exercises . 120 11 Subgroups 125 11.1 Subgroups . 125 11.2 Examples of Subgroups . 126 11.3 The Center of a Group . 127 11.4 Lagrange’s Theorem . 128 11.5 Cyclic Groups Revisited . 129 11.6 Cayley’s Theorem . 130 11.7 Exercises . 131 12 Puzzle Groups 135 12.1 Puzzle Groups . 135 12.2 Rubik’s Cube . 136 12.3 Hungarian Rings . 143 12.4 15-Puzzle . 144 12.5 Exercises . 145 13 Commutators 147 13.1 Commutators . 147 13.2 Creating Puzzle moves with Commutators . 148 13.3 Exercises . 156 14 Conjugates 161 14.1 Conjugates . 161 14.2 Modifying Puzzle moves with Conjugates . 163 14.3 Exercises . 168 15 The Oval Track Puzzle 173 iv CONTENTS 15.1 Oval Track with T = (1 4)(2 3) ..................................... 173 15.2 Variations of the Oval Track T move . 184 15.3 Exercises . 185 16 The Hungarian Rings Puzzle 189 16.1 Hungarian Rings - Numbered version . 189 16.2 Building Small Cycles: Tools for Our End-Game Toolbox . 191 16.3 Solving the end-game . 196 16.4 Hungarian Rings - Coloured version ..