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 May 7, 2012 Contents Contents i Preface ix Greek Alphabet xi 1 Permutation Puzzles 1 1.1 Introduction . .1 1.2 A Collection of Puzzles . .2 1.2.1 A basic game, let’s call it Swap ................................2 1.2.2 The 15-Puzzle . .4 1.2.3 The Oval Track Puzzle (or TopSpinTM)............................5 1.2.4 Hungarian Rings . .7 1.2.5 Rubik’s Cube . .8 1.3 Which brings us to the Definition of a Permutation Puzzle . 12 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 Sage . 17 2.5 Exercises . 19 3 Permutations 21 3.1 Permutation: Preliminary Definition . 21 3.2 Permutation: Mathematical Definition . 23 3.2.1 Functions . 23 3.2.2 Permutations . 24 3.3 Composing Permutations . 26 i ii CONTENTS 3.4 Associativity of Permutation Composition . 28 3.5 Inverses of Permutations . 29 3.5.1 Inverse of a Product . 31 3.5.2 Cancellation Property . 32 3.6 The Symmetric Group Sn ........................................ 33 3.7 Rules for Exponents . 33 3.8 Order of a Permutation . 34 3.9 Exercises . 35 4 Permutations: Cycle Notation 37 4.1 Permutations: Cycle Notation . 37 4.2 Products of Permutations: Revisited . 39 4.3 Properties of Cycle Form . 40 4.4 Order of a Permutation: Revisited . 41 4.5 Inverse of a Permutation: Revisited . 42 4.6 Summary of Permutations . 44 4.7 Working with Permutations in Sage . 44 4.8 Exercises . 45 5 From Puzzles To Permutations 49 5.1 Introduction . 49 5.2 Swap ................................................... 50 5.3 15-Puzzle . 52 5.4 Oval Track Puzzle . 53 5.5 Hungarian Rings . 56 5.6 Rubik’s Cube . 58 5.6.1 2 × 2 × 2 Cube.......................................... 58 5.6.2 3 × 3 × 3 Cube.......................................... 59 5.7 Exercises . 60 6 Permutations: Products of 2-Cycles 65 6.1 Introduction . 65 6.2 Product of 2-Cycles . 66 6.3 Solvability of Swap . 67 6.4 Exercises . 68 7 Permutations: The Parity Theorem 69 7.1 Introduction . 69 CONTENTS iii 7.2 Variation of Swap . 71 7.3 Proof of the Parity Theorem . 72 7.3.1 Proof 1 of Claim 7.2 . 72 7.3.2 Proof 2 of Claim 7.2 . 74 7.4 Exercises . 77 8 Permutations: An and 3-Cycles 81 8.1 Swap Variation: A Challenge . 81 8.2 The Alternating Group An ....................................... 81 8.3 Products of 3-cycles . 83 8.4 Variations of Swap: Revisited . 85 8.5 Exercises . 86 9 Mastering the 15-Puzzle 89 9.1 Solvability Criteria . 89 9.2 Proof of Solvability Criteria . 91 9.3 Strategy for Solution . 94 9.4 Exercises . 95 10 Groups 101 10.1 Group: Definition . 101 10.1.1 Multiplication (Cayley) Table . 103 10.2 Some Everyday Examples of Groups . 103 10.3 Further Examples of Groups . 106 10.3.1 Symmetric and Alternating Groups . 106 10.3.2 Finite Cyclic Groups . 108 10.3.3 Group of Integers Modulo n: Cn ................................ 109 10.3.4 Group of Units Modulo n: U(n) ................................ 112 10.3.5 Dihedral Groups: Dn ...................................... 115 10.3.6 Notation for Dn ......................................... 117 10.4 Exercises . 118 11 Subgroups 123 11.1 Subgroups . 123 11.2 Examples of Subgroups . 124 11.3 The Center of a Group . 125 11.4 Lagrange’s Theorem . 126 11.5 Cyclic Groups Revisited . 127 iv CONTENTS 11.6 Cayley’s Theorem . 128 11.7 Exercises . 129 12 Puzzle Groups 133 12.1 Puzzle Groups . 133 12.2 Rubik’s Cube . 134 12.2.1 3-Cube Group . 134 12.2.2 2-Cube Group . ..