Permutation Puzzles. A Mathematical Perspective

Part I — Part One: Foundations
	1 Permutation Puzzles
		1.1 Introduction
		1.2 A Collection of Puzzles
		1.3 Which brings us to the Definition of a Permutation Puzzle
		1.4 Exercises
	2 A Bit of Set Theory
		2.1 Introduction
		2.2 Sets and Subsets
		2.3 Laws of Set Theory
		2.4 Examples Using SageMath
		2.5 Exercises
Part II — Part Two: Permutations
	3 Permutations
		3.1 Permutation: Preliminary Definition
		3.2 Permutation: Mathematical Definition
		3.3 Composing Permutations
		3.4 Associativity of Permutation Composition
		3.5 Inverses of Permutations
		3.6 The Symmetric Group Sn
		3.7 Rules for Exponents
		3.8 Order of a Permutation
		3.9 Exercises
	4 Permutations: Cycle Notation
		4.1 Permutations: Cycle Notation
		4.2 Products of Permutations: Revisited
		4.3 Properties of Cycle Form
		4.4 Order of a Permutation: Revisited
		4.5 Inverse of a Permutation: Revisited
		4.6 Summary of Permutations
		4.7 Working with Permutations in SageMath
		4.8 Exercises
	5 From Puzzles To Permutations
		5.1 Introduction
		5.2 Swap
		5.3 15-Puzzle
		5.4 Oval Track Puzzle
		5.5 Hungarian Rings
		5.6 Rubik\'s Cube
		5.7 Exercises
	6 Permutations: Products of 2-Cycles
		6.1 Introduction
		6.2 Product of 2-Cycles
		6.3 Solvability of Swap
		6.4 Exercises
	7 Permutations: The Parity Theorem
		7.1 Introduction
		7.2 Variation of Swap
		7.3 Proof of the Parity Theorem
		7.4 Exercises
	8 Permutations: An and 3-Cycles
		8.1 Swap Variation: A Challenge
		8.2 The Alternating Group An
		8.3 Products of 3-cycles
		8.4 Variations of Swap: Revisited
		8.5 Exercises
	9 The 15-Puzzle
		9.1 Solvability Criteria
		9.2 Proof of Solvability Criteria
		9.3 Strategy for Solution
		9.4 Exercises
Part III — Part Three: Group Theory
	10 Groups
		10.1 Group: Definition
		10.2 Some Everyday Examples of Groups
		10.3 Further Examples of Groups
		10.4 Exercises
	11 Subgroups
		11.1 Subgroups
		11.2 Examples of Subgroups
		11.3 The Center of a Group
		11.4 Lagrange\'s Theorem
		11.5 Cyclic Groups Revisited
		11.6 Cayley\'s Theorem
		11.7 Exercises
	12 Puzzle Groups
		12.1 Puzzle Groups
		12.2 Rubik\'s Cube
		12.3 Hungarian Rings
		12.4 15-Puzzle
		12.5 Exercises
	13 Commutators
		13.1 Commutators
		13.2 Creating Puzzle moves with Commutators
		13.3 Exercises
	14 Conjugates
		14.1 Conjugates
		14.2 Modifying Puzzle moves with Conjugates
		14.3 Exercises
	15 The Oval Track Puzzle
		15.1 Oval Track with T=(14)(23)
		15.2 Variations of the Oval Track T move
		15.3 Exercises
	16 The Hungarian Rings Puzzle
		16.1 Hungarian Rings - Numbered version
		16.2 Building Small Cycles: Tools for Our End-Game Toolbox
		16.3 Solving the end-game
		16.4 Hungarian Rings - Coloured version
		16.5 Exercises
	17 Partitions & Equivalence Relations
		17.1 Partitions of a Set
		17.2 Relations
		17.3 Equivalence Relation
		17.4 Exercises
	18 Cosets & Lagrange\'s Theorem
		18.1 Cosets
		18.2 Lagrange\'s Theorem
		18.3 Exercises
Part IV — Part Four: Rubiks\' Cube
	19 Rubik\'s Cube: Beginnings
		19.1 Rubik\'s Cube terminology and notation
		19.2 Impossible Moves
		19.3 A Catalog of Basic Move Sequences
		19.4 Strategy for Solution
		19.5 Exercises
	20 Rubik\'s Cube: The Fundamental Theorem
		20.1 Rubik\'s Cube - A Model
		20.2 The Fundamental Theorem of Cubology
		20.3 When are two assembled cubes equivalent?
		20.4 Exercises
	21 Rubik\'s Cube: Subgroups of the Cube Group
		21.1 Building Big Groups from Smaller Ones
		21.2 Some Subgroups of RC3
		21.3 Structure of the Cube Group RC3
		21.4 Exercises
Part V — Part Five: Symmetry & Counting
	22 The Orbit-Stabilizer Theorem
		22.1 Orbits & Stabilizers
		22.2 Permutations Acting on Sets: Application of the Orbit-Stabilizer Theorem
		22.3 Exercises
	23 Burnside\'s Theorem
		23.1 A Motivating Example
		23.2 Burnside\'s Theorem
		23.3 Applications of Burnside\'s Theorem
		23.4 Exercises
Part VI — Part Six: Light\'s Out
	24 Lights Out
		24.1 Lights Out
		24.2 Lights Out: A Matrix Model
		24.3 Summary of 55 lights out puzzle
		24.4 Eigenvalues and Eigenvectors
		24.5 Other sized game boards
		24.6 Light-Chasing Strategy
		24.7 Exercises
Part VII — Appendix
	A SageMath
		A.1 SageMath Basics
		A.2 Variables and Statements
		A.3 Lists
		A.4 Sets
		A.5 Commands/Functions
		A.6 if, while, and for statements
		A.7 Exercises
	B Basic Properties of Integers
		B.1 Divisibility and the Euclidean Algorithm
		B.2 Prime Numbers
		B.3 Euler\'s -function
		B.4 Modular Arithmetic
		B.5 Exercises
	Bibliography
		Articles
		Books
		Web Sites
	Index