## Puzzle It Out: Cubes, Groups and PuzzlesExplains the mathematical theory of groups and how it can be used to solve Rubik's Cube and similar puzzles |

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### Contents

The Cube | 4 |

Permutations of 4 coloured squares | 6 |

Permutations of the Cube | 16 |

Commutators | 24 |

Understanding Commutators | 26 |

Conjugation | 28 |

Solving that Cube | 30 |

Groups | 32 |

Groups within groups Subgroups | 34 |

Cycles | 36 |

Even and Odd Permutations Impossible Rearrangements | 40 |

The Twelve Cubes | 42 |

Super Moves | 44 |

### Common terms and phrases

anticlockwise arrow diagram basic moves beads in rows black bead centre space centre wheel stationary COLOURED SQUARES continued combination of permutations commutator completed cube conjugate corner pieces correct colour correctly in position CUBE continued cube group cycle notation cycle the three disassemble disjoint cycles double slice group effect empty square example flipped four coloured squares four squares front-right group theory identity permutation INSTANT INSANITY inverse layer leaving line segment near-corner pieces number of transpositions o o o o obtain odd crossing number odd number odd permutation permutation is called permutations of four permute the squares piece in position pieces correctly place the four position c4 positions w rearrange Rotate the slider rotational symmetry Rubik Cube Sam Loyd sequence of moves simple moves single plunger solution solve the cube solve the puzzle square in position stage subcollection subgroup Super Moves twists w x y z write wxyz