The Mathematics of the Rubik’s Cube Dr Pamela Docherty School of Mathematics University of Edinburgh

The Rubik's Cube Invented in 1974 by Ernő Rubik 350 million cubes have been sold worldwide

Question How can Mathematics describe the moves of a Rubik’s cube?

Algebra: the study of symmetry What are the symmetries of a Rubik’s cube?

Simpler Example Symmetries of an equilateral triangle

Triangle symmetries A symmetry of a triangle are the moves you can make to the triangle that leaves it unchanged

IR1R1 R2R2 S1S1 S2S2 S3S3 IIR1R1 R2R2 S1S1 S2S2 S3S3 R1R1 R1R1 R2R2 IS3S3 S1S1 S2S2 R2R2 R2R2 IR1R1 S2S2 S3S3 S1S1 S1S1 S1S1 S2S2 S3S3 IR1R1 R2R2 S2S2 S2S2 S3S3 S1S1 R2R2 IR2R2 S3S3 S3S3 S1S1 S2S2 R1R1 R2R2 I

Groups Notice that the symmetries of a triangle have three special properties. There is an “Identity” symmetry Each symmetry has an “Inverse” Closure - combining two symmetries make another symmetry

Groups A group is a set of objects with an operation.

Groups Groups have three special properties. Identity - does nothing Inverse - reverses a move Closure – combining two moves gives another move

Rubik's Cube 6 faces 9 facets per face 54 facets altogether

The Rubik's Cube Group The set of all moves of a Rubik's cube is a group. - Identity - Inverse - Closure

The Rubik's Cube Group The order of the group is 4.3 x There are 4.3 x possible positions Only one of these is the solution!

The Rubik's Cube Group The maximum order of a move is 1260 The combination of simple moves RU has order 105

How to solve a Rubik's Cube There are lots of algorithms to solve a Rubik's Cube. But group theory can help you prove that you only need 20 moves to solve it from any starting point!

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