The Mathematics of the Rubik’s Cube

Dr Pamela Docherty

School of MathematicsUniversity 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 thetriangle that leaves

it unchanged

Combining SymmetriesCombining Symmetries

I R1

R2

S1

S2

S3

I I R1

R2

S1

S2

S3

R1

R1

R2

I S3

S1

S2

R2

R2

I R1

S2

S3

S1

S1

S1 S

2S3

I R1

R2

S2

S2

S3

S1

R2

I R2

S3

S3

S1 S

2R1

R2

I

GroupsNotice that the symmetries of a trianglehave 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.

GroupsGroups have three special properties.

Identity - does nothingInverse - reverses a moveClosure – combining two moves gives another move

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 is4.3 x 1019

There are 4.3 x 1019

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!

Thanks for listening!