Solving ‘Rubik’s Polyhedra’ Using Three-Cycles

Jerzy Wieczorek

Franklin W. Olin College of Engineering

Assisted by Dr. Sarah Spence

Wellesley MAA Conference, 11/21/03

Cube Terminology

Center

(immobile)

Edge

Corner

Face

Groups

A group is a nonempty set G closed under a binary operation with the associative property; it contains an identity element and inverses for each element.

Rubik’s Cube – its group uses the set of face rotation sequences; each sequence has an inverse.

Permutations

A permutation is a function that rearranges some or all of the elements in a set.

Any permutation can be written as a series of transpositions (two-piece

permutations).

Rubik’s Cube – each sequence of face rotations is a permutation that rearranges some pieces on the Cube.

Even and Odd Permutations

A permutation is even if it can be broken down into an even number of transpositions; otherwise, it odd.

Combining even and odd permutations works like adding even and odd numbers:

even + even = eveneven + odd = oddodd + odd = even

Permutation Example

To cycle these four elements counterclockwise, perform three transpositions: switch 1 and 2, then 1 and 3, then 1 and 4.

This permutations is odd, since it uses three transpositions.

1 2

4 3

2 1

4 3

2 3

1 4

2 3

4 1

= + +

Cube Permutations

A single face rotation on the Cube performs two odd permutations (similar to the previous example), resulting in an even permutation overall.

1↔2, 1↔3, 1↔4A↔B, A↔C, A↔D

6 transpositions:an even permutation

1

2

3

4

A

B

C

D

Alternating GroupsThe alternating group An uses only the set of even permutations of n pieces.In An, odd permutations (including single transpositions) are impossible.

Rubik’s Cube – since any face rotation is even, and even permutations combine to make even ones, it belongs in A20 (12 edges, 8 corners) and performing a single transposition is impossible.

Commutators and 3-CyclesThe commutator of permutations α and β, written [α, β], is the sequence αβα-1β-1; it is always an even permutation.Theorem: [α, β] permutes exactly three pieces if there is exactly one piece x such that both α and β independently permute x.No pieces other than x, α(x), or β(x) will be affected by αβα-1β-1: anything changed by one function but not the other will be fixed by its inverse. All three will be permuted since permuting only two pieces would be odd.

Solving the CubeUsing the above theorem, play around with the Cube to form a library of rotation sequences that result in various 3-cycles.

On a scrambled Cube with more than 3 unsolved pieces, use 3-cycles to solve up to 3 pieces at a time until exactly 3 are left. Solve them with the appropriate cycle.

Since transpositions are impossible, you will never need a 2-cycle.

(Of course, I did not address orientations…)

Other PolyhedraThe same methodology applies to theTetrahedron and Dodecahedron puzzles, belonging to A6 (6 edges, 0 corners) andA50 (30 edges, 20 corners), respectively.

τ = (123) = (13)(12) ←cycle notation

δ = (12345)(678910) = (15)(14)(13)(12)(610)(69)(68)(67)

Resources

Bump, Daniel. “Mathematics of theRubik’s Cube.” http://match.stanford.edu/rubik.html

Rubik’s Online Home Page.http://www.rubiks.com