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 = even even + odd = odd odd + 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 = + +

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↔4 A↔B, A↔C, A↔D 6 transpositions: an even permutation A B C D

Alternating Groups The alternating group A n uses only the set of even permutations of n pieces. In A n, 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 A 20 (12 edges, 8 corners) and performing a single transposition is impossible.

Commutators and 3-Cycles The commutator of permutations α and β, written [α, β], is the sequence αβα -1 β -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 -1 β -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 Cube Using 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 Polyhedra The same methodology applies to the Tetrahedron and Dodecahedron puzzles, belonging to A 6 (6 edges, 0 corners) and A 50 (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 the Rubik’s Cube.” Rubik’s Online Home Page.