By Mariah Sakaeda and Alex Jeppson The Rubik’s Cube By Mariah Sakaeda and Alex Jeppson

Notation - Sides No notation for the middle because you don’t move it

Notation - “Cubies” By cubie, we mean 1 of the 26 small cubes Red is FUR Yellow is RUF Blue is URF Lime green is ULB Pink is RF Dark green if FR Black is F

Notation - Rotations Lowercase letter to refer to what side you move f is the rotation of the side F 90 degrees clockwise f’ is the rotation of the side F 90 degrees counter-clockwise

Bounds on solving 8 corners means 8!, 3 orientations means 3^8 (but once 7 are decided, the 1 has no choice so really 3^7) 12 edges means 12!, 2 orientations means 2^12 (but once 11 are decided, the last has no choice so really 2^11) ½ correct cubie-rearrangment parity (?) To put this into perspective, if one had as many standard sized Rubik's Cubes as there are permutations, one could cover the Earth's surface 275 times. The preceding figure is limited to permutations that can be reached solely by turning the sides of the cube. If one considers permutations reached through disassembly of the cube, the number becomes twelve times as large, which is approximately 519 quintillion.

Parity Theorem: The cube always has even parity, or an even number of cubies exchanged from the starting position.

Group Closed Associative Identity Inverse

Subgroups

Cosets Recal: we define a coset and note some properties of cosets. If G is a group and H is a subgroup of G, then for an element g of G: gH = {gh : h ∈ H} is a left coset of H in G. Hg = {hg : h ∈ H} is a right coset of H in G.

Theorem: If the cube starts at the solved state, and one move sequence P is performed successively, then eventually the cube will return to its solved state.

Isomorphisms

C Commutativity with support and commutator

fudlluuddru flips exactly one edge cubie on the top face

r3drfdf3 twists one cubie on a face

ff swaps a pair of edges in a slice

r3dr cycles three corners