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

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

3. 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

4. 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

5. 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.

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

7. Group
Closed
Associative
Identity
Inverse

8. Subgroups

9. 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.

10. 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.

11. Isomorphisms

12. C
Commutativity with support and commutator

13. fudlluuddru flips exactly one edge cubie on the top face

14. r3drfdf3 twists one cubie on a face

15. ff swaps a pair of edges in a slice

16. r3dr cycles three corners