RUBIK'S CUBE PROJECT

The Rubik's Cube group G₀ = < R,L,F,B,U,D>. In our project we looked at a subgroup of G₀ called G. G = < R,L,U,D,F²,B²>. Looking at this group G, we first began by developing a catalog of moves that are considered to be legal within this group. A legal move in the group G is defined by any move in G₀ with the exception of any odd exponent of 'F' & 'B'. We came up with 25 moves from the catalog in [B] belonging to G.

• Corner Twists
  1. RU^-1(RU)²(R^-1U^-1)²R²U^-1RUR^-1U^-1RU^-1R^-1 (+urf)(-ufl)(+ulb)(-ubi)
  2. R^-1U²RUR^-1U^-1RUR^-1UR (+urf)(-ufl)(+ulb)(-ubr)(uf,ul,ur)
  3. R²U²RU²R²U (+urf)(-ufl)(+ulb)(-ubr)(uf,ul,fr)(ub,ur,br)
  4. L²D²RD²L²U (+urf)(-ufl)(+ulb)(-ubr)(+uf)(+ul)(+ub)(+ur)
  5. RU²(R²U^-1)²R²U²RU² (+urf)(+ufl)(-ulb)(-ubr)(ul,ub,ur)
  6. RU^-1(R²U)²R²U^-1R * U² (+urf)(+ufl)(-ulb)(-ubr)(ul,ur)(uf,br,ub,fr)
  7. (U²L²R^-1D)⁴ (+urf)(+ufl)(-ulb)(-ubr)(+dfr)(-dlf)(+dbl)(-drb)
• 3-Cycle Corner Permutations
  1. R²F²UB²U^-1F²UB²U^-1R² (ufl,urf,ubr)
  2. L²B²L^-1F²LB²L^-1F²L^-1 (ufl,fur,ubr)
  3. (R²DL²D^-1)² (ufl,urf,drb)
  4. ULU^-1 * R² * UL^-1U^-1 * R² (ufl,rfu,ubd)
  5. L²DLU²L^-1D^-1LU²L (ufl,fur,drb)
  6. D²RU²RD²R^-1U²RD²R²D² (ufl,fur,bdr)
  7. LD^-1L^-1 * U² * LDL^-1 * U² (ufl,ubr,rdf)
  8. R² * D^-1LD * R² * D^-1L^-1D (ufl,rub,rdf)
• Other Corner Manuevers
  1. U²R(UR^-1)²(U^-1R²)²U²R (+ufl,rub)(-urf,bul)
  2. (R²U^-1R²U)³ (ufl,ubr,drb,urf,dfr)
  3. R²U²F²(U²F²)²F²R²U² (ufl,ubr,dfr)(ulb,drb,dlf)
• 3-Cycle Edge Permutations
  1. U^-1R^-1UR (uf,ur,br)(-ufl,urf)(+ubr,bdr)
• Other Edge Manuever
  1. (R²U²)³ (rf,rb)(uf,ub)
• Face Cubie Turn ("Invisible Operations")
  1. (URLU²R^-1L^-1)² (++u)
• 2-Cycle Corner & Edge Permutations
  1. F²U^-1F²UF²R²UR²D^-1F²D (ufl,urf)(uf,ub)
  2. RD² * UL^-1ULU²F²DRD^-1F² * D²R^-1 (ufl,ubr)(uf,ur)
• OTHER CORNER TWISTS
  1. (R^-1F²RD^-1B²D)² (+ubr)(-fdl)
• OTHER 3-CYCLE EDGE PERMUTATIONS
  1. F²ULR^-1F²L^-1RUF² (ur,uf,ul)

After we came up with the catalog of moves for our group G, we then figured out the orientation for each of the corners and each of the edges with respect to the move used. First, we will show the corner orientations and then the edge orientation. The corner orientations are the following 8-tuples...

Note: For the corner orientation we orient the cube as per the lecture notes.

MOVE  			CORNER ORIENTATION

1.			(1,2,1,2,0,0,0,0)
2.			(1,2,1,2,0,0,0,0)
3.			(1,2,1,2,0,0,0,0)
4.			(1,2,1,2,0,0,0,0)
5.			(1,2,2,1,0,0,0,0)
6.			(1,2,2,1,0,0,0,0)
7.			(1,2,2,1,1,2,1,2)
8.			(0,0,0,0,0,0,0,0)
9.			(1,0,0,2,0,0,0,0)
10.			(0,0,0,0,0,0,0,0)
11.			(0,0,0,1,0,2,0,0)
12.			(1,0,0,2,0,0,0,0)
13.			(0,0,0,2,0,1,0,0)
14.			(0,2,0,0,1,0,0,0)
15.                     (0,0,0,2,1,0,0,0)
16.                     (2,2,0,2,0,0,0,0)
17.                     (0,0,0,0,0,0,0,0)
18.                     (0,0,0,0,0,0,0,0) 
19.                     (2,2,0,0,0,2,0,0) 
20.                     (0,0,0,0,0,0,0,0) 
21.                     (0,0,0,0,0,0,0,0)
22.                     (0,0,0,0,0,0,0,0)
23.                     (0,0,0,0,0,0,0,0)
24.                     (0,1,0,0,0,0,0,2)
25.                     (0,0,0,0,0,0,0,0)

And now for the 12-tuple edge orientations...

By labeling the cube in the following manner we will get an edge orientation of all zero 12-tuples. First, we label each edge of the up face and the down face with a + sign. Second, we label the left side of the bl and the fl edges and the right side of the br and the fr edges with a + sign.

MOVE			EDGE ORIENTATION
1-25 			(0,0,0,0,0,0,0,0,0,0,0,0)

After orienting the cube according to the moves in our catalog, we next determined that with our group < U,D,R,L,F²,B²> we are able to do the following manuevers:

• 3-CYCLE EDGE PERMUTATIONS
• 3-CYCLE CORNER PERMUTATIONS
• CORNER TWISTS
• CORNER & EDGE 2-CYCLE PERMUTATIONS

Note: In order to do these manuevers it might be necessary to introduce a set-up move.

Examples of the previous mentioned manuevers & possible set-up moves:

• 3-CYCLE EDGE PERMUTATIONS F²ULR^-1F²L^-1RUF² It permutes the edges ur,uf,ul SET-UP MOVE: edges that need to be permuted: (ur,ub,fr) MOVE: F²L^-1U²
• 3-CYCLE CORNER PERMUTATIONS
• R²F²UB²U^-1F²UB²U^-1R² It permutes the corners ufl, ufr, ubr SET-UP MOVE: corners that need to be permuted: (ubr,ufl,dbl) MOVE: D²L^-1F²U²
• CORNER TWISTS (R^-1F²RD^-1B²D)² It twists the ubr corner clockwise and the fdl corner counter-clockwise each by 120 degrees SET-UP MOVE: corners that need to be twisted: (ubr, ufl) MOVE: F²D^-1
• corner & edge 2-cycle permutations F²U^-1F²UF²R²UR²D^-1F²D This move permutes the ufl & urf corners and the uf & ub edges. SET-UP MOVE: corner & edges that need to be permuted: (ub,dr)(ubl,dbr) MOVE: D^-1F²B²R²B²

To summarize, using our group G, there are 4 basic properties which apply to the Rubik's Cube:

• Given any 3 edges subcubes, there is a move which is a 3-cycle on these edges and preserves the orientations and positions of all other subcubes.
• Given any 3 corners, there is a move which is a 3-cycle on these corners and preserves the orientations and positions of all other subcubes.
• Given any pair of corners, there is a move which twists one clockwise and one counterclockwise by 120 degrees and preserves the orientations and positions of all other subcubes.
• Given any pair of edges and any pair of corners, there is a move which is a two cycle on these edges, a two cycle on these corners, and preserves the orientations and positions of all other subcubes.

Basically, using these four properties one can easily solve the cube from any mixed position. However, one can only solve the cube up to the point where the edges need to be flipped. Like we stated before edge flips are the only moves that cannot be used using the group G. We determined edge flips were an unavailable move in the group G by using the computer program "GAP".

To determine edge flips were not available, we first generated the group G₀ and then generated the group G. Using the numbering system of the cube in GAP, we attempted to flip edges (21,28) & (20,13). When using the group G₀ the flip was possible however when using the group G, the flip was not possible. Thus, we determined that all edge flips are not possible in the group G.

To conclude our project, we determined the size of our group G compared to the size of the Rubik's Cube group G₀. The following is our result...

We know that in the Rubik'S cube group, G₀ = < U,D,R,L,F,B>, the number of elements can be counted as:

However, our group, G = < U,D,R,L, F², B² >, will not contain as many elements. |C₂¹¹| will be left out because, as mentioned before, our group does not allow edge flips. Thus, the number of elements in our group is counted as:

From this we came up with the following Theorem:

THEOREM: G is isomorphic to the subgroup of the Rubik's group of all 4-tuples (v,r,w,s) such that w is the zero vector. it is also isomorphic to the kernel of the map f which sends

note: The proof of this theorem is verified by using the computer program GAP. This index is also shown by SINGMASTER.