Rubik Cubes

Skew B Ultimate

This cube is a variation of SkewB. Although it is shaped as a dodecahedron, it is really a cube. It has the same pieces and movements as SkewB, but one additional detail: its center pieces now have an orientation of order 2. This means the group of permutations are a refinement of the group of permutations of SkewB.

Here an interactive 3D version of the cube. Drag around the canvas to rotate. Tab on corners to make cube movements.
Use the slider to see the transition from the cube to the dodecahedron.

Notation

We will use the same notation than for SkewB, but with a refinement for the center faces.

Now, each center piece will have two faces. We will use the notation A_B, where A indicates the face at which is the piece and B the halve this face is closer to.

Then, those are: L_U, L_D, R_U, R_D, U_F, U_B, D_F, D_B, F_L, F_R, B_L and B_R.

Here is an ASCII art diagram:

               .+------------:------------+
             .'      .  ·  '  '.        .'|
           .'_·  '         U_B  '.    .'  |
         .''. '''''-----..._______'..'    |
       .'    '.  U_F      _ . · ' .':'.   |
     .'        '.  . · '        .' .'   '.|
    +------------:------------+'   : R_U.:|
    |          .'|'.          |   :   .'· |
    |       .'   |   '.       |  :  .' :  |
    |    .'      |      '.    | : .'  ·   |
    | .'         |         '. |..' R_D:   +
    |:     F_L   |   F_R     :|:     .' .'
    | '.         |         .' |  ' . '.'
    |    '.      |      .'    |     :'
    |       '.   |   .'       |   .'
    |          '.|.'          | .'
    +------------:------------+'

Permutations

We will use same notation. We will use the name of the center piece for permutations swapping both faces of a center piece.

Example U is defined to be (U_F U_B).

Computations

As expected, we obtained the same results but with a refinement of the centers. We already now how to solve that cube up to center orientations.

Ideally, in this occasion we would like to achieve a way to describe as simple as possible expressions of center orientations in terms of the "greek permutations".

We have:

α := a (c_U b_B e_L) (c_F b_R e_D) (c_L b_U e_B)
     (L_U U_B B_L) (L_D U_F B_R)

β := b (a_U d_R f_B) (a_L d_F f_D) (a_B d_U f_R)
     (B_R U_B R_U) (B_L U_F R_D)

δ := d (b_U c_F h_R) (b_B c_L h_D) (b_R c_U h_F)
     (R_U U_F F_R) (R_D U_B F_L)

Alpha - beta pair

We have:

α β = (a_L d_R f_B a_U d_U f_R a_B d_F f_D)
      (b_B e_L c_U b_U e_B c_L b_R e_D c_F)
      U (L_U R_U B_R L_D R_D B_L)

β α = (a_L d_F f_D a_U d_R f_B a_B d_U f_R)
      (b_B e_B c_L b_U e_D c_F b_R e_L c_U)
      B (R_U L_D U_F R_D L_U U_B)

which, at powering to 3, results into

(α β)^3 = (β α)^3 = a b c d e f L R U B

Three times that permutation orient four centers meanwhile four times this permutation will orient six corners.

Another interesting combination is:

α β (β α)^{-1} = a b^{-1} c f^{-1} (U_F L_U U_B L_D) (R_U B_L R_D B_R)

Four times this permutation will orient four corners while time six times this permutation will orient four centers.

You can use the last one twice with physical rotation of the cube to orient two centers.

Alpha - delta pair

If we multiply them we obtain:

α δ = a d c (b_B e_L h_F b_R e_D h_R b_U e_B h_D)
      (L_U F_L R_D U_B B_L) (L_D F_R R_U U_F B_R)

δ α = a d b (c_U h_F e_D c_F h_R e_B c_L h_D e_L)
      (R_U B_R L_D U_F F_R) (R_D B_L L_U U_B F_L)

Solving strategy

A possible solving strategy would be

  • Solving the cube as it were the SkewB, ignoring face orentations.
  • Use the above alpha-beta combinations to orient the faces.
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