From:

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1. No faces stuck. The familiar cube group.

2. D face stuck. As previously noted, all positions can be reached.

In addition, all Supergroup positions that fix the orientation of

the D face center are achievable.

3. B and D faces stuck. All Supergroup positions that fix the BD

edge and the B and D face centers are achievable.

4. U and D faces stuck. Edges cannot be flipped. If we define edge

orientation by marking the F and B facelets of the F and B edges,

and the U and D facelets of the others [cf Jim Saxe's message of 3

September 1980], then all Supergroup positions that fix the

orientation of all edges and the U and D face centers are

achievable.

5. L, B, and D faces stuck. All Supergroup positions that fix the

BLD corner, the LB, BD, and DL edges, and the L, B, and D face

centers are achievable.

6. U, B, and D faces stuck. Again, edges cannot be flipped. All

Supergroup positions that fix the orientation of all edges, the

position of the UB and BD edges, and the orientation of the U, B,

and D face centers are achievable.

7. U, L, B, and D faces stuck. Singmaster has a very nice

description of this group [indexed as Group, Two Generators]. The

group of achievable permutations of the six movable corners is

isomorphic to the group of all permutations on five letters. All

Supergroup positions that permute the corners in an achievable

permutation, fix edge orientation, and fix the unmovable two

corners, five edges, and four face centers are achievable.

8. U, L, D, and R faces stuck. Sixteen positions

9. U, L, D, B, and R faces stuck. Four positions.

10. All faces stuck. One position.