[next] [prev] [up] Date: Tue, 21 Jul 81 23:50:00 -0400 (EDT)
[next] [prev] [up] From: Dan Hoey <Hoey@CMU-10A >
~~~ ~~~ [up] Subject: The ten stuck-axle subgroups

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.


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