Date: 21 July 1981 2350-EDT (Tuesday) From: Dan Hoey at CMU-10A To: Cube-Hackers at MIT-MC Subject: The ten stuck-axle subgroups In-Reply-To: ISAACS@SRI-KL's message of 21 Jul 81 11:08-EST Message-Id: <21Jul81 235052 DH51@CMU-10A> 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.