Further Work
The results presented here indicate that there is a relationship between the groups of these puzzles. Since not all of the work to complete the proof of this relationship was feasible to do by hand, below are some additional points that might be interesting to look into.
- Find the final sequence of moves that relate the move on the cube to the cuboctahedron as well as the sequence of moves to relate the remaining generators of the cube group.
- Find the final sequence of moves that relate the move on the octahedron to the cuboctahedron as well as the sequence of moves to relate the remaining generators of the octahedron group.
The calculations for these sequences are long and tedious. A type of computer algebra software would be useful in solving for them. We suppose that there will be some cancelations and the sequence will simplify somehow.
- Once the generators of the cube and octahedron are mapped to the cuboctahedron, find the overlap in the two groups. This might be useful in finding a relationship between the cube and the octahedron groups.
- Find the sequence of moves that relate the 3-cycle on edges and vertices on the cube to the cuboctahedron.
- Find the sequence of moves that relate the 3-cycle on edges and vertices on the octahedron to the cuboctahedron.
- Find the sequence of moves that could be used in Proposition 4.3 to change only the orientations of two vertex pieces on the octahedron.
- Find the sequence of moves that could be used in Proposition 6.3 to change only the orientations of two vertex pieces on the cuboctahedron.
- Find the final sequence of moves that relate the move on the cube to the cuboctahedron as well as the sequence of moves to relate the remaining generators of the cube group.
- Find the final sequence of moves that relate the move on the octahedron to the cuboctahedron as well as the sequence of moves to relate the remaining generators of the octahedron group.
The calculations for these sequences are long and tedious. A type of computer algebra software would be useful in solving for them. We suppose that there will be some cancelations and the sequence will simplify somehow.
- Once the generators of the cube and octahedron are mapped to the cuboctahedron, find the overlap in the two groups. This might be useful in finding a relationship between the cube and the octahedron groups.
- Find the sequence of moves that relate the 3-cycle on edges and vertices on the cube to the cuboctahedron.
- Find the sequence of moves that relate the 3-cycle on edges and vertices on the octahedron to the cuboctahedron.
- Find the sequence of moves that could be used in Proposition 4.3 to change only the orientations of two vertex pieces on the octahedron.
- Find the sequence of moves that could be used in Proposition 6.3 to change only the orientations of two vertex pieces on the cuboctahedron.