- Evaluate the size of the Rubik's Cube group
- Prove that a single corner twist is impossible
- Prove that a single edge flip is impossible
- Prove lemmas about conjugates (setup moves for Old Pochmann), perhaps using some normal subgroup?
- Define an Old Pochmann-style algorithm for solving any valid scramble
- Prove that the algorithm always terminates
- Prove that the algorithm in fact solves the cube
- Prove move-based & permutation/orientation-based group definitions to be equivalent
- Extending to other puzzles
- 2x2
- 5x5
- Prove that repeating the same sequence of moves eventually returns to solved state