Tonight I was having dinner in the kitchen of my dorm and one of my neighbours who also studies math was with me. This led to a discussion on "the Rubik's cube group". Whether it's any interesting I don't know, but we asked ourselves:

• What if you take a sufficiently interesting subgroup (that is, not necessarily generated by a subset of standard moves) and only allow these moves to be used? They could be combinations of standard moves, hence you always do multiple moves in the same time. You'd need a computer program to play such a cube, but it might lead to interesting results. You'd have to train yourself to only consider moves which are part of the given subgroup. Would this be entertaining?
• Is the "minimal subgroup generated by moves to solve a state" an interesting invariant? This might be related to the diameter of the group. For instance I wonder whether there are two states for which the minimal number of moves satisfies a strict inequality, while the order of the minimal subgroups of moves satisfies the other.