I will define the game as follows:
There are two players, the "Solver" and the "Unsolver". The Solver's goal is to return the cube to it's standard solved state. Likewise, the Unsolver's goal is to prevent the Solver from solving the cube.
Start the game by scrambling a standard 3x3 Rubik's cube.
Starting with the Solver, each player is allowed one 90 degree face twist. However, the current player's move may not be the simple inverse of the previous move (e.g. you aren't allowed to simply "undo" the other's).
My question is: are there any legal initial Rubik's cube states such that the Solver can win, assuming the Unsolver plays perfectly? I will exclude "scrambles" that involve only a single twist, since the solver would win in one move.
Bonus question: What is the behavior of this game given symmetry-breaking rule changes? The Solver might be allowed 180 degree twists, for example, or two moves before the Unsolver gets one.