I tried doing a little bit of googling about this, but I am not able to find a decent answer. Let a configuration of a face be a choice of color for each of the 9 squares on that face of the cube. Is it possible to reach every configuration on some face starting from the solved Rubik's cube? Is there a general method for finding the sequence of moves to reach the possible configurations?
- 6 corners are independently orientable*
- 7 corners are independently positionable*
- 11 edges are independently orientable
- 10 edges are independently positionable
- One face only has 4 edges and 4 corners
We will be able to place all 8 non-centre cubies without interference.
The only remaining 'cubie' is the centre, which can only move if we allow 'slice' moves or rotating the entire cube.
Thus, one face is solvable (in this sense) in general iff the centre cubie can be moved to a different face.
As for the algorithm to do so, it will merely be an adapted version of the 'first-layer' algorithms in many beginner methods. There are several variations, all simple.
*meaning they can each be positioned/oriented any way without affecting previous cubies
An obvious way to do this (without changing the center piece), is to put that face on top (or bottom, if that's how you usually do it), and create the cross with the colors you want, just like when you are solving the cube normally. Then just insert corners with the colors you want, and that particular face can thus be manipulated however you want (not including the center piece).