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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?

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You cannot change the color of the center square once you choose a face. –  Pratyush Sarkar Jan 6 at 4:24
    
Good point, thank you. –  user119335 Jan 6 at 4:26
    
I suggest that you actually play with a Rubiks Cube. What you are asking is the one thing everyone manages to master very quickly without instructions while it is unpleasant to write down a "general method" for something that simple that relies on lots of cases. (Apart from the center issue, the answer is Yes, you can have any color combination on a single face.) –  Phira Jan 6 at 10:16
    
Thanks Phira. I will have to take your word for it. I don't have a Rubik's cube handy, so I will try tomorrow and hope it will be as trivial as you suggest. –  user119335 Jan 6 at 10:36
    
It's certainly possible to have any combination of colours on a single face provided you don't care what's on the other faces. First rotate the cube so the centre square is the colour you want then put the other pieces in place. As for a general method there are plenty of books on solving the Rubiks cube –  Warren Hill Jan 6 at 12:36

2 Answers 2

Given that:

  • 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

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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).

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