A standard Rubik's cube is initially unscrambled (say per picture below, green facing observer)
The same manipulation is repeated:
- rotation of front face by 1/4 turn (say clockwise)
- rotation of the whole cube 1/4 turn around vertical axis (say anticlockwise seen from top)
(with the proposed orientations, the center of the rotated face at 1 will cyclically be green, orange, blue, red; in standard notation these moves cycle between F L B R; the center of the upper face always remain white).
After how many manipulations will the cube be first unscrambled again? (note: by symmetry, this is independent of the direction of the rotations, as long as they remain the same across manipulations).
Is there a simple argument to tell which face is facing the observer at that point (equivalently, to determine the answer modulo 4)?
Note: I'm interested in the reasoning to solve that kind of problems, rather than in the answer for that particular manipulation.