## 305 – Cube moves

Here is a small catalogue of moves of the Rubik’s cube. Appropriately combining them and their natural analogues under rotations or reflections, allow us to solve Rubik’s cube starting from any (legal) position. I show the effect the moves have when applied to the solved cube.

For each move listed below, the Sage code to generate the displayed graph is

RubiksCube().move(“***”).show3d()

where the move is listed between the quotation marks, instead of ***.

Default position:  This move only changes the orientation of the two cubes displayed:

• L*F*R^(-1)*F^(-1)*L^(-1)*U^2*R*U*R*U^(-1)*R*R*U*U*R This move exchanges one of the cubes in the middle tier with one of the bottom cubes, as shown (it also exchanges some cubes on the bottom tier):

• F*D^(-1)*F^(-1)*D^(-1)*F*D*D*F^(-1)*D*F*D*F^(-1)*D*D This moves swaps the position of the two bottom corners of the front face (and does some additional damage to orientations or middle cubes in the bottom tier):

• R^(-1)*D^(-1)*R*F*D*F^(-1)*R^(-1)*D*R*D*D This move does not affect the top or middle tiers, and changes the orientation (but not the position) of the three bottom corners other than the botton left one of the front face:

• R^(-1)*D^(-1)*R*D^(-1)*R^(-1)*D*D*R*D*D This move only permutes 3 middle cubes in the bottom tier (and affects their orientation as shown):

• L^(-1)*R*F^(-1)*L*R^(-1)*D^2*L^(-1)*R*F^(-1)*L*R^(-1) Finally, this move has a similar effect, but the direction of the permutation is different:

• F*L*B^(-1)*L^(-1)*B*F^(-1)*D*B*D^(-1)*B^(-1) As an exercise, try to compute an upper bound on how many basic moves (in the half-turn metric) are needed to solve the cube using the moves just listed.