MathNet 2011 Question 33

Consider the $8 \times 8 \times 8$ Rubik's cube below. Each face is painted with a different color, and it is possible to turn any layer, as you can with smaller Rubik's cubes. Let $X$ denote the move that turns the shaded layer shown (indicated by arrows going from the top to the right of the cube) clockwise by $90$ degrees, about the axis labeled $X$. When move $X$ is performed, the only layer that moves is the shaded layer. Likewise, define move $Y$ to be a clockwise $90$-degree turn about the axis labeled $Y$, of just the shaded layer shown (indicated by the arrows going from the front to the top, where the front is the side pierced by the $X$ rotation axis). Let $M$ denote the move "perform $X$, then perform $Y$."



Imagine that the cube starts out in "solved" form (so each face has just one color), and we start doing move $M$ repeatedly. What is the least number of repeats of $M$ in order for the cube to be restored to its original colors?