# \(\mbox{R}_2\mbox{U}_2\) Back To The Future

**Discrete Mathematics**Level 3

To start, let us consider the two moves \(\mbox{R}_2\) and \(\mbox{U}_2\). When \(\mbox{R}_2\) is performed, the right side of the cube is spun two quarter rotations clockwise. Similarly, \(\mbox{U}_2\) indicates that the top layer (\(\mbox{U}=\) **U**p) is spun two quarter rotations clockwise. Whenever a face gets two quarter turns, each piece in that face ends up diagonal to its original position on the face.

We start transforming the cube by performing \(\mbox{R}_2\) followed by \(\mbox{U}_2\). We call this sequence of events the \(\mbox{R}_2\mbox{U}_2\) permutation. How many \(\mbox{R}_2\mbox{U}_2\) permutations do we go through before the \(2\times 2\times 2\) cube is back to its original state?

**Note**:
The \(\mbox{R}_2\) and \(\mbox{U}_2\) moves are displayed below on a \(3\times3\times3\) cube.

- \(\mbox{R}_2\)
- \(\mbox{U}_2\)