When \(\mbox{RU}\) coming back to me?

Algebra Level pending

Now, let's spice things up a bit. In the last problem, we did \(\mbox{R}_2\) and \(\mbox{U}_2\) moves to transform the cube. This time, let's do \(\mbox{R}\) and \(\mbox{U}\) moves (only one clockwise quarter turn each time). If we perform \(\mbox{R}\) followed by \(\mbox{U}\), we call that the \(\mbox{RU}\) permutation.

How many \(\mbox{RU}\) permutations must we perform before the \(2\times 2\times 2\) cube is back to its original state?

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