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

Discrete Mathematics Level 3

Consider the \(2\times 2\times 2\) cube puzzle, shown above. It consists of 8 pieces that start out in the solved orientation and can be transformed into alternative orientations by rotating any of the six faces.

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 (\(\mbox{U}=\) Up) 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.

  • \(\mbox{R}_2\)
  • \(\mbox{U}_2\)

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