R2U2\mbox{R}_2\mbox{U}_2 Back To The Future

Probability Level 2

Consider the 2×2×22\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 R2\mbox{R}_2 and U2\mbox{U}_2. When R2\mbox{R}_2 is performed, the right side of the cube is spun two quarter rotations clockwise. Similarly, U2\mbox{U}_2 indicates that the top layer (U=\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 R2\mbox{R}_2 followed by U2\mbox{U}_2. We call this sequence of events the R2U2\mbox{R}_2\mbox{U}_2 permutation. How many R2U2\mbox{R}_2\mbox{U}_2 permutations do we go through before the 2×2×22\times 2\times 2 cube is back to its original state?

Note: The R2\mbox{R}_2 and U2\mbox{U}_2 moves are displayed below on a 3×3×33\times3\times3 cube.

  • R2\mbox{R}_2
  • U2\mbox{U}_2

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