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BrainteasersJane Street

How many positions does a Rubik's cube have?

countingpermutationsestimationparity

A Rubik's cube has 8 corner pieces (each of which can sit in any corner in 3 orientations) and 12 edge pieces (each in any edge slot in 2 orientations). The 6 centre pieces never move relative to each other.

  1. Count the arrangements that could be assembled by taking the cube apart and rebuilding it.
  2. Only some of those can be reached by turning faces. Which fraction, and why? Give the final number.
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Corners: 8! placements times 3 to the 8 orientations. Edges: 12! times 2 to the 12. Then three constraints, each cutting the count by a factor, come from the fact that a face turn is a particular kind of permutation.

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