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The upper bound for the number of moves required to solve a regular Rubik's cube has been shown to be 20.

Two questions come to mind:

  1. Does this result have more general significance?

  2. What are the most pressing issue with regards to Rubik's cube (or generalizations) or its group?

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    http://www.math.harvard.edu/~jjchen/docs/Group%20Theory%20and%20the%20Rubik's%20Cube.pdf here's a pretty good paper on the Rubik's Cube Group, for reference2011-03-01
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    @Eugene: Nice reference, thanks!2011-03-01

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