The Group C_{1} |
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The Group C_{1} is the whole cube group which contains 43,252,003,274,489,856,000 elements. Dan Hoey showed in 1994 that this group contains 901,083,404,981,813,616 elements up to M-symmetry. Mike Godfrey and I showed in 2005 that there are 450,541,810,590,509,978 elements up to M-symmetry and antisymmetry. We have the following table (see sequence A080601 in the On-Line Encyclopedia of Integer Sequences) for the distances in this subgroup:
Because we have the tables for all other symmetry groups except C_{1} which exactly have a given symmetry, we can compute the table for the number of cubes wich are symmetric:
This are altogether 164,604,041,664 cubes. By subtracting the second table from the first we can compute the number of cubes which exactly have C_{1}-symmetry (and hence are unsymmetric):
If we want to know the number of cubes which exactly have C_{1}-symmetry mod M, we have to divide all numbers in the last table by 48. It is interesting to notice that the the probability to get a symmetric cube by random is 164,604,041,664 / 43,252,003,274,489,856,000 which is only about 3.81*10^{-9}. There are myriads of nice unsymmetric cubes. Some examples are given below. |
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Cube display with AnimCubeJS |
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© 2017 Herbert Kociemba |