Rubik's cube and its generalizations [attracts certain attention][1] of mathematical community. It is somehow ["noteworthy"][2]  that [it has been proved that diameter  of the Rubik's cube group is 20][3], i.e. cubik can be turned into initial position at worst at 20 moves, it rises certain interesting questions e.g. [MO139469][4].

It is not only the diameter has been calculated, but the position count for all [distances presented][3]. If one plots it at logarithmic scale one sees linear dependence  almost everywhere:

[![Log(number of positions) ][5]][5]

So looking more carefully we see that number of positions at next step is approximately 13 times greater (see details below). 

**Question 1** Is there some explanation of such exponential grow ?

**Question 2** What are the other examples of groups with similar growth patter ? 


**Remarks:** What seems puzzling for me, that exponential growth corresponds to (for example) for free group (obviously number of words of length n in k-letter alphabet is k^n), but for groups+generators which are similar to abelian groups the growth should be similar to normal distribution (a version of central limit theorem) - for example for $(Z/2Z)^n$ it corresponds to the setup of the most classical central limit theorem, which can be visualized by the [Galton board (Bean machine)][6]. Similar results has been extented to permutation groups and metrics of them see [MO320497][7]. So it is somehow strange for me that Rubik's group+generators looks like free group, rather than abelian. 

--------------

Datum (from [http://www.cube20.org][3]) + Pyhon code:

    # Distance		Count of Positions	
    datum  =[ 0	,	1	,
    1	,	18	,
    2	,	243	,
    3	,	324	,
    4	,	43239	,
    5	,	574908	,
    6	,	7618438	,
    7	,	100803036	,
    8	,	1332343288	,
    9	,	17596479795	,
    10	,	2.32248E+11	,
    11	,	3.06329E+12	,
    12	,	4.03744E+13	,
    13	,	5.31653E+14	,
    14	,	6.98932E+15	,
    15	,	9.13651E+16	,
    16	,	1.1E+18	,
    17	,	1.2E+19	,
    18	,	2.9E+19	,
    19	,	1.5E+18	,
    20	,	490000000 ]
    
    import matplotlib.pyplot as plt
    import numpy as np
    plt.plot(np.log( datumCount),'*-' )
    plt.xlabel('Distance')
    plt.ylabel('Log(number of positions)')
    plt.show()

    datumCount = datum[1::2]
    np.array(datumCount[1:])/np.array( datumCount[:-1])
    
    array([1.80000000e+01, 1.35000000e+01, 1.33333333e+00, 1.33453704e+02,
           1.32960522e+01, 1.32515776e+01, 1.32314572e+01, 1.32172933e+01,
           1.32071666e+01, 1.31985490e+01, 1.31897368e+01, 1.31800776e+01,
           1.31680718e+01, 1.31463944e+01, 1.30721014e+01, 1.20396081e+01,
           1.09090909e+01, 2.41666667e+00, 5.17241379e-02, 3.26666667e-10])


  [1]: https://mathoverflow.net/search?q=rubik%20cube
  [2]: https://mathoverflow.net/a/83387/10446
  [3]: http://www.cube20.org
  [4]: https://mathoverflow.net/q/139469/10446
  [5]: https://i.sstatic.net/6XGTo.png
  [6]: https://en.wikipedia.org/wiki/Bean_machine
  [7]: https://mathoverflow.net/questions/320497/metrics-on-finite-groups-and-generalizations-of-central-limit-theorems-for-balls