Here it is folks, the Ultimate Expression of Cubism! Welcome to...

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God's Algorithm Calculations for Rubik's Cube, Rubik's Subgroups,
 and Related Puzzles!
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This is the work of Dan Hoey, Jerry Bryan, Dik Winter, Micheal Reid,
Martin Schoenert and Mark Longridge. Some of the efforts are incomplete,
but you never know what the future will determine.

Real size of Cube Space = 901,083,404,981,813,616 (approx 901 quadrillion)

The Real Size of Cube Space was first calculated on Fri, 4 Nov 94
 by Dan Hoey.


       Analysis of <T2, D2, F2>         Analysis of <T2, D2, F2, B2>
       ------------------------         ----------------------------

        Level   Number of               Level   Number of Positions
                Positions

         0          1                     0             1
         1          3                     1             4
         2          5                     2            10
         3          8                     3            24
         4         13                     4            53
         5         21                     5            64
         6         23                     6            31
         7         13                     7             4
         8          5                     8             1
         9          3                                 ---
        10          1                                 192
                   --
                   96       Antipode 2 X order 2 = R2 T2 L2 R2 T2 R2 D2 T2

Antipode  2 H order 2 = (R2 D2)^2 (T2 R2)^2 D2 T2

        Analysis of <T2, F2, R2>

        Level   Number of
                Positions

        0           1
        1           3
        2           6
        3          12
        4          24
        5          48
        6          93
        7         180
        8         315
        9         489
       10         604
       11         522
       12         250
       13          42
       14           3
                -----
                2,592

        Analysis of the <M_R, D> Group
        ------------------------------

     Level         Number of      Time       Branching
                   Positions                  Factor

       0               1           0 s          --
       1               4           0 s           4
       2              10           0 s           2.5
       3              24           0 s           2.4
       4              58           0 s           2.416
       5             140           2 s           2.414
       6             338          11 s           2.414
       7             816          67 s           2.414
       8            1909         433 s           2.339
       9            4296        2793 s           2.250
      10            8893       17355 s           2.070

10-ply for Pentium 100 Mhz = 10486 s

        Analysis of the 3x3x3 5 Generator Group
        ---------------------------------------

     Level         Number of       Local      Branching
                   Positions        Max        Factor

       0                  1           0
       1                 10           0           10.000
       2                 77           0            7.700
       3                584           0            7.584
       4              4,434           0            7.592
       5             33,664           0            7.592
       6            255,320           0            7.584
       7          1,933,936                        7.575
       8         14,635,503                        7.568

        Analysis of the 3x3x3 Slice & Anti-Slice Groups
        -----------------------------------------------
                  arrangements       M          arrangements           M
Moves Deep     (2q or slice)     conjugates (4q or double slice)   conjugates

  0                    1             1               1                 1
  1                    6             1               9                 2
  2                   27             2              51                 4
  3                  120             6             247                15
  4                  287            16             428                25
  5                  258            15              32                 3
  6                   69             9             ---                --
                     ---            --             768                50
                     768            50

                  arrangements           arrangements
Moves Deep   (2q or anti-slice moves)   (4q or double anti-slice moves)

    0                   1       1             1        1
    1                   6       1             9        2
    2                  27       3            51        5
    3                 120      10           265       25
    4                 423      37           864       75
    5               1,098      93         1,785      152
    6               1,770     166         2,017      184
    7               1,650     147         1,008      108
    8                 851      89           144       16
    9                 198      21
                    -----     ---         -----      ---
                    6,144     568         6,144      568


        Analysis of the 2x2x2 cube group
        --------------------------------

Originally computed on a DEC VAX 11/780 in over 51 hours of CPU time
 on Sept. 9, 1981
Moves Deep     arrangements (q+h)   arrangements (q)  loc max (q+h) loc max (q)

  0                    1                   1                 0               0
  1                    9                   6                 0               0
  2                   54                  27                 0               0
  3                  321                 120                 0               0
  4                1,847                 534                11               0
  5                9,992               2,256                 8               0
  6               50,136               8,969                96               0
  7              227,536              33,058               904              16
  8              870,072             114,149            13,212              53
  9            1,887,748             360,508           413,392             260
 10              623,800             930,588           604,516           1,460
 11                2,644           1,350,852             2,644          34,088
 12                                  782,536                           402,260
 13                                   90,280                            88,636
 14                                      276                               276
               ---------           ---------         ---------         -------
               3,674,160           3,674,160         1,034,783         527,049

        Analysis of the full 3x3x3 cube group
        -------------------------------------

Moves Deep   arrangements (q+h)  bf        arrangements (q only) *

  0                    1         --                     1
  1                   18         18                    12
  2                  243         13.5                 114
  3                3,240         13.33              1,068
  4               43,239         13.34             10,011
  5              574,908         13.29             93,840 (March   22, 1981)
  6            7,618,438         13.25            878,880 (August  14, 1981)
  7          100,803,036         13.23          8,221,632 (December 7, 1981)
  8        1,332,343,288         13.217        76,843,595 (July    18, 1994)
  9       17,596,479,795         13.207       717,789,576
 10                                         6,701,836,858
 11                                        62,549,615,248 (February 4, 1995)

	PH[0]  = 1
	PH[1] <= 6*3*PH[0]
	PH[2] <= 6*2*PH[1]   + 9*3*PH[0]
        PH[n] <= 6*2*PH[n-1] + 9*2*PH[n-2] for n > 2

Solving yields the following upper bounds:

htw        new        total      htw        new         total
 0           1            1       10    2.447*10^11   2.646*10^11
 1          18           19       11    3.267*10^12   3.531*10^12
 2         243          262       12    4.360*10^13   4.713*10^13
 3        3240         3502       13    5.820*10^14   6.292*10^14
 4       43254        46756       14    7.769*10^15   8.398*10^15
 5      577368       624124       15    1.037*10^17   1.121*10^17
 6     7706988      8331112       16    1.385*10^18   1.497*10^18
 7   102876480    111207592       17    1.848*10^19   1.998*10^19
 8  1373243544   1484451136       18    2.467*10^20   2.667*10^20
 9 18330699168  19815150304

        Analysis of the 3x3x3 squares group
        -----------------------------------

                                          branching
Moves Deep       arrangements (h only)     factor      loc max (h only)

  0                    1                      --             0
  1                    6                      6              0
  2                   27                      4.5            0
  3                  120                      4.444          0
  4                  519                      4.325          0
  5                1,932                      3.722          0
  6                6,484                      3.356          1  (6 X pattern)
  7               20,310                      3.132          0
  8               55,034                      2.709         65
  9              113,892                      2.069      1,482
 10              178,495                      1.567      7,379
 11              179,196                      1.004     25,980
 12               89,728                      0.501     50,320
 13               16,176                      0.180     11,328
 14                1,488                      0.092        912
 15                  144                      0.096        144
                 -------                                ------
                 663,552                                97,611

        Analysis of the 3x3x3 <U, R> group
        ----------------------------------
ML's Conjecture: The < U, R > group is >=20 turns deep in qt metric
                 Now confirmed, Sept 1, 1994

                                          branching
Moves Deep       arrangements (q only)     factor

  0                    1                      --
  1                    4                       4
  2                   10                       2.5
  3                   24                       2.4
  4                   58                       2.416
  5                  140                       2.413
  6                  338                       2.414
  7                  816                       2.414
  8                1,970                       2.414
  9                4,756                       2.414
 10               11,448                       2.407
 11               27,448                       2.401
 12               65,260                       2.378
 13              154,192                       2.363
 14              360,692                       2.339
 15              827,540                       2.294
 16            1,851,345                       2.237
 17            3,968,840                       2.144
 18            7,891,990                       1.988
 19           13,659,821                       1.755
 20           18,471,682                       1.352
 21           16,586,822                       0.898
 22            8,039,455                       0.485
 23            1,511,110                       0.188
 24               47,351                       0.031
 25                   87                       0.002
              ----------
              73,483,200


        Analysis of 3x3x3 corners only
        ------------------------------

Moves Deep   arrangements (q+h)   arrangements (q only) *  loc max (q only)

  0                    1                       1                   0
  1                   18                      12                   0
  2                  243                     114                   0
  3                2,874                     924                   0
  4               28,000                   6,539                   0
  5              205,416                  39,528                   0
  6            1,168,516                 199,926                 114
  7            5,402,628                 806,136                 600
  8           20,776,176               2,761,740              17,916
  9           45,391,616               8,656,152              10,200
 10           15,139,616              22,334,112              35,040
 11               64,736              32,420,448             818,112
 12                                   18,780,864           9,654,240
 13                                    2,166,720           2,127,264
 14                                        6,624               6,624
              ----------              ----------          ----------
              88,179,840              88,179,840          11,870,110

        Analysis of 3x3x3 edges only
        ----------------------------

    Distance    Number of  Branching    Number of  Branching
        from   M-Conjugate   Factor    M-Conjugate   Factor
       Start     Classes                 Classes
                Without                  With
                Centers                 Centers

           0            1                       1
           1            1       1.00            1         1.00
           2            5       5.00            5         5.00
           3           25       5.00           25         5.00
           4          215       8.60          215         8.60
           5        1,860       8.65        1,886         8.77
           6       16,481       8.86       16,902         8.96
           7      144,334       8.76      150,442         8.90
           8    1,242,992       8.61    1,326,326         8.81
           9   10,324,847       8.31   11,505,339         8.67
          10   76,993,295       7.46   96,755,918         8.40
          11  371,975,385       4.83  750,089,528         7.75
          12  382,690,120       1.03      ....
          13    8,235,392       0.02      work
          14           54       0.00       in
          15            1       0.02    progress

              -----------
    Total     851,625,008

Total number of positions on edges-only 3x3x3:
 (2 ^ 12 / 2 ) * 12! =  980,995,276,800

*Note* that normally there would be only half the number of positions
 since on a real 3x3x3 cube you can't exchange one pair of edges alone.

              Analysis of Pyraminx
              --------------------

Moves Deep       arrangements      branching
                                    factor

  0                    1             --
  1                    8              8
  2                   48              6
  3                  288              6
  4                1,728              6
  5                9,896              5.726
  6               51,808              5.235
  7              220,111              4.248
  8              480,467              2.183
  9              166,276              0.346
 10                2,457              0.015
 11                   32              0.013
                 -------
                 933,120

(If tips are included: 933,120 * 3^4 = 75,582,720)

               Analysis of the Skewb
               ---------------------

The correct numbers for H = < RUF, LUB, RDB, LDF > are as follows

 Moves Deep    Arrangements
 ----------    ------------
     0                  1
     1                  8
     2                 48
     3                288
     4              1,728
     5             10,248
     6             59,304
     7            315,198
     8          1,225,483
     9          1,455,856
    10             81,028
    11                 90
