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

                                          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
