Karl Dahlke, the author of the pocket cube program I mentioned on 1
August, sent me a note about the appearance of the unusual constant 870
in his program. It turns out that the program is correct, and the
constant arises in an interesting way.
Recall that the pocket cube has 729 orientations and 5040 permutations
of the pieces. Dahlke had noticed that the ``reflections and
rotations'' of a position need not be stored, since they are the same
distance from start. By reflections and rotations, he means the
S-conjugates, where S is the six-element symmetry group of the pocket
cube with one corner fixed.
It turns out that the pocket cube has 2 permutations with a six-element
symmetry group, 16 permutations with a three-element symmetry group,
138 permutations with a two-element symmetry group, and 4884
permutations with a one-element symmetry group. Thus the number of
permutations that are distinct up to S-conjugacy is 2 + 16/2 + 138/3 +
4884/6 = 870.
This discussion of symmetry recalls a question I have meant to propose
to Cube-Lovers for some time: How many positions are there in Rubik's
Cube? We know from Ideal that the number is somewhat over three
billion. Most cube lovers will tell you a number of about 43
quintillion. But I really don't see why we should count twelve
distinct positions at one quarter-twist from solved--all twelve are
essentially the same position. So the question, suitably rephrased, is
of the number of positions that are distinct up to conjugacy in M, the
48-element symmetry group of the cube. I think this is an interesting
question, but I don't see any particularly easy way of answering it.
My best guess is that it involves a case-by-case analysis of the 98
subgroups of M, or at least the 33 conjugacy classes of those
subgroups. In ``Symmetry and Local Maxima'', Jim Saxe and I examined
five of the classes, which we called M, C, AM, H, and T.
Even finding the numbers for the pocket cube is a little tricky. If we
limit ourselves to symmetry in S, I believe the pocket cube has 2
positions with a six-element symmetry group, 160 positions with a
three-element symmetry group, 3882 positions with a two-element
symmetry group, and 3670116 positions with a one-element symmetry
group, for 613062 positions distinct up to S-conjugacy. But the
numbers for M-conjugacy are still elusive; I am not even sure how to
deal with factoring out whole-cube moves in the analysis. I hope to
find time to write a program for it.
I expanded my pocket cube program to deal with the corner group of
Rubik's cube. This group is 24 times as large as the group of the
pocket cube, having 3^7 * 8! = 88179840 elements. The number of
elements P(N) and local maxima L(N) at each (quarter-twist) distance N
from solved are given below.
N P(N) L(N) 0 1 0 1 12 0 2 114 0 3 924 0 4 6539 0 5 39528 0 6 199926 114 7 806136 600 8 2761740 17916 9 8656152 10200 10 22334112 35040 11 32420448 818112 12 18780864 9654240 13 2166720 2127264 14 6624 6624
The alert reader will notice that rows 10 through 14 contain values
exactly 24 times as large as those for the pocket cube. This is not
surprising, given that the groups are identical except for the position
of the entire assembly in space, and each generator of the corner cube
is identical to the inverse of the corresponding generator for the
opposite face except for the whole-cube position. Thus when solving a
corner-cube position at 10 qtw or more from solved, it can be solved as
a pocket cube, making the choice between opposite faces in such a way
that the whole-cube position comes out right with no extra moves.