From:

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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.

Dan