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Two more classes with exacly 4 symmetries done - most 20f* are antisymmetric

Submitted by Herbert Kociemba on Thu, 03/30/2006 - 12:09.
I finished the analysis of two more classes with 4 symmetries now. The computation took more than two weeks. All can be solved within 20 moves.
The definition of the classes D2(edge) and C2v(b) are explained on this page. Here you also can get some more information about these and other classes.

What is interesting, that from the 12 20f*-cubes of the class D2 (edge), 10 also have antisymmetry and from the 94 20f*-cubes of the class C2v(b) 92 also are antisymmetric.
From all cubes in D2(edge), only 44% have antisymmetry, from all cubes in C2v(b) it are 53%. Is there an explanation for the exceptional high number of antisymmetric cubes in the 20f* cases?
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First things first...

Submitted by Mike G on Fri, 03/31/2006 - 12:39.
> Is there an explanation for the exceptional high number of antisymmetric cubes in the 20f* cases?

Can we even explain clearly why the cubes of high symmetry should have a disproportionately large number of 20f cases?
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I can give you a stochastic a

Submitted by rokicki on Fri, 03/31/2006 - 13:18.
I can give you a stochastic argument.

Let's say the group is random, for some definition of a random group that I won't get into here. A position with low symmetry has more "distinct" moves to bring it closer to a solution; a position with high symmetry has fewer "distinct" moves.

Take a center position, like superflip. In the quarter-turn metric, there is only a single move wrt M; in the half-turn metric, only two moves wrt M. A random assymetrical position, on the other hand, has 12/18 moves (QTM and HTM respectively), and thus has more "chances" to get closer to identity.

Antisymmetry works in a similar fashion.

Another perspective is to consider the tree of positions reachable from the initial position, with respect to M+inv. From superflip and QTM, this tree has only a single position at depth 1, and (I think) five positions at depth 2; thus, a random position (say, identity) is more likely to be pretty deep in that tree. For an assymetrical position, the tree has 18 moves at depth 1, and a few hundred at depth 2, and so on; the tree is shallower and thus identity probably is closer.
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Thanks -- that's clearer. I

Submitted by Mike G on Fri, 03/31/2006 - 15:07.
Thanks -- that's clearer. I can see now that others have alluded to arguments of this kind, without making them explicit.

> Antisymmetry works in a similar fashion.

Except, perhaps, that the M+inv reduction is a little harder to visualize. It would be quite a subtle "thinning out" if you were starting from a self-inverse, unsymmetrical cube position.
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Wow, good point! I tend to o

Submitted by rokicki on Fri, 03/31/2006 - 16:26.
Wow, good point! I tend to overlook this because frequently I think in terms of appending moves on both "ends" of a position (i.e., moves are not just position.U and position.F but also U.position and F.position) but you're quite right. I'll have to think some more about this.
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Great work! Of the 771 kno

Submitted by rokicki on Thu, 03/30/2006 - 15:27.
Great work!

Of the 771 known depth-20 positions (known to me, that is, with contributions from both Kociemba and Silviu), 535 are
antisymmetric but not self-inverse, and 142 are self-inverse,
so a total of 677 of the 771 are antisymmetric.

Are there more antisymmetric cubes than symmetric cubes? Maybe this would be an interesting set to study.
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