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

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.

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...

> 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?

### I can give you a stochastic a

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.

### Thanks -- that's clearer. I

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.

### Wow, good point! I tend to o

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.

### Great work! Of the 771 kno

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.