44 million cubes

Yessir; here are the results of those (33,970 cubes):

16	1
17	23
18	283
19	2336
20	11300
21	14543
22	5465
23	19

In doing various analyses, particularly the huge permutation-of-the-cubies analysis in QTM, I have noticed that the deepest positions tend to have symmetry, even though symmetric positions represent a rather small amount of the total positions. From my permutations analysis, I noticed that all depth 17q positions have symmetry in the corner configuration. Of these 11 (unique wrt M+inv) positions, all but one have symmetry in the edge configuration. That position is the only one that does not have symmetry when both edges and corners are considered. Below I list these 11 positions and the degree of symmetry (how many elements in M do not change the position under conjugation) for the corner configuration, edge configuration, and the overall position. I also show the order of each position.

Depth 17q positions in corner and edge permutations analysis

     Position        Symmetries    Order
    CP         EP     C  E All    C  E All
 -----  ---------    --  - ---    -  - ---
    54  130377481    12  3  3     2  4  4
   127  163710727     4  1  1     2  6  6
   415    3653859     4  4  4     2  2  2
  4761  123758783    12  8  4     2  2  2
  5209  127387463     4  4  2     4  4  4
  9800  127389022     4  2  2     4  4  4
  9800  127390445     4  2  2     4  4  4
 12322  299547493     4  8  4     4  4  4
 14128  409720015     4  4  4     8  8  8
 14128  449636809     4  4  4     8  8  8
 14854  248577983     4  2  2     4  8  8

- Bruce Norskog

Here's the distribution of symmetry for the 74 length-20 positions for the earlier analysis I posted (four sets of 44M solutions):

  count symmetry
      9 1
     22 2
      6 3
     19 4
      5 6
     10 8
      2 24
      1 48

Here a symmetry of 1 means *no* symmetry, and a symmetry of 48 means full M-symmetry. Certainly symmetrical positions are overrepresented, but there are a surprisingly large number of asymmetrical positions, too.

Over the last two weeks time I made a list of *all* possible cube positions which have more than 4 symmetries and ran my optimal solver over them.
Less than 200 have length 20 and there are no 21 maneuvers. I can give detailed results and also post the maneuvers, if desired.
So it is proven that a cube position with length 21 must have 4 or less symmetries.

I'd be interested to see the details, if you have time.

> positions which have more than 4 symmetries

Do the lower-symmetry 20f cases tend to have antisymmetry? [Suggested by one (!) example,
D' F2 L2 B2 R F' L2 D R2 D2 B U F2 L2 R2 D' L2 F L' U' (20f).]

I wondered if you had considered looking at the antisymmetric cases with 3 and 4 symmetries.

I'm not sure I understand the question. I believe that the way antisymmetry has been used on this list is to refer to the fact that for a position x, the length of the inverse x^-1 is the same as the length of x. This approach allows a search space to be reduced by a factor of approximately 96, rather than by a factor of approximately 48 if only symmetry is taken into account. Do you mean that if a position "has antisymmetry" that it is not equal to its inverse. Obviously, if a position is equal to its inverse then you can't get a 96 times reduction for that particular position.

If we take (as per Cube-Lovers) Symm(x) to mean the set (and group) of all m in M such that m^-1xm=x, then it is the case that Symm(x^-1)=Symm(x). So if Herbert has already checked all positions that have more than four symmetries, then he has automatically also checked their inverses. I assume that by "more than 4 symmetries", Herbert means |Symm(x)|>4.

We have h^-1xh=x for h in the symmetry group H of x. In addition there is a group G, also a subgroup of M. H is a subgroup of index 2 in G and for all g in G which are not in H we have g^-1xg=x^-1.
If x is selfinverse, this definition is quite useless, but there are many examples with x^-1<>x which have antisymmetry. In the symmetry editor of Cube Explorer you can check the antisymmetry radiobutton and then define the groups H and G. In addition, if you have a cube in the main window and select it by clicking with the mouse on it while having the symmetry page open, all symmetries will show up with the symmetry icons pressed and the antisymmetries will show up with the symmetry icons raised.

To clarify just one thing:

> If x is selfinverse, this definition is quite useless

...in the sense that there are two distinct forms that AntiSymm(x) may have:

(1) the trivial form H+T.H (where H is the symmetry group of a self-inverse x), and

(2) the more interesting form H+T.H' (where H' is the coset of H in G), which you describe.

Positions with either kind of antisymmetry group /might/ be worth investigating further, depending on whether the symmetric 20f-positions tend also to be antisymmetric.

Though I did not analyse it carefully it seems to me, that antisymmetry does not happen more often for 20f-positions than for other positions.
>(2) the more interesting form H+T.H' (where H' is the coset of H in G), which you describe.

I am sure you do not mean coset, but the complement of H in G.

> I am sure you do not mean coset

True indeed; "coset" would allow H'=H...

Certainly in the restricted test I performed, it does happen more frequently (given that the restricted test has the edges already in a fully M-symmetric position); indeed, fully 66 of the 74 length-20 M-unique positions were antisymmetric; the remaining 8 were 4 pairs of "inverses" (i.e., the inverse was M-congruent to the other member of the pair).

Maybe we should start a database of 20f positions, especially since they are so relatively difficult to find?

Maybe I expressed not clear enough. What I wanted to say, that it seems to me that for example with all cubes which have symmetry S_6 there is no correlation between antisymmetry and the maneuver lenght. There are also many antisymmetric positions with S_6 symmetry which are 19f or 18f.

The database is a good idea. How should we organize this?

I believe there is a strong correlation between antisymmetry and maneuver length; almost all the 20f* cubes we've found have been antisymmetric, but very few random cube positions are antisymmetric.

> Do you mean that if a position "has antisymmetry" that it is not
> equal to its inverse.

If m'xm = x', where m is an ordinary symmetry element, then we say that x has the antisymmetry Tm. This is the same terminology that Herbert and I have used earlier on this list.

The combined set of symmetries and antisymmetries of x also form a group, which you might call AntiSymm(x).

> by "more than 4 symmetries", Herbert means |Symm(x)|>4.

That would be my understanding, too.

I simply wondered whether he had thought about extending his calculation to include all of the cases with |AntiSymm(x)| > 4. So far, I think, his results will have included all those with |AntiSymm(x)| > 8.