# 44 million cubes

12 | 14 |

13 | 235 |

14 | 3823 |

15 | 58086 |

16 | 835372 |

17 | 9817695 |

18 | 31335765 |

19 | 2038930 |

14 | 1 |

15 | 37 |

16 | 579 |

17 | 6062 |

18 | 15263 |

19 | 811 |

## Comment viewing options

### Very interesting!!! I wonder

### symmetry and deep positions

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 sy

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

### 20 move positions with symmetries

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

> 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

^{-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

^{-1}xm=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.

### Antisymmetry

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.

### Tendency to antisymmetry for 20f-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.

### Certainly in the restricted t

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

### Maybe I expressed not clear e

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

### I believe there is a strong c

### > Do you mean that if a posit

> 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.

## Have you ever done a similar