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Regarding God's Algorithm Calculations and lower bounds on face metric

Submitted by xintax on Mon, 03/06/2006 - 10:35.
Hi...
Im kinda new here. Just a few intros: Im sheelah from the philippines and im doing an overnight research on the length of God's Algorithm... due tomorrow.

I was kinda ready but then I encountered this page that says that the real size of the cube universe is not 43 quintillion but rather 901 quadrillion.

For one, the repercussions of this on my project are:
1.) Theoretically less runtime and
2.) A different criteria for 'similarity'.

I was trying to read the calculations but I got lost between the numbers. Can someone please clarify this for me?

Secondly, I want to ask about the current lower-bound of the length of God's Algorithm in face turn metric. I keep seeing 20f* but I was wondering if there is a higher lowerbound...

Thank you for any information that you might provide...
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I can say that the actual low

Submitted by silviu on Mon, 03/06/2006 - 16:09.
I can say that the actual lower bound is 20f.
The 901 quadrilion number means how many equivalence classes there are when the group M acts on the cube by conjugation. This action has the property that the elements in an equivalence class have the same length. This number is reduced to half if you also allow inversion see the Godfrey Kociemba article on this forum.
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Clarifications

Submitted by xintax on Thu, 03/09/2006 - 03:12.
1.)what is group M?

2.)Regarding equivalence classes, does it mean, for example that LRU and R' LRU R are of the same equivalence classes?

I'm really sorry, i'm hopelessly a nooB...
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To expand just a little bit m

Submitted by Mike G on Fri, 03/10/2006 - 07:02.
To expand just a little bit more -- I'll try not to introduce any inaccuracies...

> 1.)what is group M?

It is the group of geometrical symmetries of an ordinary cube (or a solved Rubik's Cube): the 48 rotations and reflections. An equivalence class defined [as above] via conjugation by elements of M is a set of positions that are "the same" as one another, apart from rotation or reflection. So the set of all equivalence classes is in 1-to-1 correspondence with the set of "distinct" positions of the Cube; i.e., a minimal set of positions, none of which is related to another by symmetry. This is why people refer to the "real" size of Cube space in this context.

> 2.)Regarding equivalence classes, does it mean, for example that LRU and R' LRU R are of the same equivalence classes?

Conjugation by the elements of the Cube group (with ~4e19 elements) itself could ALSO be used to define equivalence classes, but they would not be the same as the ones referred to above. In this case, the members of an equivalence class would all have the same cycle structure.
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An equivalence class containi

Submitted by silviu on Thu, 03/09/2006 - 03:37.
An equivalence class containing a certain elemet x in the cube group contains all elements m'xm with m in M. Typically 48 elements because the group M has 48 elements. For a general definition of an equivalence class you can consult the Mathworld site.
You can define the generators of the cube in GAP i.e. U,L,F,B,R,D. And you can also define the generators of M. There is a post about this on the forum submitted by Joe Miller. In the comment of this post I submitted the generators of M with respect to the generators I use. If you conjugate a generator by an element of M then you get another generator of the cube.
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oh...

Submitted by xintax on Sun, 03/12/2006 - 11:12.
Thank you so much for the clarifications. :)
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