2gen question

Calling a 21QTM alg "the OH version" when there's a nice one with 12QTM seems weird.

Usually 2gen means <U,R> but it's not precisely defined.
 
In 2-gen you don't move 2 faces, you move 2 "planes". For a 5x5x5 that could mean something as stupid as l and r slices.
In general, people mean RU when they are talking about 2-gen, but sometimes LU (for righthanded OH) or MU (speed) is also meant.
Technically EB and SD are 2-gen, but why would you use that?

Jacco said:
R2 U2 R2 U2 R2 U' R2 U2 R2 U2 R2 U
Click to expand...
R2 U2
R
U2 R2 U2 R2 U2
R
U2 R2
:p
 
Last edited:
2-gen refers to <U,R> most commonly (or <F,R> or other isomorphic group). When you say "2-gen", people think <U,R>. So to avoid confusion, if you want to talk about <M,U>, you should call it something else.
 
From a purely geometrical standpoint, I don't think anything that changes the orientation of the centers can be 2-gen, since it transforms your coordinate axes (or planes, as AvGalen refers to it). Lucas' argumentum ad absurdum points out exactly why: r u r is the same as L B D (I think I got that right), but that is clearly not 2-gen. However, with that said, r u f is 2-gen, so my statement really only refers to the notation. F M U is 2-gen, F M F is not. The proof of the last assertion is left as an exercise for the reader.

However, on thing that is not clear to me: is R U L 2-gen? According to Macky's glossary 2-gen means two faces, not two planes. To be completely rigorous it should probably be defined as two faces on different planes, since L R L' R' is just silly. :-)
 
brunson said:
From a purely geometrical standpoint, I don't think anything that changes the orientation of the centers can be 2-gen, since it transforms your coordinate axes (or planes, as AvGalen refers to it). Lucas' argumentum ad absurdum points out exactly why: r u r is the same as L B D (I think I got that right), but that is clearly not 2-gen. However, with that said, r u f is 2-gen, so my statement really only refers to the notation. F M U is 2-gen, F M F is not. The proof of the last assertion is left as an exercise for the reader.

However, on thing that is not clear to me: is R U L 2-gen? According to Macky's glossary 2-gen means two faces, not two planes. To be completely rigorous it should probably be defined as two faces on different planes, since L R L' R' is just silly. :-)
Click to expand...
Well, I don't agree. 2-gen is 2 generators.

<r,u> is 2-gen.

< (RUR'URU2R') , (RUR'U'R'FR2U'R'U'RUR'F') > is 2-gen!

R and U are our traditional cube state generators. If we consider the cube group (is it still a group?) on the stickers, nothing's wrong with using M as a generator. The Sune is a perfectly good state to use as a generator, too...

But how about < U , (toss the cube up in the air and catch randomly) > ?
(What's (toss the cube up in the air and catch randomly)' ?)
It has a finite expected solve length. I should try it...
Actually, now that I think about it, it's a very interesting problem. I think optimal would be to consider the few shortest solutions after every U-turn, and continue on whichever's first move face lands on U first (then recompute, etc.)...
 
I agree with Lucas, 2-gen is 2 generators. Be it a whole alg or just a single turn (F, R, U, B, L, D, M, E, S). As long as you have two generators its 2 gen. And, I don't see why ppl are contesting the definition seeing as its name is its own definition...

2-gen = 2-generator
 
I also agree with Lucas (and CanadianPires) that < (RUR'URU2R') , (RUR'U'R'FR2U'R'U'RUR'F') > is quite literally (in group theory) "2-gen." Of course, that means something as simple as a U layer adjustment is forbidden for that 2-gen group. There are only 216 total elements in < (RUR'URU2R') , (RUR'U'R'FR2U'R'U'RUR'F') >.

I guess taking "2-gen" too literally in cubing can be kind of silly, though. Any alg could be split into two parts, and each part defined to be a generator, simply so it could be called a "2-gen" alg. This would make "2-gen" a rather useless term (in cubing). Usually, for a mechanical puzzle, generators are chosen to correspond with the simplest mechanical movements of the puzzle that perform some useful transformation of the puzzle. Then algs can show how more complicated movements can be generated from the most basic ones.

I am not clear about Lucas's notation: < U , "(toss the cube up in the air and catch randomly)" >. I don't see that as a group, and thus, the notation is nonsense, in my view. Lucas needs to extend the the definition of the group operation so that operations with this new element are well defined. Then show that the group operation still satisfies all the various properties necessary for satisfying the definition of a group. For example, what's the inverse of "(toss the cube up in the air and catch randomly)"?

I also noticed this claim in the "A Collection of Permutation Algorithms" thread recently:
I found a 1-gen T-perm.
z' U z U z' U U U z U U U z' U U U z x U x' z' U U z U U U z' U U U z U U U z' U z U z' U U U x U U U
Click to expand...

Of course, mathematically speaking anyway, this is not 1-gen. Obviously, the group <U> only has four elements, none of which is a T-perm. The above alg relies on cube rotations (x and z) to change which center's layer is actually rotated by U. So it's actually 3-gen: < U, x, z >. Using GAP, I find that a T-perm configuration can actually be generated using only < U, x > or < U, z >. So the "3-gen" alg could be replaced with a "2-gen" alg.
 
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