Things that are 'n-gen'?

[StachuK1992]

WE NEED TO CLEAR UP THIS DEFINITION.

This is how I see it:

By default, many assume the term 2-gen to default as <R,U> solving or scrambling, meaning doing either by only turning the R and U faces.

2-gen can refer to any two faces/slices being turned, so <M,U> or <R,D> are both 2-gen, while <R,U,D> is 3-gen.

2-gen can, being as correct as I know, refer also to algs solved/scrambled with <R U R', D> meaning you're allowed to apply the short sequence "R U R'", its inverse, D, or its inverse.

For this reason, the following algs should be considered 2-gen:
I'll just use | as a random separator
M U M' | <M,U>
R U R' | <R,U>
R U R' D R U' R' D' | <R U R', D>

while this is 3gen
r (R U R' U') r' U2 (R U R U' R2) <R,r,U>


I'd just like to clear up some confusion concerning the n-gen term -
it seems as though people overlook some applications of it.




Just as a note, any 3x3 cube solution should be 3-gen at max:

Spoiler

<R,x,y>

Discuss.

 

[5BLD]

Well, theres some common sense involved I guess.
Technically, <R U F2 U' B S2, D> is also 2-gen. Therefore, maybe a solution given to a scrambled cube could be 1-gen, i.e. <solution to the cube>
However, for our uses this isn't exactly practical right?

By the way, I'm just pondering...

 

[StachuK1992]

Clarification:
ALL 3x3 scrambled positions can be solved with the same 3-gen restrictions.
I think that's better?

 

[Escher]

<R U R' U R U2 R', R U R' F' R U R' U' R' F R2 U' R'>

You make it sound as though <R U R', D> is something special...

 

[StachuK1992]

Of course it's nothing special. I just like my <R,U,D> algs and that's the first generator combination that came to mind.

 

[Escher]

Of course it's nothing special. I just like my <R,U,D> algs and that's the first generator combination that came to mind.

Sure, I was just trying to point out it didn't sound like an analogy/example of something bigger.

 

[Sa967St]

Is there another term that can be used to describe a sequence of turns that involve N faces?

e.g.
Person A: "R U R' D R U' R' D' is 3-gen because it uses 3 different faces."
Person B: "No it's not 3-gen because it can be notated as <R U R', D>, you mean it's 3-______."

 

[5BLD]

Clarification:
ALL 3x3 scrambled positions can be solved with the same 3-gen restrictions.
I think that's better?

Yah.
But still, with your definition... ANY alg could be 2gen.

As I said there's common sense involved...
My point is that ANY sequence of moves can be 2gen by definition.
Also, if RUL is 2-gen, it means on two axes then, not just two layers?
The definition is a bit vague in my opinion.

 

[Escher]

Yah.
But still, with your definition... ANY alg could be 2gen.

As I said there's common sense involved...
My point is that ANY sequence of moves can be 2gen by definition.
Also, if RUL is 2-gen, it means on two axes then, not just two layers?
The definition is a bit vague in my opinion.

How is it vague?

Don't use Stachu's post as a perfect definition of '2-gen'. It doesn't really tell you what a generator is.

<R, U, L> is not '2-gen', for example, but that moveset can be used to write a 2-generator sequence, such as L U' R' U L' R.

The first 'generator' in that instance is the sequence L U'. It cannot be re-written as L U or L' U'. Not 'any' alg can be 2-gen.


Correct me if I'm wrong, I've never studied mathematics beyond the age of 16 and my knowledge comes from a few people on this forum.

 

[JustinJ]

while this is 3gen
r (R U R' U') r' U2 (R U R U' R2) <R,r,U>

No, it's 1-gen.

<r R U R' U' r' U2 R U R U' R2>

 

[5BLD]

How is it vague?

Don't use Stachu's post as a perfect definition of '2-gen'. It doesn't really tell you what a generator is.

<R, U, L> is not '2-gen', for example, but that moveset can be used to write a 2-generator sequence, such as L U' R' U L' R.

The first 'generator' in that instance is the sequence L U'. It cannot be re-written as L U or L' U'. Not 'any' alg can be 2-gen.


Correct me if I'm wrong, I've never studied mathematics beyond the age of 16 and my knowledge comes from a few people on this forum.

Hm. I guess. But my point is that RUL is sometimes referred to as 2gen because it's on two axes...
And I'm not sure of Justin was joking, but yes, his statement DOES fit the definition. Therefore, in this way, ANY alg can be 2gen.
Furthermore, we could even call it 2gen due to it being on two axes...
That's what I mean by vague...

Please do argue against my points if necessary as that's the point of this thread. To discuss.

 

[aronpm]

Hm. I guess. But my point is that RUL is sometimes referred to as 2gen because it's on two axes...

???

 

[StachuK1992]

A single alg can always be 1-gen.

An alg set can always be 3-gen (<R,x,y>).

I favorite alg sets usually fall under <R,U,D>.

I think this could be a more clear way to state this?

 

[whauk]

what is an "alg set"?

 

[StachuK1992]

PLL.

 

[DaKrazedKyubizt]

A single alg can always be 1-gen.

An alg set can always be 3-gen (<R,x,y>).

I favorite alg sets usually fall under <R,U,D>.

I think this could be a more clear way to state this?

Here's a pretty straightforward way to state it: You're saying that the alg set, or whatever it's called (<A, B, C, etc.>) must be defined before defining an algorithm as "?-gen". It cannot be done the other way around.

For example, R2 B2 R F R' B2 R F' R is ?-gen (or I guess you could call it "an undefined algorithm set"), but R2 B2 R F R' B2 R F' R given the generator <R, B, F> is 3-gen, or if you're given the generator <R2, B2 R, F R'> is also 3-gen, and if you're given <R2 B2 R F, R' B2 R F' R>, it becomes 2-gen.

It's similar to saying that you can't have a blob of human skin keep its shape without a skeleton to hold it up.

It's not vague at all.

 

[Stefan]

An alg set can always be 3-gen (<R,x,y>).

I'd say an algorithm like y y y R y isn't the algorithm F, so F isn't in <R,x,y>. Talk about "cube group elements" or maybe "cases" instead of "algorithms", and I'd agree.

 

[5BLD]

This is what I mean. There are many interpretations...
I agree with the crazed cubist... There's no set way to tell is there?
But we use our discretion to tell...

 

[cuBerBruce]

Mathematically speaking, <RUR', D> is a two-generator group. If I write "<RUR', D>", then by definition I am considering RUR' and D to be generators of a group.

Historically, mathematicians have used {U, D, L, R, F, B} as a set of generators for what is generally called the "Rubik's Cube group." It is obvious from these 6 "moves" we can create any legal cube position by combining them in some way. RUR', for example, can be generated by RURRR. The "Rubik's Cube group" was defined as <U, D, L, R, F, B>. (The order of the generators is unimportant). Of course, there are many other ways the same group can be defined in terms of some generators.

Any group using only two of the above generators, such as <U, R>, then logically became to be called two-generator as only two of the six standard generators were being used.

A problem arises in the speedsolving community, because we often use moves that don't keep the centers in a fixed location. (In other words U may not always turn the layer with the same color center piece). We are now dealing with a mathematical group that is 24 times larger than the group that is usually termed "the Rubik's Cube group." Within this larger group, we have other moves that are "basic" enough to think of as "standard generators": M, x, r or Rw, etc. There is really no "standard" set of generators for this group. We could write <U, D, L, R, F, B, x, y, z> or <U, D, L, R, F, B, M, E, S> or <U, x, y>. Logically it tends to make sense to think of any move that can be performed as a single simple action like M, x, or R as a "standard generator" and then calling any combination of the original six generators and this extended set of standard generators as "2-gen". This does not change the fact, of course, that it is always possible (mathemtically speaking, anyway) to use as generators, elements that are combinations of the "standard" generators like RUR' or MUM'.

In any case, <R, U, L> is definitely three-gen. I suppose there exist two elements within this group that will generate the whole group, but simply writing it as <R, U, L> means we are treating this group as a group generated by 3 elements, and thus, we can not say it is two-generator even though only two axes are turned. Anybody who does not understand this simply doesn't understand the concept of n-generator groups.

 

[5BLD]

Yay. A detailed explanation. Makes sense now.