All cube states can be solved using only one of these PLLs: J(a), J(b), R(a), R(b)

[blah]

Edit: Found

Spoiler

Odd vs even.

Am I right?

I suspect you mean the right thing, but it's not clear what you mean.

Permutation parity.

To clarify:

Spoiler

Even parity is achievable by performing a finite number of an algorithm that achieves odd parity.
Odd parity is not achievable by performing any number of an algorithm that achieves even parity.

The simplest example is this:
Alg that achieves odd parity: U
Alg that achieves even parity: U2

 

[cuBerBruce]

@Stefan: I did notice that the Rs and the Js are all corner 2-cycles even before you told me about the Gs, but it didn't occur to me that it had anything to do with the result; thought it was just a coincidence

This J has a corner 3-cycle: L R U2 R' U' R U2 L' U R'

Any PLL can have either even or odd parity in the corners, depending upon the AUF case you use.

 

[blah]

I think you missed this:

And the G perms work as well.

Ah, yes. Thanks for that But that's if you define a G-perm as a corner 2-cycle and an edge 4-cycle When most cubers say G-perm, they mean a corner 3-cycle and on edge 3-cycle. Just like when they say J-perms or R-perms, they mean a corner 2-cycle and an edge 2-cycle, not a corner 3-cycle and an edge 3-cycle.

 

[Stefan]

Odd parity is not achievable by performing an infinite number of an algorithm that achieves even parity.

Please write "any number" rather than "an infinite number". I don't even have time to perform an infinite number at all.

Alg that achieves odd parity: U

Sorry, I'd say that's even parity.

 

[blah]

Odd parity is not achievable by performing an infinite number of an algorithm that achieves even parity.

Please write "any number" rather than "an infinite number". I don't even have time to perform an infinite number at all.

Sorry, bad choice of words

Alg that achieves odd parity: U

Sorry, I'd say that's even parity.

Okay okay, so I phrased it poorly Odd parity for corners and for edges. Sum of parities is even. Was this the problem in my original post?

 

[cuBerBruce]

I think you missed this:

And the G perms work as well.

Ah, yes. Thanks for that But that's if you define a G-perm as a corner 2-cycle and an edge 4-cycle When most cubers say G-perm, they mean a corner 3-cycle and on edge 3-cycle. Just like when they say J-perms or R-perms, they mean a corner 2-cycle and an edge 2-cycle, not a corner 3-cycle and an edge 3-cycle.

And I would basically disagree with that comment...

Most speedsolvers would would say something like "I had a J-Perm" regardless of which angle case and which AUF case they actually had.

In BLD solving, yes, it's important to realize (and usually clear in such discussions) which AUF case and which angle case is meant.

 

[blah]

Most speedsolvers would would say something like "I had a J-Perm" regardless of which angle case and which AUF case they actually had.

I meant it in a different sense.

The reason it's called a J-perm is because in a PLL diagram, the arrows look like a J. When was the last time you saw a J-perm PLL diagram with a corner 3-cycle and an edge 3-cycle?

When people learn algorithms for the J-perm, they learn one that swaps 2 corners and 2 edges, not one that cycles 3 and 3. Likewise, when people learn algorithms for the U-perm, they learn one that cycles 3 edges, not one that cycles 4 corners and swaps 2 edges.

 

[brunson]

I was constipating really bad in the bathroom, so to ease the pain, my mind wandered off into the realm of cubes

Familiar with the concept of TMI?

 

[cuBerBruce]

Most speedsolvers would would say something like "I had a J-Perm" regardless of which angle case and which AUF case they actually had.

I meant it in a different sense.

The reason it's called a J-perm is because in a PLL diagram, the arrows look like a J. When was the last time you saw a J-perm PLL diagram with a corner 3-cycle and an edge 3-cycle?

When people learn algorithms for the J-perm, they learn one that swaps 2 corners and 2 edges, not one that cycles 3 and 3. Likewise, when people learn algorithms for the U-perm, they learn one that cycles 3 edges, not one that cycles 4 corners and swaps 2 edges.

No, when people learn PLLs, they often learn one that needs an AUF move to make it correspond the canonical arrows diagram. These algs are usually written with an additional U, U' or U2 move in parentheses to indicate it's for a different AUF case. Often, there may also be a parenthesized y, y' or y2 at the beginning to indicate it's for a different angle case.

 

[StachuK1992]

I was constipating really bad in the bathroom, so to ease the pain, my mind wandered off into the realm of cubes

Familiar with the concept of TMI?

+1

No, when people learn PLLs, they often learn one that needs an AUF move to make it correspond the canonical arrows diagram. These algs are usually written with an additional U, U' or U2 move in parentheses to indicate it's for a different AUF case. Often, there may also be a parenthesized y, y' or y2 at the beginning to indicate it's for a different angle case.

Much like the 'standard' E perm

 

[Lord Voldemort]

Now you prove why the same goes for T-perms

Well, I don't know how lay out a formal proof of this, but T-Perm has an only an opposite swap of edges, so there are some cubes (most) that won't be solvable because they need an adjacent edge swap, which you can't do with pure opposite swaps (again, I don't know how to prove this formally). Same goes for any other PLL that swaps two opposite pieces.

Someone correct me if I'm wrong.

 

[blah]

I was constipating really bad in the bathroom, so to ease the pain, my mind wandered off into the realm of cubes

Familiar with the concept of TMI?

Wikipedia gave me a whole list of stuff with this acronym. Only Transmarginal Inhibition seems to fit the context Too Much Information arguably fits the context of this thread too

Patients who have reached this shutdown point often become socially dysfunctional or develop one of several personality disorders.

I don't consider cubing a "social dysfunction" or a "personality disorder" More like - *thinks hard* - a disease

 

[Tim Reynolds]

Now you prove why the same goes for T-perms

Well, I don't know how lay out a formal proof of this, but T-Perm has an only an opposite swap of edges, so there are some cubes (most) that won't be solvable because they need an adjacent edge swap, which you can't do with pure opposite swaps (again, I don't know how to prove this formally). Same goes for any other PLL that swaps two opposite pieces.

Someone correct me if I'm wrong.

Pieces in the M slice stay in the M slice.
Pieces in the E slice stay in the E slice.
Pieces in the S slice stay in the S slice.

Good enough?

 

[blah]

Now you prove why the same goes for T-perms

Well, I don't know how lay out a formal proof of this, but T-Perm has an only an opposite swap of edges, so there are some cubes (most) that won't be solvable because they need an adjacent edge swap, which you can't do with pure opposite swaps (again, I don't know how to prove this formally). Same goes for any other PLL that swaps two opposite pieces.

Someone correct me if I'm wrong.

Pieces in the M slice stay in the M slice.
Pieces in the E slice stay in the E slice.
Pieces in the S slice stay in the S slice.

Good enough?

For me? Yep I don't remember who coined the term, but these are known as "circuits".

 

[Lucas Garron]

Now you prove why the same goes for T-perms

Well, I don't know how lay out a formal proof of this, but T-Perm has an only an opposite swap of edges, so there are some cubes (most) that won't be solvable because they need an adjacent edge swap, which you can't do with pure opposite swaps (again, I don't know how to prove this formally). Same goes for any other PLL that swaps two opposite pieces.

Someone correct me if I'm wrong.

Pieces in the M slice stay in the M slice.
Pieces in the E slice stay in the E slice.
Pieces in the S slice stay in the S slice.

Good enough?

For me? Yep I don't remember who coined the term, but these are known as "circuits".

Orbits? That's the group theory term, and it means what you're probably thinking of.

 

[blah]

For me? Yep I don't remember who coined the term, but these are known as "circuits".

Orbits? That's the group theory term, and it means what you're probably thinking of.

Ryan Heise's Human Thistlethwaite Algorithm page (I think this is where I got the term):

Put cube into <L2,R2,F2,B2,U2,D2> group, named G3 ... *some text* ... In G3, each corner can only move between one of 4 positions, which I call a circuit.