Cube Research/Math/Theory subforum

[badmephisto]

We lead a lot of similar discussion in the Off-Topic/Speedcubing sections of the forum, but I think this deserves a section of its own;

For example, we have threads on the least number of moves to solve any cube. We also had threads on God's algorithm, and the 2phase algorithms. We had a lot of threads on Commutators and Conjugates. We also had many threads that discuss new possible methods, and their short-comings and strengths. Some people were also interested in knowing how to solve the cube completely intuitively.

I really do believe Research/Math/Theory deserves a section of its own because there was a lot of discussion surrounding it, there are many people on the forum interested in it, and it is most certainly not off-topic.

 

[Lucas Garron]

Well, we don't have a forum for general cubing topics.
(Which, as is evident from twistypuzzles.com, wouldn't really work)

I'm not sure if having such a forum will cause it to be used well, but I'm in favor.
As long as we don't get a weekly beginner asking about this cube math they'd been told about, we might actually get something done!

You know about http://cubezzz.homelinux.org/drupal/, right?

 

[badmephisto]

wow, bookmarked!! How did I never find out about this? I read through half of Stefan's paper, his program Hume seems very interesting. Can't wait to wake up tomorrow and read more

anyway, we need a place to discuss exactly that stuff, somewhere.

 

[Dene]

Ooooh I love how he named his program after my favouritest philosopher ever!!!

As for the subforum, it would be nice, but I don't think it would get enough traffic.

EDIT: After quickly skimming over the thesis, I see it is in English. So, to keep it simple, why? Why not German? Surely it could have a lot more meaning (or is English sufficient and used for a wider audience?)

EDIT 2: I think this is a great idea, btw. You have motivated me to find any excuse I can to do postgraduate study on the Rubik's Cube.

 

[fanwuq]

That would be nice. More to read and nice opportunities to get confused. The best part about cubing!

 

[immortalcube]

Ah, but being confused is a good chunk of learning something new, right? This looks like a good idea, I'd love to learn more about 'cube theory'

 

[pjk]

I like the idea, but I highly doubt we would have enough interest. The speedcubing sub-forum should be sufficient in the meantime. Plus we have added several new forums recently, so we probably want to let the site grow before adding more sub-forums.

 

[ruffleduck]

What is the centralizer of the Rubik's cube group in itself? i.e. the set of elements g such that g*a*g^(-1) = a (for all a), or in other words the set of elements which commute with every other element.

Of course the identity and super-flip is in the set. What else? Is checkerboard in the set?

(A big bump considering that this thread is only a couple years younger than yours truly. I couldn't find a better thread to post this question, and I felt that it is too minor a question to warrant its own thread. Also, it would be cool to see this thread revived.)

 

[Renslay]

What is the centralizer of the Rubik's cube group in itself? i.e. the set of elements g such that g*a*g^(-1) = a (for all a), or in other words the set of elements which commute with every other element.

Of course the identity and super-flip is in the set. What else? Is checkerboard in the set?

(A big bump considering that this thread is only a couple years younger than yours truly. I couldn't find a better thread to post this question, and I felt that it is too minor a question to warrant its own thread. Also, it would be cool to see this thread revived.)

Only the id and the superflip, nothing else.
g cannot contain any permutation, otherwise there can be an a such that gag' is something different than a, because permutation is non-commutative generally.
Similarly, g cannot contain a strict subset of edge flips.
Also, g cannot contain corner twists, because they have 3 states (oriented, cw, ccw).

Not a rigid proof, more like an outline of a proof. But if any of these are not clear, and can try to explain it in more detail.

The remaining states are the id and the superflip, and both works.

 

[edward_9x]

darn i was writing my attempt


Let g\in Z(G). Then, gxg'=x,\forall x\in G.

EO: Assume g flips 1, 2 or 3 edges on at least one face f, then we can face the cube so f is on the U layer for convenience. We can then set x=U, so gUg'=U, i.e. flipping 1, 2 or 3 edges in the LL, doing a quarter-turn AUF then unflipping the new edges that are in the same spot as the old ones is the same to the EO as not doing anything. This is obviously false. So for any face f, g flips either 0 or 4 edges, i.e. it either flips all edges on the cube or none.

EP & CP: i actually messed this up

CO: idk

 

[xyzzy]

darn i was writing my attempt


Let g\in Z(G). Then, gxg'=x,\forall x\in G.

EO: Assume g flips 1, 2 or 3 edges on at least one face f, then we can face the cube so f is on the U layer for convenience. We can then set x=U, so gUg'=U, i.e. flipping 1, 2 or 3 edges in the LL, doing a quarter-turn AUF then unflipping the new edges that are in the same spot as the old ones is the same to the EO as not doing anything. This is obviously false. So for any face f, g flips either 0 or 4 edges, i.e. it either flips all edges on the cube or none.
[...]
CO: idk

EO and CO can be handled similarly. Once we know that EP and CP are solved, this forces g to be a pure orientation state.

If this commutes with every piece permutation (in particular, with the transpositions), then every edge must have the same orientation and likewise every corner must have the same orientation.

For edges, the only possibilities are solved and all-flip. For corners, the only possibility is solved; there's no all-cw-twist or all-ccw-twist because those are unsolvable CO states.