Why does 4x4 Parity happen? (2016 lost media found)

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Күн бұрын

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@paulmahvi3302 Ай бұрын

If you ask other questions about the cube such as "why can't I flip only one edge?", the reason almost always comes down to the theorem (which does not seem to have a name despite being fundamental to all of group theory) that states that no permutation can be obtained with both an even and odd number of transpositions (or swaps). I am currently working on a video series that explains the entire group theory of the Rubik's Cube and similar puzzles.

@thephysicistcuber175 Ай бұрын

EO and CO invariants can usually be explained without the theorem, by just counting how each move affects EO or CO. This, very remarkably, does not work for the very interesting case of the skewb, where CO and CP of two distinct tetrads interact in a nontrivial way.

@paulmahvi3302 Ай бұрын

@@thephysicistcuber175 Yes that is true. Skewb has a different algebraic structure than the 3x3, which is why is has these weird properties. It also has a unique property that is also found on the FTO

@cubykh Ай бұрын

oh wow, that's interesting that the theorem is so pivotal, yet doesn't have a name. Yeah, I'd love to see a video series about group theory of the Rubik's Cube, I remember I signed up for that college class in 2016 because I thought it would help me with cubing! (Also, nice to see you here Paul!)

@thephysicistcuber175 Ай бұрын

@@cubykh you could refer to the theorem as "well-definedness of permutation parity".

@karlkewbz Ай бұрын

Ooo this video is so Early 2016, like it looks visually the same as early TWOW and the Special Relativity Videos! Pretty neat time capsule of sorts!

@cubykh Ай бұрын

Oh my gosh, hi karl! Yeah, I feel like all videos that I made in 2016 have a similar vibe, and I'm happy you remember my Special Relativity Videos (that's one of my proudest video series that is sort of "underground", haha)

@Planes_stacksandcubes Ай бұрын

cary plz return your old haircut

@fayil23 Ай бұрын

i've lowkey always wondered how this works, you're the first one to explain it in a way that makes sense to me 💀

@kiwwicube Ай бұрын

My 3-style knowledge is kicking in right now

@iamcurun1r Ай бұрын

same

@kiwwicube Ай бұрын

@@iamcurun1r hmmmmm I think I know you from somewhere

@iamcurun1r Ай бұрын

@@kiwwicube i totally dont know what ur talkin about :3

@kiwwicube Ай бұрын

@@iamcurun1r my bad, mistook you for someone on the speedsolving forum who has the same name as you and had the same pfp for a while, what a coincidence…

@HotL1ne Ай бұрын

This is the most Cuby cubykh video about cubes and colored cubes, total cube!

@b1oodzy Ай бұрын

I knew this guy was from BFDI a long time ago, the voice is just too familiar.

@PurpulPancakes Ай бұрын

this is such a cool video! i always like videos that make hard to understand equations, much easier to not only understand, but visualize as well! and putting it in a real scenario like speed cubing is super cool too!

@xycrypt Ай бұрын

I have a pretty decent understanding of CSP and still don't understand why parity happens on square-1. I know how to check for it, I know how to solve it for many cases, but the way in which square-1 turns (especially during CS) makes it a lot more difficult to understand

@thephysicistcuber175 Ай бұрын

Basically if you stay in cubeshape every slice turn is a 2c2e and U/D turns are like in 3x3, so you won't solve parity by staying in cubeshape. But if you go out of cubeshape there are ways to do an odd parity permutation. Example: on star cases doing a 2 on the star is a 6-cycle, odd parity.

@cubykh Ай бұрын

Oooh, I've always thought CSP was a clever concept, and I'm impressed more and more cubers have learned it! I think thePhysicistCuber explained it pretty well, but yeah - if you want to do a single swap, you're gonna have to switch between an odd permutation and an even one. But that can't happen in cubeshape, because every slice move swaps two pairs of edges and two pairs of corners! So the shortest "parity alg" (an alg that takes you from odd to even) involves going from Square-Square to Shield-Shield (3 slices), then swapping the 3 pairs of corners of the shields (since the corners are now all on the same side), and then undo-ing the 3-move setup. (7 slices total)

@jch123-r6v Ай бұрын

only slightly related question, but does anyone know why 2gen does not affect corner permutation? unlike this parity case I can't really put my finger on what is being "preserved"

@happygamer787 Ай бұрын

That's actually a really good question, wish I had the answer

@ulrikbredland6965 Ай бұрын

This is a really good question, and I would love some deeper insight into it myself. I've developed and learned some 2gen reduction methods, but they all seem to rely on the fact that 2gen has some specific ordering of the pieces rather than a deep understanding of what 2gen really is. The CEOR/YruRU tracing really is just a fancy and efficient way of figuring out where you can do a 2swap (or tperm) to force 2gen. I think I saw some article describing the mathematics behind it, although I didn't feel like it gave me any tangible information that I could use. More like a formal proof that 2gen is preserved. I don't know how well you know 2gen, but there are 6 different permutation states. If you have a 2x2 and solve the DL 2x1x1, you have a 1/6 chance that the case can be solved 2gen. It's easy to verify that there exists 6 cases, you can solve the D layer using and AUF for example the UFL-corner correctly. From here, you have 6 possible permutations of the last 3 pieces: solved, diag, 2 adj cases, and 2 a perm cases. These are also preserved in the same way, so if you do a Tperm on a solved cube, then you cannot reach Tperm from any other AUF by simply doing RU-moves.

@cupoinn8t_jl Ай бұрын

What’s the 4x4 oll parity algorithm you used in the video?

@P.25000 Ай бұрын

Man, I love the first song. It reminds me of the amazing marble race's

@cebucubes Ай бұрын

all i know about parity is that it makes my 4x4 explode

@Vioviovio_341 Ай бұрын

What is your best speedrun yet? (I havent seen it yet)

@vesmacubing Ай бұрын

Thanks for this cuby knowledge CubyKH

@troliskimosko Ай бұрын

Cubing Cary makes me happy

@DereadLordLives Ай бұрын

This is great, thank you!

@FourPlushie Ай бұрын

I didn't know you were still in college wth

@tinxsstuff Ай бұрын

im pretty sure its apart of the 2016 video XD

@skmgeek Ай бұрын

@@tinxsstuffr/woooosh

@Sutlahpai-i7i Ай бұрын

this helped alot if my 4x4 didnt break

@2763ec Ай бұрын

Cary you're so smart and awesome sauce!

@JohnDaCuber Ай бұрын

Cary pls ask BenjixScarlet to remove the “song”

@FannyAndBella_AhaWorld_OSC Ай бұрын

Good idea!

@xibuxkoax Ай бұрын

you are yet to remember me

@thephysicistcuber175 Ай бұрын

0:44 no, it really is that simple. There might be also a statement about this being a group homomorphism, but by decomposing the permutations into swaps this is trivial to see, given what you wrote on the screen.