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.
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!
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"
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.
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
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.
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.
![How to Turn FASTER [ft. Matty Hiroto]](http://i.ytimg.com/vi/__q-5MwlOiU/mqdefault.jpg)
i've lowkey always wondered how this works, you're the first one to explain it in a way that makes sense to me 💀
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!
cary plz return your old haircut
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.
I knew this guy was from BFDI a long time ago, the voice is just too familiar.
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!
This is the most Cuby cubykh video about cubes and colored cubes, total cube!
Man, I love the first song. It reminds me of the amazing marble race's
What is your best speedrun yet? (I havent seen it yet)
all i know about parity is that it makes my 4x4 explode
Thanks for this cuby knowledge CubyKH
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"
That's actually a really good question, wish I had the answer
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.
Cubing Cary makes me happy
What’s the 4x4 oll parity algorithm you used in the video?
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
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.
I didn't know you were still in college wth
im pretty sure its apart of the 2016 video XD
@@tinxsstuffr/woooosh
This is great, thank you!
Cary you're so smart and awesome sauce!
this helped alot if my 4x4 didnt break
Amazing
greetings cary
Cary pls ask BenjixScarlet to remove the “song”
you are yet to remember me
Yes Rubik’s cube
makes sense
YOU ARE CARYKH???
under 1 hour gang
👇
i was literally thinking about this like 5 hours ago....
Good morning sir
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.