Rubik's Cube: Why are some cases impossible to solve?

A full explanation on parity, and why it happens on the void cube and 4x4 and not 3x3!

Using the code "JPERM" supports my channel and also gives you a discount on your order.

Beginner:

Special Tutorials:

Advanced Tutorials:

Special thanks to these generous Patrons:
Jake Maul, Dcs Banana, Thisis alexclarkfan, Asteroid, Leland Hoagland, Broxson Barnhill, Ivan Marinkovic, someone 1, Jeremy Johnson, QueenSquare1, Derek Worth, Inventor of Loob46, Eisen Vigilia, Dawson Wilson, Sesha Sai Patro, Gregory Martin, Michael Mendoza, Pete Ryland, Jan Strohbeck, KingIceCream, Victor Giaquinto, Piotr, Leszek Gawron, Leo Li, Ed Connell, Omri Riss chbarbi, Chris Bonnello, Edan Maor, Petrified, Xiaotian Li, Keian Inouye, Riley Lee, Michael Conard, Johnathan Starkey, Those Nerds, whichonemi, Benjamin Force, Isaac Farnsworth, Li Jiang, Hudson, Dakota, James Moore, Csaba Daday, Dmitri Shabes, Voidedifbroken, Rayan Al Hobayb, Ram Thakkar, kavish, Mirzo, Aidan Gerick, Adrin Esterabadeyan, Bernhard Kofler, Cade Watson, Adam Fenn, Paul Thorley, Amay Saxena, Weisselberger Timothee, Philippe Schwartz, Jonas Rongé, Pontus Uggla, Takeshi Yanagita, Jane Qiao, Adam Prato, Ricky Robber, The Slick Cuber, DLyu, Letin, Ruimin Yan, Nina, Benjamin Jonassen, Walter, RedKB, Christopher Robin, Sebastian Szafran, Tipa Reimer, ZHU SHENJIE, Fred Strauss, Danny Williams, Rick Lush, Chris Cobb, Adam Rosales, Martijn Schiedon, David Brown, Taco, Mike Sampson, Siege Wolfe, Rob Peters, Miranda Manning

[kaikyc.]

Damn, I remember that when I started to learn how to solve a 3x3 I got parity, and then I would always restart the solve thinking that I did something wrong *for hours* . Just to later realise that it was impossible and I just had to do a corner twist.
Definetely one of my most frustrating life experiences.

[Spatzna]

"1 is a multiple of 3 if you're clever"

Great vid, loved it

[rahulsahay]

Dude i legit have been thinking about this for a while. My best guess was that centers were not fixed. But couldn't figure out for odd layered cubes, and of course the void cube

[puerlatinophilus]

Legends say that the number of times he says "swap" in this video has to have an even parity.

[leo]

"Every odd number is a multiple of 3" -J Perm

[eyeturtle]

For megaminx, using the same logic in the video, a turn is 4 edge swaps, so you can swap the two edges, then swap edges twice somewhere else, which doesn't do anything, and there would still be one swap, so it is impossible to have to only have two edges swapped using outer turns on a megaminx.

[vinuthabhandary]

You make the types of videos I can watch over and over again without getting bored.

[crimson]

i absolutely love this, i live for this kind of nerdy stuff, i hope you do more cube theory videos in the future

[danielsaad]

once i was doing a solve in public and someone came up and told me to solve their cube.
me, being the humble cuber i am, obliged.
after i got to pll, though, i realized i had parity. on a 3x3.


i guess that solve took a *turn for the worse*

[maxcrenshaw]

I genuinely enjoy the "cube theory" format, keep doing them my guy!

[Inspirator_AG]

AT TIMESTAMP 10:40 )
The number of *corner swaps* and *edge swaps* must be _both_ either odd or even on a *Megaminx.* The reason this only applies to odd-layered puzzles (-Minus the *Void Cube)* and not even-layered puzzles is because any puzzle with *centers* has this rule. An *A-perm, G-perm* or *U-perm* is possible because 2 intersecting *swaps* form a *cycle.* A *cycle* is formed when intersecting *swaps* of the same type of *piece (-Edge/Corner)* share a *last piece* (-This is why a *cycle* must have an odd number of *pieces.).*

[Sora-iukc]

someone need to count how offten he said "swap"

[VictorArets]

This is a multipel of 3, iF YoU’Re CLevEr.

-jperm 2020

[pathfindernutsinurface]

My mom: So what did you learn today?
Me: I learned Dylan's law of slice turns and parity explanation

[baileycooper]

This was such a helpful video. I’m getting parities on my 4x4 all the time and I don’t know why. Previously, I thought it was meant to be “impossible”, but it turns out it is not.

[sinpi]

This video is super interesting, I really love cubing + maths vids. Make more of these kind!

[natanfurman]

I think swapping two pieces on the megamix is impossible for the same reason it's impossible on a 3x3. When you turn a face (which is the only legal move), you swap 5 corners and 5 edges. You can probably swap two edges and two corners, or two edges and two sets of two corners, or two edges and any swapping of corners that can be seen as an even number of corner swaps.
This is my reasoning:
When you turn a face, you do five edge swaps. With some manipulation, you can use this to swap two corners, then swap two other corners four times. This would give you two edges swapped while keeping the rest of the edges undisturbed. However, because this took the equivalent of five edge swaps, you need five corner swaps. It's impossible to have five corner swaps that cycle back to their original state, but you can swap two corners and leave the rest undisturbed. You can do this the same way you do edges, by swapping two with one swap and then using the other four swaps to cancel themselves out.
I used the word "swap" 17 times in this comment.

[JediJess]

This is the single most comprehensive and intuitive explanation for parity on twisty puzzle I have seen to date! Thank you!

[MinutemenSyndrome]

Sorry Professor Dylan,
I forgot my megaminx homework at home.

[radoaahmed]

Whenever I am unable to solve a weird state of my 4x4x4 cube I just scramble it a lot, start over and hope for the best.

I’m not a speedcuber or anything but it is in my interest.