The parity of permutations and the Futurama theorem

The Mathologer has a go at showing Fry & Co how to sort out their mind-switching mess in the best possible way and gets sidetracked into ying and yang territory--the parity of messes, shuffles, and permutations.

Enjoy :)

[MrRubikraft]

You basically described the Pochmann method to solve the Rubik's cube (blindfolded). The "naive" approach that consists in solving one person at a time is the Pochmann approach, that solves one piece at a time.
And here, parities play an important role: if you solved the corners with an even permutation, then you need to do another even permutation to put the edges back in place, then you solve the edges. You need the total permutations to be odd, so if you had to do a parity-fixing permutation, and if the permutation for the edges was odd, then you also need to do an extra "parity-fixing".
The parity-fixing algorithm is a double 2-cycle: it does a 2-cycle of corners and a 2-cycle or edges.
More formally, the product of the signature of the permutations of corners and of the signature of the permutations of edges is necessarily 1 (cannot be -1), because any sequence of moves on the Rubik's cube turns a face which is basically a 4-cycle of edges and a 4-cycle of corners, both having a signature of -1, the product of which is 1.

[maximelectron]

Fun fact:
At 7:31, there is an "impossible" Rubik's cube picture.
There is an RBY corner but it is oriented in the wrong way. (In this case, if it goes clockwise and we see red after blue then the next color should match the center, which goes "clockwise" after R and B centers. And it is green. Meanwhile, the sticker on the corner is yellow)

[Holobrine]

If math class looked like this, many more people would be educated. This doesn't trick you to sit through a proof; rather, text books trick you into getting bored, zoning out, missing the important parts, getting confused, and finally giving up.

[NoriMori]

Did you ever make the video you mentioned at the end? I don't know what it's called and it's not linked in the description or the info card, so if you made it I can't find it.

[samstep]

Working from home in lockdown. My favourite time of day: office laptop off, own laptop on, glass of beer, piano chords and "Welcome to another Mathologer video."

[hvadhvem]

Wow, I never thought the signatures of permutations I computed (with great boredom) in algebra class had such an application ! Group theory has sooo much to offer...

[Gefo__]

With this one being the last, I've watched all of your videos and thoroughly enjoyed them, you obviously put a lot of work into them and they are highly interesting as well as entertaining (and your humor hits it in the nail for me). I hope this channel picks up and gets a good viewership, because this is definitely top notch material. Keep it up! Cheers from an Argentinian biochemist!

[povilasrackauskas]

this is a great channel and a great video, indeed .

[dannygjk]

Another thing I noticed when I used to do the Rubik's cube is some operations rotate the center pieces relative to the rest of the pieces. Therefore you can add a little difficulty to solving it if you dictate that the center pieces must be returned to their original orientation relative to the other colors. If the solvers involved are being timed this adds an interesting twist (ha ha) to the competition.

[DaanSnqn]

It's 1:35AM and I'm watching mathologer videos in my underwear while I just had some ice cream. I'm awesome.

[bsodcat]

I love anything dealing with the chirality of the universe its crazy how everything even down to molecules with mirror versions can help or hinder the work we do in everyday life.

[TECHNORULEZ]

This channel is a real gem and you sir, are a genius really.. I fell in love with this channel ..

[IncaTrails]

Thanks and excellent job.  I hope that you continue to ignore the advice from your colleagues and continue to produce videos like this in the future!

[codingbyday]

I just wanted to thank you for all amazing content you provide. I wish more people would find your videos, they are amazing. and you are truly awesome presenter. :)

[lowellcrook]

This is sort of unrelated. Mathologer should do an episode on the strategies for solving the jump all but one game that you can find in Cracker Barrel. There are already some videos about it on YouTube, but none of them really try to explain the game.

[afterugirl]

Such an elegant proof and explanation simultaneously

[Sussylizzy]

There is in fact a use for the theorem, however it is outside of mathematics. It is useful for storing algorithms in computer science, as it states that at minimum 2 extra variables are needed to swap values in a dataset/array, as otherwise the original value will be overwritten with no memory index to give it its new position. The episode likely wasn’t trying to solve this as this problem had already been solved for sorting in code but it is a use for the theorem

[hernanipereira]

i would have been a very happy person if i had someone like you to teach me math like this in school. My area of expertise is not mathematics (its actually Social Sciences), but i always learn something either interesting, different, curious or just intellectually rewarding. Thank you for your videos! :)

[sadkritx]

Such a beautiful and wonderful video. I wish I could like it more than once. Sad that it doesn't have as many views as your many other beautiful videos.

[pulsefel]

the purpose of science and mathmatics is to take the complexity of the world and make it understandable by any.