Parity of permutations, impossible puzzles and the magical determinant

This is a video I've been meaning to do for a long, long time. I've used the parity of permutations in quite a few videos and I've repeatedly promised to give a proper explanation in a separate video. This is it!

Parity of permutations, the distinction between even and odd permutations or rearrangements, is one of the simplest nontrivial invariants in mathematics, yet it has far-reaching consequences across many areas. This seemingly modest idea underpins the structure of the alternating group, a fundamental object in group theory that plays a key role in understanding symmetry.

In linear algebra, parity is built directly into the definition of the determinant, where alternating signs ensure that the determinant correctly captures orientation and volume. Without this distinction, the determinant would lose its essential properties. In geometry and topology, parity governs orientation: even permutations preserve orientation, while odd ones reverse it, a concept central to integration and manifold theory.

In combinatorics, parity enables powerful cancellation arguments, where terms paired by opposite parity eliminate each other, simplifying complex counts. It also appears in algorithms and puzzles, where parity acts as a hidden invariant determining whether certain configurations are reachable. In algebra, it influences objects such as polynomial discriminants and the structure of Galois groups.

Overall, parity serves as a unifying principle, linking symmetry, orientation, and invariance across mathematics. (Part of) the ying and yang of mathematics :)

Here is the link to my javascript app:
http://www.qedcat.com/parity

Things to watch out for: In the permutation diagrams you sometimes get less crossings than inversions. This is because of the presence of multi-crossings (more than two arrows forming a crossing). Usually you can resolve these multi-crossings with the "resolve multicross" button.

One thing that I missed out on mentioning in the present video is that originally the 15-puzzle was sold with the 14 and 15 swapped, thereby making it into an impossible puzzle that took the world by storm very much like the Rubkik's cube one hundred years later. Find out about the history of the 15-puzzle in the early Mathologer video mentioned below.

00:00 Intro
01:21 Permutations, inversions and parity
03:55 Identity permutation and swaps flip parity
08:48 odd + odd = even
11:08 The 15-puzzle
16:41 My permutation visualiser app
18:49 The Rubik's cube (corners)
22:38 The Rubik's cube (edges)
25:00 The Rubik's cube (corners & edges)
29:27 The determinant
31:54 The proof
35:56 Postscript
36:43 Thank you!

Here are relevant earlier Mathologer videos for you to check out:

I Built an Original One-Glance Proof from Dice https://youtu.be/QbKMSH5CLZ8
If you take out the corner and edge pieces from a Rubik's cube and fit them in randomly into the leftover 3d cross there is only a 1/12 chance that the resulting permutation is solvable with legal moves/twists. 12=2x2x3. Among other things, this video justifies where the 3 comes from.

Why did they prove this amazing theorem in 200 different ways? Quadratic Reciprocity MASTERCLASS
https://youtu.be/X63MWZIN3gM
In this video I demonstrate how the quadratic reciprocity has the parity of permutations at its core.

The 15 puzzle - solving the unsolvable 19th century Rubik's square
https://youtu.be/GXJOVoyZcXQ
A whole video about the 15-puzzle and its history.

The parity of permutations and the Futurama theorem
https://youtu.be/w0mxdo5ur_A
A different visual proof for why parities add like numbers motivated by an argument I used in one of the earliest Mathologer videos.

The Futurama Theorem
https://youtu.be/J65GNFfL94c
One of the earliest Mathologer videos. All about permutations ... and Futurama :)

Enjoy!

Burkard Receive SMS online on sms24.me

Watch on YouTube

Subscribe on YouTubeReaderBot

TubeReader video aggregator is a website that collects and organizes online videos from the YouTube source. Video aggregation is done for different purposes, and TubeReader take different approaches to achieve their purpose.

Our try to collect videos of high quality or interest for visitors to view; the collection may be made by editors or may be based on community votes.

Another method is to base the collection on those videos most viewed, either at the aggregator site or at various popular video hosting sites.

TubeReader site exists to allow users to collect their own sets of videos, for personal use as well as for browsing and viewing by others; TubeReader can develop online communities around video sharing.

Our site allow users to create a personalized video playlist, for personal use as well as for browsing and viewing by others.

@YouTubeReaderBot allows you to subscribe to Youtube channels.

By using @YouTubeReaderBot Bot you agree with YouTube Terms of Service.

Use the @YouTubeReaderBot telegram bot to be the first to be notified when new videos are released on your favorite channels.

Look for new videos or channels and share them with your friends.

You can start using our bot from this video, subscribe now to Parity of permutations, impossible puzzles and the magical determinant

What is YouTube?

YouTube is a free video sharing website that makes it easy to watch online videos. You can even create and upload your own videos to share with others. Originally created in 2005, YouTube is now one of the most popular sites on the Web, with visitors watching around 6 billion hours of video every month.