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.
At some point I say that no matter how many tiles we are shuffling there will always be the same number of odd and even permutations. Of course that's not true if there is only one tile.
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 :)
Music at the end by Ian Post: Dream instrumantal version T-shirt: https://www.zazzle.com.au/math_i_cant_even_t_shirt-235002184636303112
Enjoy!
Burkard