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The Mysterious Number That Took 100 Years To Compute

PurpleMind

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In this video, we cover an exciting new result in mathematics - the "Empty Hexagon Problem," a 100-year old geometry problem has finally been solved, bringing a happy ending to a century-long mathematical journey! It started with a simple idea from the 1930s, and evolved into a rich history of geometric questions, culminating in a spectacular and lucky solution in 2024 that makes use of the newest developments in SAT solvers, a computerized proof method, filling in the last piece of the puzzle.

Special thanks to Professor Marijn Heule and Carnegie Mellon University for sponsoring this video, to Joe Horton, Bernardo Subercaseaux, and Cayden Codel for their help with the project.

This video is part of "Forefront," a collection of videos on this channel that features the cutting edge of research in math and computer science. Full playlist here: Forefront
If you or your institution is interested in sponsoring the topic of a future PurpleMind video, please contact me via purplemindcs@gmail.com!

References:
Heule Paper: https://arxiv.org/pdf/2403.00737
Subercaseaux et al. Paper: https://arxiv.org/abs/2403.17370
Horton Paper: https://tinyurl.com/md48uf3u
Overmars Paper: https://link.springer.com/content/pdf...
Nicolás Paper: https://link.springer.com/article/10....
Gerken Paper: https://link.springer.com/article/10....
Happy Ending Problem: https://en.wikipedia.org/wiki/Happy_e...

Point Realizer Game: https://w-realizer.surge.sh/ (special shoutout to Bernardo Subercaseaux for making this!) You can upload your own constraints for this game as a text file on the website. See this example for reference (it’s the unrealizable set of constraints I used as an example in the video): https://pastebin.com/BvWqcd1d

Another game from Bernardo where you can try to place points without making convex/empty polygons: https://k-hole-tester.surge.sh/

Math animations are made using Manim, by 3Blue1Brown.
Character animation done by Divyesh Patel.

Discord Server: Discord: discord. Feel free to join!
Business Inquiries: purplemindcs@gmail.com

20 Comments

@PurpleMindCreations Creator 1 year ago Pinned

To watch the full interview with Professor Heule and Bernardo Subercaseaux, check out my Patreon! https://www.patreon.com/PurpleMindCS

@Sam_on_YouTube 1 year ago

They also got really lucky that the number WAS 30. They may have just ended up proving that it was greater than 30. And then we would have to try to prove it's 31.

@rocky_wang 1 year ago

Like the “Four Color Theorem”, the proof is a tedious phone book rather than an elegant argument 😂😂😂

@SellymeYT 1 year ago

36:24 "So if no human can read it, how can we know for sure it's valid? The short answer is that instead of making a human do it, we used grad students"

@cube_cup 1 year ago

That part of the 'if the triangle is empty, there must be an empty hexagon' is gorgeous! And that was Marijn's big breakthrough.
The way you faded him out felt like you slighted him a bit. I really wanted him to explain it

@furbyfubar 1 year ago (edited)

Minor nitpick: Writing "Image from Flickr" is not giving credit to the maker of the image. In the case of the DNA Sequencing image it was by Darryl Leja, NHGRI. The image is in the public domain (at least as per how it's listed on Flickr) so legally you don't have to credit anyone for it. But crediting Flickr and not the person who made it rubs me the wrong way. It would be like if someone wrote an article about the contents of this video and credited the research and ideas in it to "YouTube". Also, for an image licensed under CC-BY it wouldn't count as properly giving credit to only write where you found it.

@IsaacDickinson-tf8sf 1 year ago

I like that they divided by 0 through the transformation

@bakamitaiyt 1 year ago

37:30 Whoa, I didn't know that the initial Four-Color Theorem proof was faulty. And I did know that the classification of finite simple groups was insanely lengthy, but I didn't know that portions were proved using computer searches. Good stuff!

@zli0122 1 year ago

The people walking in the background are actually endermen.

@kylewood4001 1 year ago

Someone HAS to formalize a constant for “atoms in the observable universe”, it’s referred to WAY too much for it not to be

@ManicShot 1 year ago

Beautifully presented and I can see the many hours of love that went into it. The interviews were insightful and helped it feel grounded. Can I make a suggestion? When you've finished a soundbite from an interviewee and you resume narrating, perhaps try cutting to something else, rather than staying on the subject and fading their sound out, because it appears as if you're talking over them. It's a minor gripe really, but thought you'd appreciate the comment that something there didn't quite feel right. Please don't let that detract from this being an excellent and truly well-crafted video. Looking forward to the next one!

@khoda81 1 year ago

OMG, sat solvers are my favorite thing. Please make a video about how the proof verification works!

@raedev 1 year ago

Did not expect to hear Mark Overmars be involved in this in any way. My gamemaker heart feels proud

@thesmilingbraniac4438 1 year ago

Love the new style, but it feels rude when you cut Marjin

@MoiraLachesis 1 year ago

Much simpler proof for quadrilateral: connect all pairs of points. If any two intersect, they form the diagonals of a convex quadrilateral. If none intersect, they you have a complete planar graph. Complete graphs with 5 or more nodes are never planar.