Puzzles with beautiful solutions - This pattern breaks, but for a good reason | Moser's circle problem

An apparent pattern that breaks, and the reason behind it. Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share the videos. This is a remake of one of the earliest videos on the channel. It's such a wonderful problem, and the audio/pacing in earlier videos was really suboptimal, so I wanted to freshen it up a little here. Timestamps 0:00 - The pattern 2:20 - Counting chords 4:03 - Counting intersection points 6:20 - Euler's characteristic formula 11:30 - Connection with Pascal's triangle 15:10 - Reflections Correction at 8:56 - The number of the regions should of course be (1, 2, 3, 4, 5), instead of (0,1,2,3,5) Thanks to these viewers for their contributions to translations Arabic: mouhsiiin French: Jonhfing Hebrew: Omer Tuchfeld Hungarian: Fabó Bence Russian: fedor Spanish: Paweł Cesar Sanjuan Szklarz Thai: @korakot ------------------ These animations are largely made using a custom python library, manim. See the FAQ comments here: https://www.3blue1brown.com/faq#manim https://github.com/3b1b/manim https://github.com/ManimCommunity/manim/ You can find code for specific videos and projects here: https://github.com/3b1b/videos/ Music by Vincent Rubinetti. https://www.vincentrubinetti.com/ Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Instagram: https://www.instagram.com/3blue1brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown