What's the average area of a cube's shadow? Numberphile video on Bertrand's paradox: https://www.youtube.com/watch?v=mZBwsm6B280 Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share the videos. There's a small error at 19:30, I say "Divide the total by 1/2", but of course meant to say "Multiply..." Curious why a sphere's surface area is exactly four times its shadow? https://www.youtube.com/watch?v=GNcFjFmqEc8 If you liked this topic you'll also enjoy Mathologer's videos on very interesting cube shadow facts: Part 1: https://www.youtube.com/watch?v=rAHcZGjKVvg Part 2: https://www.youtube.com/watch?v=cEhLNS5AHss I first heard this puzzle in a problem-solving seminar at Stanford, but the general result about all convex solids was originally proved by Cauchy. Mémoire sur la rectification des courbes et la quadrature des surfaces courbes par M. Augustin Cauchy https://ia600208.us.archive.org/27/items/bub_gb_EomNI7m8__UC/bub_gb_EomNI7m8__UC.pdf The artwork in this video was done by Kurt Bruns Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld Hindi: rajeshwar-pandey ------------------- Timestamps 0:00 - The players 5:22 - How to start 9:12 - Alice's initial thoughts 13:37 - Piecing together the cube 22:11 - Bob's conclusion 29:58 - Alice's conclusion 34:09 - Which is better? 38:59 - Homework ------------------ 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_animations/ Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown