A tale of two problem solvers | Average cube shadow area

What's the average area of a cube's shadow? Numberphile video on Bertrand's paradox:    • Bertrand's Paradox (with 3blue1brown) - Nu...   Help fund future projects:   / 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?    • But why is a sphere's surface area four ti...   If you liked this topic you'll also enjoy Mathologer's videos on very interesting cube shadow facts: Part 1:    • The cube shadow theorem (pt.1): Prince Rup...   Part 2:    • The cube shadow theorem (pt.2): The best h...   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/it... 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/a... Stream the music on Spotify: https://open.spotify.com/album/1dVyjw... ------------------ 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:   / 3blue1brown   Reddit:   / 3blue1brown   Instagram:   / 3blue1brown_animations   Patreon:   / 3blue1brown   Facebook:   / 3blue1brown