# Alice, Bob, and the average shadow of a cube

A tale of two problem solvers.
Numberphile video on Bertrand’s paradox: https://youtu.be/mZBwsm6B280
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Special thanks to these supporters: https://3b1b.co/lessons/newtons-fractal#thanks
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Curious why a sphere’s surface area is exactly four times its shadow?

If you liked this topic you’ll also enjoy Mathologer’s videos on very interesting cube shadow facts:
Part 1: https://youtu.be/rAHcZGjKVvg
Part 2: https://youtu.be/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

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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

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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/

https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown

Stream the music on Spotify:

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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