A tale of two problem solvers

What's the average area of a cube's shadow?
Numberphile video on Bertrand's paradox: https://youtu.be/mZBwsm6B280
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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://youtu.be/GNcFjFmqEc8

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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https://www.3blue1brown.com/faq#manim
https://github.com/3b1b/manim
https://github.com/ManimCommunity/manim/

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Music by Vincent Rubinetti.
https://www.vincentrubinetti.com/

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Видео A tale of two problem solvers канала 3Blue1Brown