Alice, Bob, and the average shadow of a cube

2021-12-20

[public] 247K views, 85.1K likes, dislikes audio only

4K

What's the average area of a cube's shadow?

Numberphile video on Bertrand's paradox: /youtube/video/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?

/youtube/video/GNcFjFmqEc8

If you liked this topic you'll also enjoy Mathologer's videos on very interesting cube shadow facts:

Part 1: /youtube/video/rAHcZGjKVvg

Part 2: /youtube/video/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

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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.vincentrubinetti.com/

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