Alice, Bob, and the average area of a cube’s shadow

From 3Blue1Brown.

A tale of two problem solvers.
Numberphile video on Bertrand’s paradox:
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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?

If you liked this topic you’ll also enjoy Mathologer’s videos on very interesting cube shadow facts:
Part 1:
Part 2:

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

The artwork in this video was done by Kurt Bruns


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:

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Music by Vincent Rubinetti.

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