Alice, Bob, and the average shadow of a cube

A tale of two problem solvers. Numberphile video on Bertrand’s paradox: Help fund future projects: ​ Special thanks to these supporters: #thanks An equally valuable form of support is to simply share the videos. 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 ------------------- Timestamps 0:00 - The pla