Archimedes Revenge Solution
Congratulations to Runze Li, who solved the Archimedes
Revenge problem. There are obviously 8 pieces which are the same.
Each of these pieces can be split into a tetrahedron of volume 1/2
and 3 pieces which each have volume given by the integral I of
[(z
2+1) (π/2-2 arctan(z)) + z
2-1 ]/2 from 0 to 1.
(The third picture shows one of those regions). To solve the integral,
note that the anti-derivative of arctan(z) is x arctan(x) - log(1+x^2)/2
(integration by parts). The answer of 1/2 + 3 I, which is log(2).
Since there are 8 pieces, the volume of our beautiful solid is 8 log(2) = log(256).
Update June 27, 2021: a much better, clearer and more detailed solution
was written down by Mark Villarino Bertram:
PDF of the writeup of Mark
