Load current computation of complicated resistance network
The solution to the 'cube body diagonal resistance: all same value resistors" is trivial to solve.
The trick is to connect up 'symmetrical' points as they all have at the same voltage. The system then collapses into standard parallel and series resistors. 

If the resistors are different you have to make, as @WhatRoughBeast says, a system of simultaneous equations. There are computer programs to solve those but YOU still have to write the equations. My guess is you need 7 equations: One for each side of the cube and a seventh for power..ground loop.

By the way: you can also solve this also if you have impedances instead of resistors. It just gets a bit more challenging.

I wrote program to solve up to 20x20 complex equations but have not used it yet