1) Twelve equal resistances R form the edges of a cube as shown.
a. A battery is connected between A and G. Can you find one set of three points that are at the same potential? Can you find another set?
b. Now imagine that you have a sheet of copper connecting the first set, and another sheet of copper connecting the second set. This does not affect the currents in the resistors. You can now show that RAG = (5/6)R.
See attached file for full problem description.
The solution finds the three points that are at the same potential on the battery. It also shows that RAG = (5/6)R.