The Resistor Cube Problem
By: David Randall - VK4TDR
1 July 2006
It’s a relaxing Saturday afternoon at your local amateur radio club and you’re busy discussing the finer points of antenna theory with some friends. The next thing you know a fellow amateur interrupts you by throwing you a small neatly soldered resistor cube. This person then challenges you to find the resistance between any two diagonally opposite corners without using an ohmmeter.

Problem: The resistor cube consists of 12 resistors, each having a value of exactly 1 ohm.
What is the total resistance between the two diagonally opposite corners labeled A and H on the diagram below?

Figure 1

Solution: There are many different approaches to solving this problem and I would encourage you to have a go at solving it for yourself before looking at the solutions I have presented.
On the pages that follow I show two very different methods for solving this particular problem. The first method, which is a more generic method, uses loop equations and simultaneous equations to solve the problem. The second method is a very elegant solution and shows what can be achieved by using a totally different approach. It makes use of simple logical reasoning with some ohms law thrown in.

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Solution 1 – Using Loop Equations
The basic steps for solving this problem using loop equations are:
1.
2.
3.
4.

Redraw the 3 dimensional resistive cube network in 2 dimensions.
Draw all the loop currents in a clockwise direction and identify them.
Write the equations for the voltage drops around each loop in turn.
Solve the equations to find the unknown currents using the method of simultaneous equations. Step 1 & 2: Draw the circuit in 2d and identify the loop currents.

Figure 2

Step 3: Write the voltage drop equations for each of the loops.
Loop i1 (orange):

V = (Ι1 +