Twelve identical wires each of resistance `6 Omega` are joined to from a skeleton cube. Find the resistance between the current of the same edge of the cube

Answer

To find the resistance between the current of the same edge of a cube formed by twelve identical wires, each with a resistance of \(6 \, \Omega\), we can follow these steps:

### Step 1: Understand the Configuration
We have a cube with 12 edges, and each edge has a resistance of \(6 \, \Omega\). We need to find the equivalent resistance between two points (let's say points A and B) that are at the ends of one edge of the cube.

### Step 2: Identify the Edge and Collapse Method
We will use the collapsing method to simplify the circuit. We will keep the edge AB (with resistance \(R = 6 \, \Omega\)) intact and collapse the other edges into a simpler equivalent circuit.

### Step 3: Analyze the Remaining Resistors
When we collapse the cube, the resistances connected to the edge AB will change. The resistors connected to the edge AB will be in parallel and series combinations.

- The two edges adjacent to AB will each have a resistance of \(6 \, \Omega\) and will be in parallel with each other. The equivalent resistance of these two resistors is:
\[ R_{parallel} = \frac{R}{2} = \frac{6}{2} = 3 \, \Omega \]

### Step 4: Combine the Resistors
Now we have:
1. The resistance of edge AB: \(R = 6 \, \Omega\)
2. The equivalent resistance of the two adjacent edges: \(R_{parallel} = 3 \, \Omega\)

The two resistances (the intact edge and the equivalent of the two adjacent edges) are in series. Therefore, the total resistance \(R_{total}\) is:
\[ R_{total} = R + R_{parallel} = 6 + 3 = 9 \, \Omega \]

### Step 5: Consider the Remaining Resistors
There are also two edges directly opposite to the edge AB. Each of these edges has a resistance of \(6 \, \Omega\) and will also be in parallel with the previous combination. Therefore, we need to find the equivalent resistance of these remaining edges.

The total resistance of the two edges opposite to AB in parallel is:
\[ R_{opposite} = \frac{R}{2} = \frac{6}{2} = 3 \, \Omega \]

### Step 6: Final Combination
Now we have:
1. The resistance from the edge AB and the two adjacent edges: \(9 \, \Omega\)
2. The resistance from the two opposite edges: \(3 \, \Omega\)

These two resistances are in parallel:
\[ \frac{1}{R_{final}} = \frac{1}{9} + \frac{1}{3} \]

Calculating this gives:
\[ \frac{1}{R_{final}} = \frac{1}{9} + \frac{3}{9} = \frac{4}{9} \]
Thus,
\[ R_{final} = \frac{9}{4} = 2.25 \, \Omega \]

### Step 7: Conclusion
The final equivalent resistance between the current of the same edge of the cube is:
\[ R_{final} = 2.25 \, \Omega \]