A wire of resistance \(R\) is bent into a triangular pyramid as shown in figure with each segment having same length. The resistance between points \(A\) and \(B\) is \(R / n\). The value of \(n\) is :

Explanation:
The equivalent resistance of the pyramidal wire framework can be evaluated using circuit symmetry:

1. Resistance of Each Segment: A triangular pyramid (tetrahedron) consists of exactly 6 identical edges. Since a single continuous wire of total resistance \( R \) is uniformly divided to form these 6 segments, the resistance \( r \) of each individual segment is given by:
\( r = \frac{R}{6} \)

2. Symmetry of the Circuit: When measuring the equivalent resistance across vertices \( A \) and \( B \), the remaining two vertices form a balanced bridge. Due to the symmetrical distribution of the resistors, these two intermediate nodes sit at the exact same electrical potential.

3. Circuit Simplification: Because no potential difference exists between the two intermediate nodes, no electrical current flows through the resistor connecting them. Removing this inactive branch simplifies the network into three parallel paths connected across terminals \( A \) and \( B \):
- One direct branch of resistance \( r \)
- Two separate series branches, each with a total resistance of \( r + r = 2r \)

4. Equivalent Resistance Calculation: Combining these three parallel paths gives:
\( \frac{1}{R_{AB}} = \frac{1}{2r} + \frac{1}{2r} + \frac{1}{r} = \frac{1}{r} + \frac{1}{r} = \frac{2}{r} \)
\( R_{AB} = \frac{r}{2} \)

5. Substituting the Value of r:
\( R_{AB} = \frac{\left(\frac{R}{6}\right)}{2} = \frac{R}{12} \)

Comparing \( \frac{R}{12} \) with the given expression \( \frac{R}{n} \) yields \( n = 12 \).

Hence, Option D is the correct choice.