1. Interpret the figure

A triangular pyramid is a tetrahedron having 4 vertices and 6 edges. Since the wire has total resistance $R$ and all 6 segments have equal length, each edge has resistance

$r=\frac{R}{6}.$

We need the equivalent resistance between two vertices $A$ and $B$ of the tetrahedron.

2. Use symmetry

Let the other two vertices be $C$ and $D$. When a potential difference is applied between $A$ and $B$, the tetrahedron is symmetric with respect to interchange of $C$ and $D$. So, the potentials at $C$ and $D$ will be equal.

Therefore, no current flows through the edge $CD$. So we can ignore the resistor between $C$ and $D$.

3. Reduce the network

Now between $A$ and $B$, there are three parallel branches:

• Direct branch $AB$ with resistance $r$
• Path $A \to C \to B$ with resistance $r+r=2r$
• Path $A \to D \to B$ with resistance $r+r=2r$

Hence,

$\frac{1}{R_{AB}}=\frac{1}{r}+\frac{1}{2r}+\frac{1}{2r}.$

So,

$\frac{1}{R_{AB}}=\frac{1}{r}+\frac{1}{r}=\frac{2}{r}$

and therefore,

$R_{AB}=\frac{r}{2}.$

Now substitute $r=\frac{R}{6}$:

$R_{AB}=\frac{1}{2}\cdot \frac{R}{6}=\frac{R}{12}.$

Thus,

$\frac{R}{n}=\frac{R}{12} \Rightarrow n=12.$

4. Check options

The correct option is:

$\boxed{12}$

So, Option C is correct.