Dodecahedron Resistance

By symmetry, if we place one vertex at the origin and the opposite one on the positive x axis, then we have 5 sets of points grouped by their similar x-coordinate:

• Group 0: 1 point, the vertex
• Group 1: 3 points, all points neighboring the initial vertex
• Group 2: 6 points
• Group 3: 6 points
• Group 4: 3 points
• Group 5: 1 point, the opposite vertex

Connecting the groups we have:

• 0-1: 3 resistors
• 1-2: 6 resistors
• 2-3: 6 resistors
• 3-4: 6 resistors
• 4-5: 3 resistors

Note, the six resistors unaccounted for connect vertices that are in the same group so by symmetry, they will draw no current.

So, by symmetry, if we inject a current $I$ into one vertex, then it will be split $3$ ways (each of the three resistors connecting to that vertex will a current of $I/3$ going through it), then six ways (three times) and then back to 3 ways once again, finally arriving at the opposite vertex. So the equivalent resistance will be given by:

$30 \mbox{ ohms} \cdot (\frac{1}{3} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{3}) = \boxed{35} \mbox{ ohms}$