Scarabs on a Dodecagon

From a problem set, but doesn't mean it's not a good problem.

Consider a regular and convex dodecagon (a polygon with 12 sides). Two scarabs initially located on two opposite vertices (summit) move at each time step along an dege. They ove independently from each other clockwise or counter-clockwise with probability 1/2. What is the proportion of time that the two scarabs spend together (on the same summit, after a large number of steps)?

Solution:

An "official" solution would be: draw your transition graph, write down your transition matrix, calculate your stationary distribution.

A much faster solution would be to simply observe that the distance between the two scarabs would always be even, and since the two scarabs move independently of each other, the expected proportion of time they spend together would simply be 1/6.