This could go on forever

Suppose we have a circle, centered at the origin, with a circumference of 12 units. Both of the \(x\)-intercepts serve as the starting point for a particle, each of which, each second, moves (independently) a distance of one unit either clockwise or counterclockwise around the circle with equal probability.

What is the expected number of seconds to elapse until the two particles simultaneously occupy the same point on the circle for the first time?


  • The particles move at the end of each second.
  • To reiterate, the particles move independently of one another.

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