Lost on an Octahedron

Alice and Bob stand on opposite vertices of a regular octahedron. At the beginning of every minute,

  • each chooses an adjacent vertex uniformly at random and moves towards it;
  • each moves at a constant rate of 1 edge per minute;
  • they will stop when they meet up.

The expected value of the number of minutes until they meet up is equal to \(\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers.

Find the value of \(p + q.\)

Note: It is possible that they would meet at a vertex or at the midpoint of an edge.


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