Lost on an Octahedron
Discrete Mathematics
Level
3
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.
by
Steven Yuan