Solution
Ant on a Cube
Show the problem again
An ant starts at one vertex of a cube and each second moves along a randomly chosen edge to an adjacent vertex. What is the expected number of moves to reach the vertex diagonally opposite its starting point?
Worked solution
The answer is 10. Classify vertices by distance from the target: 3 (start), 2, 1, 0. Let E₃, E₂, E₁ be expected steps from each class. From distance 1: E₁ = 1 + (2/3)E₂. From distance 2: E₂ = 1 + (2/3)E₁ + (1/3)E₃. From distance 3: E₃ = 1 + E₂. Solving the system gives E₁ = 7, E₂ = 9, E₃ = 10.
Source: Classic random-walk-on-a-graph exercise from standard Markov chain texts. Statement written for AxiomIQ.