The key insight I kept fumbling around before landing on: you can reduce this to tracking the relative gap between the two particles, which itself does a random walk on a cycle.
Model the problem as a Markov chain on the distance between the particles modulo 8, then solve for the expected hitting time using first-step analysis. Alternatively, use the fact that the difference process is a simple random walk on a cycle and compute the expected time to hit 0 from distance 4.
Pro tip: After deriving the answer, sanity-check it by considering the symmetry and the expected time for a single particle to return to its start (8 steps). The answer should be less than 8 because the particles are moving toward each other half the time.
Let the state be the clockwise distance from particle A to particle B modulo 8. Initially, the distance is 4 (since they are opposite).
Each step, the distance changes by +1, -1, or 0 with probabilities 1/4, 1/4, and 1/2 respectively, because each particle moves independently.
Let E_i be the expected number of steps to reach state 0 from state i. Write equations using first-step analysis: E_i = 1 + (1/4)E_{i+1} + (1/4)E_{i-1} + (1/2)E_i, with E_0 = 0.
Simplify the equations to 2E_i - E_{i+1} - E_{i-1} = 4, and solve the linear system for E_4, using symmetry E_i = E_{8-i}.
Solve to get E_4 = 8. Verify by checking that the expected time is reasonable and consistent with the random walk properties.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.