In this paper we extend the random walk example of Sutton and Barto to a multistage dynamic programming optimization setting with discounted reward. Using Bellman equations on presumed action, the optimal values are derived for general transition probability ρ and discount rate ϒ, and include the original random walk as a special case. Temporal difference methods with eligibility traces, TD(λ), are effective in predicting the optimal values for different ρ and ϒ; but their performances are found to depend critically on the choice of truncated return in the formulation when ϒ is less than 1.