Movement costs −1, including blocked moves and traffic delays. At the top right, each attempted movement stays put with probability 0.2; at the bottom left, this probability is 0.5. Otherwise the intended move happens. Moves into a boundary stay in the same cell. Merely reaching the patient does not rescue them.

Repeatable rescue: the first rescue at the patient gives +100. Every rescue elsewhere, and every subsequent rescue at the patient, gives −100. Rescue does not move the robot. All actions remain available after success, with movement still costing −1. The state contains the position and whether the patient has been rescued.

For 0 < γ < 1, V*(s) = maxₐ [r(s,a) + γ Σₛ′ P(s′|s,a)V*(s′)]. The app solves this Bellman equation with value iteration. Discounted return is Σₜ γᵗrₜ, with the first reward undiscounted. Changing γ recalculates the policy and the whole recorded return. Equal best actions are all shown; automatic play picks the first.