Optimal Stopping with a Ten-Sided Die and a Reset

Roll a fair ten-sided die labeled 0 through 9. A positive roll adds its value to your running total. A zero ends the game and loses the entire total. After each non-ending roll, you may stop and receive your current total. What strategy maximizes expected winnings? Now change the rule: the first zero resets your total to zero but allows you to continue; the second zero ends the game with no winnings. How does the optimal strategy change? ### Constraints & Assumptions Rolls are independent, there is no cost per roll or finite horizon, and the objective is expected cash payout. In the variant, a reset does not bank any previous total. The state must include whether the first zero has already occurred. ### Clarifying Questions Can you stop after a reset? Does the first zero reset or preserve the total? Is the objective expected value rather than probability of exceeding a target? ### What a Strong Answer Covers A stopping-versus-continuing comparison, a proof of the threshold policy, and a state-aware treatment of the extra reset. Define any constant needed for the second threshold precisely. ### Follow-up Questions Why is comparing the next roll's average increment alone insufficient after a reset option is introduced? What happens at an exact indifference threshold? How would a per-roll cost alter the Bellman equation?