What I think makes this especially hard is that your strategy (of whether to roll again or not) depends on the relative score as things go along. You can't just choose the strategy with the fewest turns, on average, to reach 100. You need to choose your strategy depending on how far you and your opponent are from 100. If they are close to 100, you probably need to try to earn 10 points no matter what the risk, because if you play safe you are almost certain to lose, and even if you have 9 points you still have a 3/4 chance of getting 10 vs. a 1/4 chance of 0. Obviously for expected value that's not such a good bet, but maximizing expected value isn't the goal. Perhaps your optimal strategy even depends on the opponent's strategy ... so I suppose we make the usual superrational assumptions. --Joshua Zucker