双骰子求和游戏最优期望得分策略求解及相关技巧咨询

Suppose we play a game with two dice where we roll and sum our rolls. We can stop any time and take the sum as our score, but if we roll a sum we’ve rolled before then we lose everything. What strategy will maximize our expected score?

我试过暴力枚举所有可能性，但还是没能找到正确的策略。有没有人能帮我解决这个问题？有没有什么捷径或技巧可以用？

我看不懂我获取这个问题的PDF里给出的解法，解法内容如下：

Let C be our current score, and S be the set of sums rolled so far, so:
$$
C = \sum_{i \in S} i.
$$
Let p(i) be the probability of rolling a sum of 'i' with a single roll of two dice.

To maximize our expected score, we should stop rolling if the expected score after rolling again is less than our current score. That is:
$$
\left( \sum_{i \in S} p(i) \times 0 \right) + \left( \sum_{i \notin S} p(i) \times (C + i) \right) < C
$$

（注：原PDF中的解法内容未完整粘贴，以上为现有可展示的部分）

备注：内容来源于stack exchange，提问作者Shs Tht