Problem 683

The Chase II

Consider the following variant of “The Chase” game. This game is played between $n$ players sitting around a circular table using two dice. It consists of $n - 1$ rounds, and at the end of each round one player is eliminated and has to pay a certain amount of money into a pot. The last player remaining is the winner and receives the entire contents of the pot.

At the beginning of a round, each die is given to a randomly selected player. A round then consists of a number of turns.

During each turn, each of the two players with a die rolls it. If a player rolls a $1$ or a $2$ , the die is passed to the neighbour on the left; if the player rolls a $5$ or a $6$ , the die is passed to the neighbour on the right; otherwise, the player keeps the die for the next turn.

The round ends when one player has both dice at the beginning of a turn. At which point that player is immediately eliminated and has to pay $s^{2}$ where $s$ is the number of completed turns in this round. Note that if both dice happen to be handed to the same player at the beginning of a round, then no turns are completed, so the player is eliminated without having to pay any money into the pot.

Let $G (n)$ be the expected amount that the winner will receive. For example $G (5)$ is approximately $96.544$ , and $G (50)$ is $2.82491788 e 6$ in scientific notation rounded to $9$ significant digits.

Find $G (500)$ , giving your answer in scientific notation rounded to $9$ significant digits.

追赶游戏II

“追赶游戏”的如下变种需要 $n$ 名玩家和两个骰子，游戏开始时这些玩家围坐在一张圆形桌子周围。游戏共分为 $n - 1$ 轮，每一轮结束时中淘汰一名玩家，被淘汰的玩家必须向奖池支付一笔特定金额的奖金，最终存活的玩家获胜并获得奖池中的所有奖金。

在每一轮开始时，两个骰子分别随机交给一名玩家。每一轮由多个回合组成：

在每一回合，当前持有骰子的两名玩家分别抛掷骰子。如果玩家掷出 $1$ 或 $2$ ，则将骰子交给其左手边的玩家；如果玩家掷出 $5$ 或 $6$ ，则将骰子交给右手边的玩家；否则，玩家保留手上的骰子进入下一回合。

如果某一回合开始时，一名玩家手上同时持有两个骰子，则本轮结束。这名玩家将被淘汰，同时，若本轮已经进行了 $s$ 个回合，则该玩家必须向奖池支付金额为 $s^{2}$ 的奖金。注意，如果在第一回合开始时，两个骰子恰好被交给了同一名玩家，由于之前没有进行过任何回合，因此这名玩家被淘汰的同时不需要支付奖金。

记 $G (n)$ 为获胜者所获得奖池中总奖金的期望。例如， $G (5)$ 约等于 $96.544$ ，在采用科学计数法并保留 $9$ 位有效数字时 $G (50)$ 约等于 $2.82491788 e 6$ 。

求 $G (500)$ ，将你的答案用科学计数法表示并保留 $9$ 位有效数字。