标题： 如何玩邓迪 显示英文标题

标题： How to Play Dundee

摘要： 我们考虑以下称为邓迪的一人游戏。 我们得到一个由s_i张价值为i的牌组成的牌堆，其中i=1,...,v，并且整数m\le s_1+...+s_v。 共有m轮。 在每一轮中，玩家指定一个介于1和v之间的数字，并从牌堆中随机抽取一张牌。 如果在至少一轮中指定的数字与抽到的数值相同，玩家就输掉游戏。 著名的十三问题，由蒙莫尔特于1708年提出，询问当v=13，s_1=...=s_{13}=4，m=13，且玩家指定序列1,...,13时获胜的概率。 这个问题及其各种推广被众多数学家研究过，包括J.和N.伯努利、德莫弗、欧拉、卡塔兰等。 然而，似乎没有人考虑过哪种玩家策略能最大化获胜的概率。 我们研究了这个问题的两个变种。 在第一个变种中，玩家在第i轮的出牌可以依赖于前几轮随机抽到的牌的值。 我们完全解决了这个版本。 在第二个变种中，玩家必须在翻开任何牌之前提前指定全部m次出牌的序列。 当s_1=...=s_v且m是任意数时，我们能够解决这个问题。 显示英文摘要

摘要： We consider the following one-player game called Dundee. We are given a deck consisting of s_i cards of Value i, where i=1,...,v, and an integer m\le s_1+...+s_v. There are m rounds. In each round, the player names a number between 1 and v and draws a random card from the deck. The player loses if the named number coincides with the drawn value in at least one round. The famous Problem of Thirteen, proposed by Monmort in 1708, asks for the probability of winning in the case when v=13, s_1=...=s_{13}=4, m=13, and the player names the sequence 1,...,13. This problem and its various generalizations were studied by numerous mathematicians, including J. and N. Bernoulli, De Moivre, Euler, Catalan, and others. However, it seems that nobody has considered which strategies of the player maximize the probability of winning. We study two variants of this problem. In the first variant, the player's bid in Round i may depend on the values of the random cards drawn in the previous rounds. We completely solve this version. In the second variant, the player has to specify the whole sequence of m bids in advance, before turning any cards. We are able to solve this problem when s_1=...=s_v and m is arbitrary.