How to Play Dundee

Abstract

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.