How to Play Dundee

Abstract

We consider the one-player game called Dundee, where a deck consists of \(s_i\) cards of value \(i\), for \(i = 1, \ldots, v\), and an integer \(m \leq s_1 + \cdots + s_v\). Over \(m\) rounds, the player names a number between \(1\) and \(v\) and draws a random card from the deck, losing if the named number matches the drawn value in at least one round. The famous Problem of Thirteen, proposed by Montmort in 1708, asks for the winning probability when \(v = 13\), \(s_1 = \cdots = s_{13} = 4\), \(m = 13\), and the player names the sequence \(1, \ldots, 13\). Studied by mathematicians including J. and N. Bernoulli, De Moivre, Euler, and Catalan, this problem’s strategic aspects remain unexplored. We investigate two variants: one where the player’s Round \(i\) bid depends on previous rounds’ drawn values, which we completely solve, and another where the player must specify all \(m\) bids in advance, solving this for \(s_1 = \cdots = s_v\) and arbitrary \(m\).