On Lasker's Card Game

We consider a two-person constant sum perfect information game, which was described in 1929 by Emanuel Lasker, the mathematician and world chess champion, who called it whistette. The game uses a deck of cards that consists of a single totally ordered suit of 2n cards. To begin play the deck is divided into two hands A and B of n cards each, held by players Left and Right and one player is designated as having the lead. The player on lead chooses one of his cards, and the other player after seeing this card selects one of his own to play. The player with the higher card wins a "trick" and obtains the lead. The cards in the trick are removed from each hand, and play then continues until all cards are exhausted. Each player strives to maximize (or minimize, in the misere version) his trick total, and the value of the game to each player is the number of tricks he takes under optimal play. While we have found an optimal strategy for the misere version, the regular game remains unsolved in general. We have derived basic properties of the regular game, found criteria under which one hand is guaranteed to be better than another, and determined the value functions in several special cases. In this paper we outline the proof of the "2 blocks vs 3" solution, which, interesting in its own right, also serves as a tool for deriving other properties, such as the equivalence of superiority and precedence.