1. The Simplest Form of the Game

Consider the following game, played by two players, Left and Right: Coins of various (positive numeric) monetary values, colored red or blue, are placed on a semi-infinite strip. We call the red coins Right's and the blue coins Left's. The playing field and possible moves are indicated in Figure 1. Each player can, on his turn, either slide one of his coins down one square, or remove one of his opponent's coins from the strip. Each player gets to keep all the money he moves off the bottom of the strip. The winner is the player who ends up with the most money, or, in the event of a tie, the player who moved last.

Since each player moves just one coin at each turn, the overall game is a disjunctive composition of subgames corresponding to the individual positions. In fact, we can assign a partizan combinatorial game, in the sense of Winning Ways [Berlekamp et al. 1982], Chapters 1-8, or On Numbers and Games [Conway 1976], to each coin on a square, and the overall game will then be the sum of these individual games, which we call terms. Figure 1 has names for these terms [Berkekamp and Wolfe 1994, §§4.1, 4.11; Wolfe 1994, §§3.1, 3.10]. We have also numbered our squares, with square 0 at the bottom.