FORMS AND VALUES OF NUMBER-LIKE AND NIMBER-LIKE GAMES

Abstract

We consider combinatorial games (positions) played by two players who move alternately. In a disjunctive sum of positions a player may play in any one summand. Plays in a particular summand may not alternate between players. Play ends in a finite number of moves when a player cannot move in any summand. The winner is determined by the last player to move. We primarily consider the case where the last player to move wins (the normal-play convention). In a sum of nimbers both players have the same available moves (options). In a sum of numbers both players would rather it not be their turn. The canonical form theory partitions positions into equivalence classes which form a group with disjunctive sum. The unique (canonical) representative of an equivalence class is called a value. Nimber-valued and number-valued positions are closed under disjunctive sum. It has long been known how to identify positions that are nimbers or numbers. There are other methods to analyze positions that are not nimber-like and number-like, such as reduced canonical forms and atomic weight. Nimbers and numbers are both hereditarily transitive (where no player would benefit from moving twice in a row) and both are Hackenbush positions. Dicotic positions (where both players have an option or neither does) are like nimbers and numbers. The dicotic hereditarily transitive positions are described using ordinal sum. We show how to recognize positions whose values are from the ruleset Hackenbush stalks (whose summands are described by ordinal sums). We then consider outcomes of Hackenbush stalks under the misère-play convention. We end by considering the ruleset Partizan Euclid, a partizan dicotic ruleset, that is like nimbers and numbers in other ways.