ON EMULATIONAL EQUIVALENCE OF IMPARTIAL GAMES AND THE GAME HACKENFORB

Abstract

We introduce a variant of the game Hackenbush, called Hackenforb. It is a class of games, each of which is determined by two parameters: a given graph, and a given set of connected graphs (called forbidden graphs). The significance of this game within the realm of impartial combinatorial games is reflected in the fact that, as we show in this article, various known combinatorial games, such as Nim, Subtraction game, Notakto, Treblecross, Chomp, are emulationally equivalent to an instance of Hackenforb (an emulational equivalence of two games is a concept stronger than Grundy-equivalence, but weaker than the isomorphism between games’ structures; our belief is that this version of equivalence is what really captures the core of the intuitive perception of what it means for two games to be “basically the same game”). At the end of our article, we show that Hackenforb is, unfortunately, not “almighty,” that is, we describe a game that is not emulationally equivalent to an instance of Hackenforb.