The Self-Referential Games Minnie and Wynnie and Some Variants

Abstract

Several self-referential impartial combinatorial games have been studied and some have been solved. However, by making small changes to the rules we can get vastly different outcomes. We look at many new games, solving some, and coming up short in others, but in particular, we look at Minnie and Wynnie. With Minnie representing a small tweak to the rules of Nim and Wynnie representing a small tweak to the rules of Wythoff, one might expect rather uninteresting, repetitive results. However, that is far from the case. We solve Wynnie and misere Wynnie, and we nearly solve Minnie up to three stacks. In addition, we solve other new games Nimb and Nimtb, and we look at Wynnie on graphs.