Tree and Tripod Nim

Abstract

This paper introduces a variant of the impartial combinatorial game nim, called tree nim, as well as a particular case of tree nim called tripod nim. A certain existence-uniqueness result and a periodicity result are proven about the distribution of $\mathcal{P}$-positions and Grundy values in tree nim. Tripod nim is associated to a family of arrays similar to those which arise in the study of sequential compound games. Using these arrays, a partial analysis is given for tripod nim. Conjectures relating to the periods of the rows of these arrays are put forward.