Tree and Tripod Nim

TL;DR

This work extends Nim by introducing Tree Nim and its simplest unsolved case, Tripod Nim, and develops a coherent framework to study $\mathcal{P}$-positions and Grundy values via the $\mathcal{P}$-completion Lemma. It introduces barriers and shadows to compare different tree-nim worlds, proves additively periodic behavior of completion sequences, and connects Tree Nim to Forest Nim through Grundy values, establishing a unifying structural view. The paper also analyzes Tripod Nim, revealing locator-like arrays with symmetry and partial equivalences (notably the two-five equivalence) and linking these patterns to a dynamical-systems perspective that distills the evolution of position distributions into $D(1,n)$ and its generalizations. Collectively, these results advance understanding of position distributions in tree-based impartial games, provide partial analyses for Tripod Nim, and propose conjectures on period structure and dynamical behavior with potential implications for sequential compound games.

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.

Figures (14)

• Figure 1: An example nim game
• Figure 2: An example tree nim position. Vertices circled in green are in play. Vertices boxed in red are not yet in play.
• Figure 3: This diagram shows an example barrier in a 2D game. The game's $\mathcal{P}$ positions are hatched. The region $U_B$ is tinted red. There is a correspondence between rays which intersect the lower set, and segments along the boundary between the upper and lower sets. One can imagine light sources to the north and west shining down on the boundary, with the $\mathcal{P}$ positions casting shadows. The shadowed segments are colored red. An arbitrary position $P$ is colored blue, and the positions it sees are tinted blue. $P$ is an $\mathcal{N}$ position because it sees the $\mathcal{P}$ position to its left. Alternatively, $P$ is an $\mathcal{N}$ position because it is in a shadowed ray, and below the barrier.
• Figure 4: On the left we have a plot of the $\mathcal{P}$ positions of 2 stack normal nim, and on the right is a plot of the $\mathcal{P}$ positions of 2 stack misère nim. The same shadow (in red) is cast onto each barrier (represented by the bold black line). Because the cast shadows are the same, the plots will look identical below the barrier.
• Figure 5: The position $B_{n,m}$ for a certain biray $B$

Theorems & Definitions (75)

• Proposition 1.1
• proof
• Definition 1.1: to see
• Proposition 1.2
• proof
• Definition 2.1: Ray
• Example 2.1
• Definition 2.2: to see, as a ray
• Remark 2.1
• Proposition 2.1