On variations of Yama Nim and Triangular Nim

TL;DR

This paper explores variations of Yama Nim and Triangular Nim, analyzing their winning strategies, Grundy numbers, and generalizations, including Wythoff variations with interesting number sequences like triangular and square numbers.

Contribution

It introduces new variants and generalizations of Yama Nim and Triangular Nim, providing detailed analysis of their strategies and connections to special number sequences.

Findings

Winning positions are characterized by triangular numbers in certain variants.

Other variants relate to square numbers, pentagonal numbers, and geometric progressions.

The paper offers explicit descriptions of P-positions for specific rule sets.

Abstract

Yama Nim is a two heaps Nim game introduced in the second author's Master Thesis, where the player takes more than $2$ tokens from one heap, and return $1$ token to the other heap. Triangular Nim is a generalization, where the player takes several tokens from one heap, and return some tokens (at least one token) to the other heap, so that the total number of the tokens in the heaps decrease strictly. In this paper, we investigate their winning strategies, Grundy numbers, and their variations and generalizations. Particularly interesting is the Wythoff variations, where in addition to the Yama/Triangular Nim moves, the player is allowed to take tokens from both heaps, say $i$ tokens from the first heap and $j$ tokens from the other, under some restrictions for $i$ and $j$. For example when we force $i=j>0$ for the Triangular Nim, then the pair of non-negative integers $(x, y)$ with $x$ less than or equal to $y$ is in the $P$-position if and only if $(x, y) is in {(0, 0), (0, 1), (1, 3), (3, 6), (6, 10), (10, 15), ...}$, namely the winning strategy is described by triangular numbers. In other rulesets, we also found examples where the square numbers, pentagonal numbers, geometric progressions, and so on.