Two Dimensional Subtraction -- Transfer Games

TL;DR

This paper extends the understanding of two-dimensional subtraction-transfer games, proving periodicity of their nim-values and calculating exact periods in many cases, advancing the theory of combinatorial game analysis.

Contribution

It generalizes previous results by Tamás Lengyel, establishing periodicity of nim-values for a broad class of two-dimensional subtraction-transfer games and introducing new notions of periodicity.

Findings

Nim-values of these games are periodic in many cases.

Exact periods are calculated for several game variants.

New concepts of periodicity are developed.

Abstract

We generalize the results and conjectures of Tam\'{a}s Lengyel, showing that the \textsc{nim}-values of a large class of two-dimensional subtraction-transfer games are periodic. These are impartial, normal-play games with two piles of tokens, where players alternate either taking some tokens from a pile or transferring tokens from one pile to the other. In many cases, we calculate the exact period. We also develop several new notions of periodicitiy.