Two Dimensional Subtraction -- Transfer Games

TL;DR

This work broadens Lengyel's results on two-pile subtraction-transfer games by establishing periodic nim-values for a large class of Lengyel transfer games $L(b;x_1,y_1;x_2,y_2)$ and introducing generalized notions of periodicity, including nim-periodicity and diagonal/horizontal periodicity. Central contributions include a proof of diagonal periodicity and explicit horizontal-period formulas for the case $L(b;x_1,0;1,1)$, together with examples of arbitrarily long preperiods and a systematic framework (via partial move functions) for generalized nim-periodicity. The paper also develops sporadic cases, proves a conjecture about multitransfer options, and proposes a unifying conjecture for the horizontal period $g(b,x_1,0;x_2,y_2)$, supported by extensive computer experiments up to small parameter ranges. Beyond providing concrete period results, the work lays groundwork for a broader theory of periodicity in vector subtraction-transfer games and highlights rich open questions, including behavior with nonzero $y_1$ and more general transfer-option families.

Abstract

We generalize the results and conjectures of Tamás 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.

Figures (16)

• Figure 1: Four periods of the array of nim-values for $L(1;1,0;1,1)$
• Figure 2: One period of the array of nim-values for $L(2;3,0;1,1)$
• Figure 3: Two initial periods of the arrays of nim-values for $L(2;1,1)$ and $L(3;1,1)$
• Figure 4: Proof of Lemma \ref{['lemma: refinement of Larsson']}
• Figure 5: The $2$-block Rule (Lemma \ref{['lemma: ac rule']})

Theorems & Definitions (75)

• Definition 1.1
• Definition 1.2
• Theorem 1.3: Lengyel
• Conjecture 1.4: Lengyel
• Theorem 1.5: First Main Theorem
• Lemma 1.6
• Theorem 1.7: Second Main Theorem
• Remark
• Lemma 2.1: Nim-Periodicity
• proof