A convergence technique for the game i-Mark

TL;DR

The paper addresses the computation of Sprague-Grundy values for multi-divisor i-Mark games and introduces a convergence-based approach that yields polynomial-time algorithms when convergence occurs. It proves convergence for all instances of the form $i\text{-Mark}(\{1\},\{d_1,d_2\})$, with detailed analysis of bottlenecks and conducive positions providing concrete step-bounds in several subcases. Experimental results support the method, showing correct Grundy values up to large $n$ and demonstrating practical efficiency of the convergence-based algorithm. The work also discusses instances with no convergence and outlines directions for extending the framework to broader classes of impartial games.

Abstract

The game of i-Mark is an impartial combinatorial game introduced by Sopena (2016). The game is parametrized by two sets of positive integers $S$, $D$, where $\min D\ge 2$. From position $n\ge 0$ one can move to any position $n-s$, $s\in S$, as long as $n-s\ge 0$, as well as to any position $n/d$, $d\in D$, as long as $n>0$ and $d$ divides $n$. The game ends when no more moves are possible, and the last player to move is the winner. Sopena, and subsequently Friman and Nivasch (2021), characterized the Sprague-Grundy sequences of many cases of i-Mark$(S,D)$ with $|D|=1$. Friman and Nivasch also obtained some partial results for the case i-Mark$(\{1\},\{2,3\})$. In this paper we present a convergence technique that gives polynomial-time algorithms for the Sprague-Grundy sequence of many instances of i-Mark with $|D|>1$. In particular, we prove our technique works for all games i-Mark$(\{1\},\{d_1,d_2\})$. Keywords: Combinatorial game, impartial game, Sprague-Grundy function, convergence, dynamic programming.

Figures (3)

• Figure 1: In the game $i\textsc{-Mark}(\{1\},\{2,3\})$, convergence starting from position $564$ occurs after $10$ steps. The four sequences at the center are the guess sequences, and the circled number in each guess sequence is its seed.
• Figure 2: Computation of guess sequences on an interval $I_m$ based on the Grundy values of previously computed intervals. In this example we have $D=\{2,3\}$. In the grayed parts of the intervals, convergence did not necessarily occur, so the guess sequences might not necessarily agree. In the white parts of the intervals, convergence already occured, so we know the actual Grundy values.
• Figure 3: Non-convergence in $i\textsc{-Mark}(\{1,3\},\{2,3\})$. The upper and lower rows contain the correct Grundy values, and the central row contains a guess sequence that differs forever from the correct Grundy sequence. Here $N$ denotes a nonzero number, $T$ denotes $2$ or $3$, $L$ denotes a number different from $1$, and $?$ denotes an arbitrary number.

Theorems & Definitions (17)

• Lemma 1
• proof
• Theorem 2
• Definition 3
• Definition 4
• Lemma 6
• proof
• Lemma 7
• proof
• Lemma 8