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Extended circular nim

Koki Suetsugu

TL;DR

This work extends circular Nim to Extended Circular Nim $ECN(m_S,k)$, where moves select up to $k$ piles at step sizes drawn from a set $S$ around a circle of $m$ piles. Using combinatorial game theory and sums of positions (disjunctive and selective), it derives closed formulas for many $m\le 8$ configurations and maps several ECN cases to well-known rulesets such as Nim, Moore's Nim, and their disjunctive/selective compositions. The results include explicit $\mathcal{P}$-position characterizations for a wide range of six-, seven-, and eight-pile ECN instances, along with numerous isomorphisms that transfer known results; numerous cases remain open, outlining a clear research agenda for larger $m$ and broader $S$. Overall, the paper deepens understanding of how move-structure in cyclic impartial games determines $P$-positions and illuminates generalizable patterns across families of ECN.

Abstract

Circular nim $CN(m, k)$ is a variant of nim, in which there are $m$ piles of tokens arranged in a circle and each player, in their turn, chooses at most $k$ consecutive piles in the circle and removes an arbitrary number of tokens from each pile. The player must remove at least one token in total. For some cases of $m$ and $k$, closed formulas to determine which player has a winning strategy have been found. Almost all cases are still open problems. In this paper, we consider a variant of circular nim, extended circular nim. In extended circular nim $ECN(m_S, k)$, there are $m$ piles of tokes arranged in a circle. $S$ is a set of positive integers less than or equal to half of $m$. Each player, in their turn, chooses at most $k$ piles selected every $s$-th pile in a circle for an $s \in S$. We find some closed formulas to determine which player has a winning strategy for the cases where the number of piles is no more than eight.

Extended circular nim

TL;DR

This work extends circular Nim to Extended Circular Nim , where moves select up to piles at step sizes drawn from a set around a circle of piles. Using combinatorial game theory and sums of positions (disjunctive and selective), it derives closed formulas for many configurations and maps several ECN cases to well-known rulesets such as Nim, Moore's Nim, and their disjunctive/selective compositions. The results include explicit -position characterizations for a wide range of six-, seven-, and eight-pile ECN instances, along with numerous isomorphisms that transfer known results; numerous cases remain open, outlining a clear research agenda for larger and broader . Overall, the paper deepens understanding of how move-structure in cyclic impartial games determines -positions and illuminates generalizable patterns across families of ECN.

Abstract

Circular nim is a variant of nim, in which there are piles of tokens arranged in a circle and each player, in their turn, chooses at most consecutive piles in the circle and removes an arbitrary number of tokens from each pile. The player must remove at least one token in total. For some cases of and , closed formulas to determine which player has a winning strategy have been found. Almost all cases are still open problems. In this paper, we consider a variant of circular nim, extended circular nim. In extended circular nim , there are piles of tokes arranged in a circle. is a set of positive integers less than or equal to half of . Each player, in their turn, chooses at most piles selected every -th pile in a circle for an . We find some closed formulas to determine which player has a winning strategy for the cases where the number of piles is no more than eight.
Paper Structure (16 sections, 19 theorems, 3 equations, 4 figures, 5 tables)

This paper contains 16 sections, 19 theorems, 3 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Combinatorial game theory
  3. Some impartial rulesets
  4. Nim and Moore's nim
  5. Simplicial nim
  6. Circular nim
  7. Sums of positions
  8. Disjunctive sum
  9. Selective sum
  10. Main results
  11. Extended circular nim
  12. Extended circular nim with less than six piles
  13. Extended circular nim with six piles
  14. Extended circular nim with seven piles
  15. Extended circular nim with eight piles
  16. ...and 1 more sections

Key Result

Theorem 1

A position $(n_0, n_1, \ldots, n_{m-1})$ in nim is a $\mathcal{P}$-position if and only if $n_0\oplus n_1 \oplus \cdots \oplus n_{m-1} = 0$.

Figures (4)

  • Figure 1: ${\rm ECN}(6_{1,2}, 3)$
  • Figure 2: ${\rm ECN}(5_{\{1\}}, i)$ and ${\rm ECN}(5_{\{2\}},i)$ are isomorphic
  • Figure 3: ${\rm ECN}(7_{\{1,2\}}, i)$ and ${\rm ECN}(7_{\{1,3\}},i)$ are isomorphic
  • Figure 4: ${\rm ECN}(8_{\{1\}}, i)$ and ${\rm ECN}(8_{\{3\}}, i)$ are isomorphic

Theorems & Definitions (45)

  • Definition 1
  • Theorem 1: Bou01
  • Definition 2
  • Theorem 2: Moo10
  • Example 1
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 3: Spr35Gru39
  • Definition 6
  • ...and 35 more