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Scoring Nim

Hiromi Oginuma, Masato Shinoda

TL;DR

This work introduces Scoring Nim, a scoring extension of Nim where each stone yields 1 point and the last stone yields an additional real-valued bonus $N$, unifying normal, greedy, and misère-like behaviors as $N$ varies. It formalizes a payoff function $f_N$ that captures the final score difference under optimal play and develops recursive, continuity, and large-scale structural properties. The authors derive explicit results for the two-pile case, establish general reduction and symmetry properties for general $n$, and perform a detailed analysis of the three-pile case, revealing intricate breakpoint structures and regime-dependent strategies. The findings illuminate how scoring incentives reshape optimal play in Nim variants and contribute to the broader study of scoring impartial games with potential for further theoretical exploration and applications in game design and analysis.

Abstract

Nim is a well-known combinatorial game, in which two players alternately remove stones from distinct piles. A player who removes the last stone wins under the normal play rule, while a player loses under the misère play rule. In this paper, we propose a new variant of Nim with scoring that generalizes both the normal and misère play versions of Nim as special cases. We study the theoretical aspects of this extended game and analyze its fundamental properties, such as optimal strategies and payoff functions.

Scoring Nim

TL;DR

This work introduces Scoring Nim, a scoring extension of Nim where each stone yields 1 point and the last stone yields an additional real-valued bonus , unifying normal, greedy, and misère-like behaviors as varies. It formalizes a payoff function that captures the final score difference under optimal play and develops recursive, continuity, and large-scale structural properties. The authors derive explicit results for the two-pile case, establish general reduction and symmetry properties for general , and perform a detailed analysis of the three-pile case, revealing intricate breakpoint structures and regime-dependent strategies. The findings illuminate how scoring incentives reshape optimal play in Nim variants and contribute to the broader study of scoring impartial games with potential for further theoretical exploration and applications in game design and analysis.

Abstract

Nim is a well-known combinatorial game, in which two players alternately remove stones from distinct piles. A player who removes the last stone wins under the normal play rule, while a player loses under the misère play rule. In this paper, we propose a new variant of Nim with scoring that generalizes both the normal and misère play versions of Nim as special cases. We study the theoretical aspects of this extended game and analyze its fundamental properties, such as optimal strategies and payoff functions.

Paper Structure

This paper contains 6 sections, 11 theorems, 28 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The Rules of Scoring Nim
  3. The Payoff Function and Recursion Formula
  4. Scoring Nim with Two Piles
  5. General Properties of the Payoff Function
  6. Changes in the Optimal Strategy Depending on the Bonus Value

Key Result

Proposition 3.1

When considering $f_{N}(\bm{p})$ as a function of $N$, the following hold: (i) $f_{N}(\bm{p})$ is continuous. (ii) If $N$ is an integer, then $f_{N}(\bm{p})$ is also an integer, and its parity coincides with that of $|\bm{p}|+N$. (iii) When $N$ is not an integer, the slope of $f_{N}(\bm{p})$ is eith

Figures (6)

  • Figure 1: A partial game tree from the initial position $(5,4,1)$
  • Figure 2: The plot of the function $f_{N}(5,4,2)$
  • Figure 3: The plot of the function $F_{3}(N)$
  • Figure 4: $f_{N}(8,7,1)=\max\{F_{3}(N),-4+|N|\}$ (bold line)
  • Figure 5: The plot of the function $f_{N}(6,4,1)$
  • ...and 1 more figures

Theorems & Definitions (15)

  • Definition 2.1: Scoring Nim
  • Proposition 3.1
  • Corollary 3.2
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 5.1
  • Proposition 5.2
  • Proposition 5.3
  • Definition 5.4: P-position set in Nim
  • Proposition 5.5
  • ...and 5 more