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One of the most striking results in the analysis of the Green Hackenbush game is the Fusion Principle [3, p. 189], a theorem that enables us to promptly determine the Sprague–Grundy value of any position by turning it into an equivalent position of the popular and well-studied Nim game. However, the constructive proof presented there is sketchy. This is the main reason behind this paper: to present a proof of the Fusion Principle without the omission of the main steps and details, building upon the work of [6].

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Simplifying Game Positions of the Green Hackenbush Combinatorial Game

  • 章节

  • 发表日期：

    2026年

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  • Dantas Serra,
  • João Luís Soares

摘要

One of the most striking results in the analysis of the Green Hackenbush game is the Fusion Principle [3, p. 189], a theorem that enables us to promptly determine the Sprague–Grundy value of any position by turning it into an equivalent position of the popular and well-studied Nim game. However, the constructive proof presented there is sketchy. This is the main reason behind this paper: to present a proof of the Fusion Principle without the omission of the main steps and details, building upon the work of [6].