AI 中文总结
本文研究Hex游戏最优玩法的复杂性，确定有限棋盘λ(5)=7、无限带5×∞棋盘λ(5×∞)=5，证实Campbell猜想并回答Stromquist问题，揭示有限与无限棋盘的根本区别。
AI 中文摘要
Hex游戏是组合数学中最著名的连接类游戏之一。尽管已知先手总能拥有制胜策略，但最优玩法的复杂性却鲜为人知。在Campbell引入参数λ(n)（分别衡量最短保证制胜路径长度）和δ(n)（先手制胜所需最少棋子数）后，我们确立了这些量的新结构性质。主要结果确定λ(5)=7，从而证实了Campbell的一个猜想。我们还证明了无限带5×∞棋盘上的对应问题呈现显著不同的表现：该场景下λ(5×∞)=5，严格小于λ(5)。据我们所知，本文首次解决了Campbell 2004年的一个猜想，并肯定地回答了Stromquist（2006）提出的问题，同时揭示了有限与无限棋盘间的根本区别，为Hex游戏最优制胜策略的本质提供了新见解。
英文摘要
The game of Hex is one of the most celebrated connection games in combinatorics. Although it is known that the first player always has a winning strategy, very little is understood about the complexity of optimal play. Following Campbell's introduction of the parameters $λ(n)$ and $δ(n)$, measuring respectively the length of the shortest guaranteed winning path and the minimum number of stones required by the first player to force a win, we establish new structural properties of these quantities. Our main result determines the exact value \[ λ(5)=7, \] thereby confirming one of Campbell's conjectures. We also show that the corresponding problem on the infinite strip $5\times\infty$ has a markedly different behavior. In this setting we prove that \[ λ(5\times\infty)=5, \] which is strictly smaller than $λ(5)$. To the best of our knowledge, this paper provides the first solution to one of Campbell's 2004 conjectures and answers a question posed by Stromquist (2006) in the affirmative. Furthermore, it uncovers a fundamental distinction between finite and infinite boards, offering new insights into the nature of optimal winning strategies in Hex.
Comments14 pages and 19 figures
