论文标题
多人游戏和多重盟友Zeckendorf游戏的赢得策略
Winning Strategy for the Multiplayer and Multialliance Zeckendorf Games
论文作者
论文摘要
Edouard Zeckendorf证明，每个正整数$ n $都可以独特地写成\ cite {ze}作为非粘合斐波那契数的总和，称为zeckendorf分解。根据Zeckendorf的分解，我们为多个玩家提供了Zeckendorf游戏。我们表明，当Zeckendorf游戏至少有3美元的玩家时，没有一个玩家以$ n \ geq 5 $的胜利策略。然后，我们将多人游戏扩展到多合规游戏，找到一些有趣的情况，在这些情况下，没有联盟具有获胜的策略。这包括两合规游戏，在某些情况下，一个联盟总是有胜利的策略。 ％我们检查了联盟或球员的组合可以赢得什么，以及他们必须做什么规模。我们还发现了关于我们的证明方法可以表现出的联盟的必要结构限制。此外，我们发现一些必须具有获胜策略的联盟结构。 ％我们还将广义的Zeckendorf游戏从$ 2 $玩家扩展到多个玩家。我们发现，当游戏拥有$ 3 $的玩家时，玩家$ 2 $从未有任何大型$ n $的胜利策略。我们还发现，当游戏具有至少$ 4 $的玩家时，没有玩家对任何大型$ n $都有获胜的策略。
Edouard Zeckendorf proved that every positive integer $n$ can be uniquely written \cite{Ze} as the sum of non-adjacent Fibonacci numbers, known as the Zeckendorf decomposition. Based on Zeckendorf's decomposition, we have the Zeckendorf game for multiple players. We show that when the Zeckendorf game has at least $3$ players, none of the players have a winning strategy for $n\geq 5$. Then we extend the multi-player game to the multi-alliance game, finding some interesting situations in which no alliance has a winning strategy. This includes the two-alliance game, and some cases in which one alliance always has a winning strategy. %We examine what alliances, or combinations of players, can win, and what size they have to be in order to do so. We also find necessary structural constraints on what alliances our method of proof can show to be winning. Furthermore, we find some alliance structures which must have winning strategies. %We also extend the Generalized Zeckendorf game from $2$-players to multiple players. We find that when the game has $3$ players, player $2$ never has a winning strategy for any significantly large $n$. We also find that when the game has at least $4$ players, no player has a winning strategy for any significantly large $n$.
