...
首页> 外文期刊>The Fibonacci quarterly >THE FIBONACCI QUILT GAME
【24h】

THE FIBONACCI QUILT GAME

机译:斐波纳契被子游戏

获取原文
获取原文并翻译 | 示例
   

获取外文期刊封面封底 >>

       

摘要

Zeckendorf [ 13] proved that every positive integer can be expressed as the sum of noncon-secutive Fibonacci numbers. This theorem inspired a beautiful game, the Zeckendorf Game [2]. Two players begin with n 1's and take turns applying rules inspired by the Fibonacci recurrence, F_(n+1) = F_n + F_(n_1). until a decomposition without consecutive terms is reached; whoever makes the last move wins. We look at a game resulting from a generalization of the Fibonacci numbers, the Fibonacci Quilt sequence [3]. This sequence arises from the two-dimensional geometric property of tiling the plane through the Fibonacci spiral. Beginning with 1 in the center, we place integers in the squares of the spiral such that each square contains the smallest positive integer that does nol have a decomposilion as the sum of previous terms that do not share a wall. This sequence eventually follows two recurrence relations, allowing us to construct a varialion on the Zeckendorf Game, the Fibonacci Quilt Game. Whereas some properties of the Fibonacci sequence are inherited by this sequence, the nature of its recurrence leads to others, such as Zeckendorf's theorem, no longer holding. Thus, it is of interest to investigate the generalization of the game in this setting to see which behaviors persist. We prove, similar to the original game, that this game also always terminates in a legal decomposition. We give a lower bound on game lengths, showing that, depending on strategies, the length of the game can vary and either player could win. Finally, we give a conjecture on the length of a random game.
机译:Zeckendorf [13]证明,每个正整数都可以表示为非通信斐波纳契数的总和。这个定理启发了一款漂亮的比赛,Zeckendorf游戏[2]。两个玩家从N 1开始,然后轮流应用灵感来自Fibonacci复发,f_(n + 1)= f_n + f_(n_1)。直到未经连续术语的分解;无论谁让最后一个举动胜利。我们看看由斐波纳契数的概括而导致的游戏,Fibonacci被子序列[3]。该序列由通过斐波纳契螺旋平铺平面的二维几何特性。从中中心的1开始,我们将整数放在螺旋的平方中,使得每个方块包含最小的正整数,NOL具有分解作为不共享墙壁的先前术语的总和。这个序列最终遵循两个复发关系,允许我们在Zeckendorf游戏中构建一个曲线,斐波纳契被子游戏。虽然斐波纳契序列的一些属性由该序列继承,但其复发性的性质导致他人,例如Zeckendorf的定理,不再持有。因此,研究该设置中游戏的概括是有意思的,以了解哪些行为持续存在。我们证明,类似于原始游戏,这个游戏也始终终止于法律分解。我们在游戏长度上给出了较低的界限,表明,根据策略,游戏的长度可能会有所不同,而且玩家可以赢得。最后,我们在随机游戏的长度上给出了猜想。

著录项

相似文献

  • 外文文献
  • 中文文献
  • 专利
获取原文

客服邮箱:kefu@zhangqiaokeyan.com

京公网安备:11010802029741号 ICP备案号:京ICP备15016152号-6 六维联合信息科技 (北京) 有限公司©版权所有
  • 客服微信

  • 服务号