康威跳棋的变种与k-nacci跳跃
Variants of Conway Checkers and k-nacci Jumping
AI总结：
本文研究了康威跳棋的变种，探讨了在不同维度棋盘上跳棋规则的变化，并推导了可达到高度的上下界，展示了在不同跳棋步数下的游戏结果。
摘要：
康威跳棋是一种在无限棋盘的下半部分每个方格上放置一个棋子的棋类游戏。棋子通过跳过相邻的棋子来移动，跳过的棋子被移除。康威证明了通过将棋盘上的每个单元格按黄金比例的幂次加权，无法在有限步内达到第五行。其他作者考虑了在不同棋盘上进行的游戏，包括将标准游戏推广到更高维度。我们研究了一个任意维度的棋盘，允许一个单元格持有多个棋子，并且每个单元格初始放置m个棋子。我们推导了可以达到的高度的上界和构造性的下界，使得上界几乎总是等于下界。我们还考虑了更一般的情况，即每个棋子不是跳过1个棋子，而是跳过k个棋子，再次展示了可达到的最大高度位于几乎总是相等的界内。
英文摘要：
Conway Checkers is a game played with a checker placed in each square of the lower half of an infinite checkerboard. Pieces move by jumping over an adjacent checker, removing the checker jumped over. Conway showed that it is not possible to reach row 5 in finitely many moves by weighting each cell in the board by powers of the golden ratio such that no move increases the total weight. Other authors have considered the game played on many different boards, including generalising the standard game to higher dimensions. We work on a board of arbitrary dimension, where we allow a cell to hold multiple checkers and begin with m checkers on each cell. We derive an upper bound and a constructive lower bound on the height that can be reached, such that the upper bound almost never fails to be equal to the lower bound. We also consider the more general case where instead of jumping over 1 checker, each checker moves by jumping over k checkers, and again show the maximum height reachable lies within bounds that are almost always equal.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
