Variants of Conway Checkers and k-Nacci Jumping

Abstract

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 generalizing 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 is almost always 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. This paper is dedicated with thanks to Peter G. Anderson, Marjorie Bicknell-Johnson and William Webb. In a similar fashion to the pagoda functions crucial to this paper, their tireless effort and leadership has allowed the journal and association to take only an upward trajectory for decades, and it is a great pleasure to acknowledge their service and mentorship.