Variants of Conway Checkers and k-nacci Jumping

TL;DR

This paper generalizes Conway Checkers to higher dimensions and multiple checkers per cell, establishing bounds on the maximum reachable height and analyzing variants with multiple jumps over checkers.

Contribution

It introduces a multi-dimensional, multi-checker version of Conway Checkers and derives bounds on the maximum height reachable, extending previous one-dimensional results.

Findings

Upper and lower bounds on maximum height are almost always equal.

Generalization to k-checker jumps and higher dimensions.

Bounds are tight and nearly always coincide.

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 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.