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Variants of Conway Checkers and k-nacci Jumping

Glenn Bruda, Joseph Cooper, Kareem Jaber, Raul Marquez, Steven J. Miller

TL;DR

The paper extends Conway Checkers to arbitrary dimensions, allowing multiple checkers per cell and k-jumps, and develops a unified energy/pagoda-function framework based on k-nacci constants φ_k. It derives tight upper and lower bounds on the maximum attainable height n_M and on the maximum number of checkers that can occupy a single square, showing these bounds hold across dimensions and are attainable for almost all values of m. The authors provide explicit 1D constructions (Lg-nacci jumping) to achieve the bounds and then project these results to higher dimensions, revealing a logarithmic dependence on m and a linear dependence on dimension d. Collectively, the work significantly generalizes Conway Soldiers, highlighting the power of pagoda-type energies and k-nacci sequences in controlling high-dimensional jumping games.

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.

Variants of Conway Checkers and k-nacci Jumping

TL;DR

The paper extends Conway Checkers to arbitrary dimensions, allowing multiple checkers per cell and k-jumps, and develops a unified energy/pagoda-function framework based on k-nacci constants φ_k. It derives tight upper and lower bounds on the maximum attainable height n_M and on the maximum number of checkers that can occupy a single square, showing these bounds hold across dimensions and are attainable for almost all values of m. The authors provide explicit 1D constructions (Lg-nacci jumping) to achieve the bounds and then project these results to higher dimensions, revealing a logarithmic dependence on m and a linear dependence on dimension d. Collectively, the work significantly generalizes Conway Soldiers, highlighting the power of pagoda-type energies and k-nacci sequences in controlling high-dimensional jumping games.

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.
Paper Structure (11 sections, 13 theorems, 73 equations, 1 figure, 6 tables)

This paper contains 11 sections, 13 theorems, 73 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. Definitions and Preliminaries
  3. Conway's Solution
  4. Generalisations
  5. Lg-nacci Jumping
  6. Solution in Lg: Lg
  7. Solution in Lg
  8. Bounds on checkers in row 1
  9. Bounds in Lg dimensions
  10. Maximum number of checkers in one square
  11. Acknowledgements

Key Result

Theorem 1.1

In the Conway $(m,k,d)$-game, with $m>1$, the maximum row attainable $n_M$ satisfies where $\phi_{k}$ is the $k$-nacci constant as defined in Lemma knacciconstant. In particular, we have that for almost all values of $m$, the upper bound is attainable. When $k=2$ we have that for all $m \in \mathbb{N}$ Again, this upper bound is attainable for almost all $m$.

Figures (1)

  • Figure 1: A move, with $k=3$.

Theorems & Definitions (38)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • proof
  • Remark 1
  • Definition 2.5
  • Lemma 2.6
  • proof
  • ...and 28 more