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A proof of the optimal leapfrogging conjecture

Sam K. Miller and Arthur T. Benjamin

Vol. 18 (2025), No. 1, 105–122
Abstract

Consider a collection of checkers that move on the integer lattice by shifting or jumping. It has been shown that a configuration of pieces can translate itself at a speed of at most 1, and this can only be achieved with a collection of 1, 2 or 4 checkers. We prove the conjecture that with any other number of checkers, the fastest obtainable speed is 2 3.

Keywords
leapfrogging, games on grid
Mathematical Subject Classification
Primary: 00A08
Milestones
Received: 15 April 2023
Revised: 15 August 2023
Accepted: 30 August 2023
Published: 24 January 2025

Communicated by Glenn Hurlbert
Authors
Sam K. Miller
Department of Mathematics
University of California Santa Cruz
Santa Cruz, CA
United States
Arthur T. Benjamin
Department of Mathematics
Harvey Mudd College
Claremont, CA
United States