A Proof of the Optimal Leapfrogging Conjecture

Abstract

Suppose we place checkers in the lower left corner of a Go board and wish to move them to the upper right corner in as few moves as possible, where the pieces move as in the game of Chinese checkers. Auslander, Benjamin, and Wilkerson in 1993 generalized this game for integer lattices and defined a measure of speed for a starting configuration of pieces. They proved that the maximum speed of any configuration is 1, and only three configurations, called "speed-of-light" configurations, attain this speed. We prove their conjecture that the maximum speed of a non-speed-of-light configuration is 2/3 in the 2-dimensional case, and present a framework that should extend to higher dimensions.