A Proof of the Optimal Leapfrogging Conjecture

TL;DR

The paper studies the problem of moving checkers on a lattice from a starting corner to the opposite corner using Chinese checkers moves, focusing on the maximal forward speed achievable by a configuration. It develops a rigorous lattice-model framework on $\mathbb{Z}^n$, introducing placements, trajectories, displacement, and speed, along with ladder structures and a weight function $\omega(M)$ to bound trajectory efficiency. The authors prove that in $\mathbb{Z}^2$, aside from the speed-of-light configurations, every configuration has maximum speed $\frac{2}{3}$, covering the cases $p=1,2,3,4$ and extending to $p>4$ via an isolating partition of movesets. This resolves the conjecture of Auslander, Benjamin, and Wilkerson for the two-dimensional case and provides a framework that is expected to extend to higher dimensions, potentially informing optimal transport-like dynamics on lattices.

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.

Figures (14)

• Figure 1: An example of the serpent's $2$-move trajectory, consisting of 2 jumps. The placements in the leftmost and rightmost diagrams are translates, and represented by the same configuration, the serpent. These two placements have displacement 2, and require 2 moves to reach one from the other. Hence, the serpent is a speed of light configuration, i.e. it has speed 1.
• Figure 2: An example of a placement $X$ which contains a ladder. The ladder is $\{p_0,p_1, p_2, p_3, p_4\}$, with $p_0$ is the base of the ladder, and pieces $p_1, p_2, p_3, p_4$ are the rungs. Here $\{p_0\} = T(X)$ and $\{p_4\} = H(X)$, so the ladder is a true ladder of $X$. $X$ has width 8.
• Figure 3: In the case of $p=4$, we cannot place $p_3$ without a contradiction.
• Figure 4: After a jump, $p$ is isolated and cannot jump.
• Figure 5: The back borders of $C$ before any moves on the left, and after the first three ascents on the right. An $\times$ indicates a location a piece cannot be located in that board state.

Theorems & Definitions (24)

• Proposition 3.0.1
• proof
• Proposition 3.0.2
• proof
• Proposition 3.0.3
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• proof