Optimal Leapfrogging, A Complete Guide

Suppose we have some checkers placed in the lower left corner of a Go board, and we wish to move them to the upper right corner in as few moves as possible. There are no opponent pieces present, and the pieces move as they would in the game of Chinese Checkers, where for one move, a piece may either shift one unit in any direction, or repeatedly leapfrog over other pieces. Let us consider the Go board as a subset of the non-negative integer lattice Z. As an example, suppose we have four pieces placed at the coordinates (0, 0), (1, 0), (0, 1), and (1, 1), and wish to move them to the squares (9, 9), (10, 9), (9, 10), and (10, 10). For the pieces to complete the task in as few moves as possible, the pieces must first be moved into a configuration such that they may jump over each other in an optimal way. We may intuitively attempt lining the checkers up diagonally in what we will call a snake configuration, that is, moving the pieces to coordinates (0, 0), (1, 1), (2, 2), and (3, 3). By repeating the three-move process of shifting the backmost piece to the right [(0, 0) −→ (1, 0)], leapfrogging that piece to the front [(1, 0) −→ (3, 4)], then shifting it right again [(3, 4) −→ (4, 4)], we can reach our destination in 4 + 4 + (3× 7) = 29 moves. However a faster method exists. We first move the pieces into what we call a serpent configuration, with the pieces on coordinates (0, 0), (1, 0), (1, 1), and (2, 1). Then we repeat the two-move process of leapfrogging the backmost piece to the front [(0, 0) −→ (2, 2)] then leapfrogging the new backmost piece to the front again [(1, 0) −→ (3, 1)], we may reach our destination in 1+1+(2×8) = 18 moves. This is indeed the fastest way of moving the checkers from the bottom left to the upper right. We define a measure of the movement efficiency of a placement of pieces, and it may be shown that under this measure, the serpent is the most efficient configuration possible. In fact, it was shown by Auslander, Benjamin, and Wilkerson that the serpent configuration is maximally efficient, with only three configurations attaining this efficiency. For any non-maximal configurations, their efficiency was conjectured to have a strict upper bound, which we prove. [1]