Board games, random boards and long boards

TL;DR

This paper studies a combinatorial game on n×n boards, analyzing the probability of solvability and the expected shortest path length for large boards, revealing that a significant fraction are solvable with surprisingly short solutions.

Contribution

It introduces a new game on square boards, analyzes solvability probabilities, and provides asymptotic results on path lengths for large boards.

Findings

Approximately one-third of large random boards are solvable.

Expected shortest path length tends to 209/96 as board size grows.

Large solvable boards typically have very short solutions.

Abstract

For any odd integer $n\geq3$ a board (of size $n$) is a square array of $n\times n$ positions with a simple rule of how to move between positions. The goal of the game we introduce is to find a path from the upper left corner of a board to the center of the square. If there exists such a path we say that the board is solvable, and we say that the length of this board is the length of a shortest such path. There are $8^{n^2}$ different boards. We discuss various properties of these boards and present some questions and conjectures. In particular, we show that for $n\gg1$ roughly $\frac{1}{3}$ of the boards are solvable, and that the expected length of a random solvable board tends to $\frac{209}{96}$, i.e., very big solvable boards tend to have extremely short solutions.