Board games, random boards and long boards
Ary Shaviv

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 a board (of size ) is a square array of 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 different boards. We discuss various properties of these boards and present some questions and conjectures. In particular, we show that for roughly of the boards are solvable, and that the expected length of a random solvable board tends to , i.e., very big solvable boards tend to have extremely short solutions.
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Taxonomy
TopicsArtificial Intelligence in Games · Algorithms and Data Compression · Mathematics and Applications
