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Board games, random boards and long boards

Ary Shaviv

TL;DR

This work introduces a minimalistic, direction-assigned $n\times n$ board game on odd $n$, where moves are dictated by the arrow in each cell and the goal is to reach the center; solvability and path length are analyzed through a graph-theoretic lens, including probabilistic results for random boards and worst-case length bounds. Notably, the probability that a random board is solvable tends to $\frac{3}{8}$ as $n$ grows, and the expected length of a random solvable board converges to $\frac{209}{96}$, indicating that large solvable boards typically have short solutions. The paper also develops a graph-structure framework $G(A)$ for boards, investigates isomorphism classes, and discusses generalized models (finite-field, torus, and 3D variants) as well as algorithmic approaches (BFS) to solvability and length, establishing a foundation for further combinatorial and probabilistic exploration of navigability on directed grids.

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.

Board games, random boards and long boards

TL;DR

This work introduces a minimalistic, direction-assigned board game on odd , where moves are dictated by the arrow in each cell and the goal is to reach the center; solvability and path length are analyzed through a graph-theoretic lens, including probabilistic results for random boards and worst-case length bounds. Notably, the probability that a random board is solvable tends to as grows, and the expected length of a random solvable board converges to , indicating that large solvable boards typically have short solutions. The paper also develops a graph-structure framework for boards, investigates isomorphism classes, and discusses generalized models (finite-field, torus, and 3D variants) as well as algorithmic approaches (BFS) to solvability and length, establishing a foundation for further combinatorial and probabilistic exploration of navigability on directed grids.

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.

Paper Structure

This paper contains 18 sections, 6 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Defining boards and games
  3. Random boards
  4. The probability of a random board to be solvable
  5. Trivial upper bound
  6. The exact asymptotic behaviour
  7. The expected length of a random solvable board
  8. Warning
  9. Longest possible board
  10. Trivial bounds
  11. The graph of a board and isomorphic boards
  12. Isomorphic boards and isomorphic graphs
  13. Some generalizations and further remarks
  14. Determining the solvability and length of a given board
  15. Generalized games on boards
  16. ...and 3 more sections

Key Result

Proposition 4

$\lim\limits_{n\to\infty}\text{Prob}_{A\in\text{Mat}_{n\times n}(\mathcal{D})}(A\text{ is solvable})=\frac{3}{8}$Whenever we write limits of the form $n\to\infty$ in this paper we mean that $n$ runs over the odd natural numbers..

Theorems & Definitions (21)

  • Definition 2
  • Example 3
  • Proposition 4: proved together with Noga Alon
  • proof
  • Lemma 5: proved together with Noga Alon
  • proof
  • Lemma 6
  • proof
  • Theorem 7
  • proof
  • ...and 11 more