A Mathematical Foundation for the Numberlink Game

TL;DR

This paper formulates the Numberlink puzzle mathematically using graph theory, providing an algorithm to analyze puzzle solutions and exploring implications and open questions in the domain.

Contribution

It introduces a graph-based mathematical framework for Numberlink and presents an algorithm to analyze puzzle solutions, advancing understanding of its structure.

Findings

Graph formulation of Numberlink puzzles

Algorithm for analyzing solutions

Open questions on puzzle complexity

Abstract

Numberlink is a puzzle game in which players are given a grid with nodes marked with a natural number, $n$, and asked to create $n$ connections with neighboring nodes. Connections can only be made with top, bottom, left and right neighbors, and one cannot have more than two connections between any neighboring nodes. In this paper, we give a mathematical formulation of the puzzles via graphs and give some immediate consequences of this formulation. The main result of this work is an algorithm which provides insight into characteristics of these puzzles and their solutions. Finally, we give a few open questions and further directions.