A Mathematical Foundation for the Numberlink Game

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.