A Mathematical Foundation for the Numberlink Game Andrea Arauza Rivera Matt McClinton David Smith October 7 2022

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A Mathematical Foundation for the Numberlink Game

Andrea Arauza Rivera, Matt McClinton, David Smith

October 7, 2022

*Andrea Arauza Rivera, Ph. D.

Assistant Professor, Mathematics

Cal State East Bay

andrea.arauzarivera@csueastbay.edu

Matt McClinton

Cal State East Bay

mmcclinton2@horizon.csueastbay.edu

David Smith

Cal State East Bay

david.smith2@csueastbay.edu

*Corresponding author.

Abstract

Numberlink is a puzzle game in which players are given a grid with nodes marked with a natural

number,

, and asked to create

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.

Contents

1 Introduction 2

2 Numbered k-Grids 3

2.1 Someinitialresults.......................................... 4

2.2 Creating connections between nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Example of computing Φk[p]and ΦΓk[p].............................. 7

3 The Path Towards a Solution 8

3.1 Wheredowestart?.......................................... 8

3.2 The τalgorithm ........................................... 9

4 Future Work 12

arXiv:2210.02888v1 [math.GM] 29 Sep 2022

1 Introduction

With millions of downloads between the Apple and Google Play stores, Puzzledom is an app that oﬀers a

variety of curious puzzles. One of these is the puzzle called Numberlink [1]. Puzzles like Numberlink have

been the subject of a number of interesting articles; the reader may enjoy any one of the following [2], [3].

The reader should note that there is another popular game called Numberlink. This version of the game is

discussed in [4], [5], [6].

We focus on the Numberlink puzzles found in the Puzzledom app. These puzzles begin with a set of

numbered boxes (nodes) conﬁgured in a grid; see Figure 1. The puzzle is solved when the player creates links

between the numbered nodes so that

•a node with number nhas nconnections to other nodes,

•there are no more than k= 2 connections between any two nodes, and

•the connections must create a path between any two given nodes (path connected).

•

The game also implicitly requires that connections be made horizontally or vertically, and that no

connections intersect.

Three sample Numberlink puzzles are shown in Figure 1. The reader is encouraged to whip out a pencil

and try to solve each puzzle! The sections in this article are set up as follows:

•

Section 1 gives a mathematical formulation of the Numberlink game via graphs. This section also

includes some initial consequences of this formulation.

Figure 1:

We show 3 sample puzzles for the reader to try and enjoy. Each of these puzzles is included in

Numberlink and listed as “Novice” (left) or “Regular”(center and right) [1].

•

Section 2 describes ways in which the player may ﬁnd the best nodes to start to create connections.

This analysis is based on the number in the node as well as the number of neighbors available for

connecting.

•

Section 3 contains the main results of this article; an algorithm which produces the “guaranteed

connections” between nodes. In this section we prove that if the algorithm we outline arrives at a

solution, then the solution is unique.

•We conclude in section 4 with some closing thoughts and open questions about these puzzles.

2 Numbered k-Grids

This section contains the deﬁnitions needed to describe the Numberlink puzzles in terms of graphs. We

begin by deﬁning the initial set up of a puzzle as a graph where the nodes are labeled with a whole number

and arranged on a grid. We call these numbered

-grids. Next, we deﬁne what it means for a node

in a numbered

-grid to have top, bottom, left and right neighbors. Finally, we deﬁne what it means for a

numbered k-grid to be solved.

The reader may be wondering what on earth this

business is. Indeed, in the original Puzzledom-

Numberlink puzzles there is a rule that no two nodes may share more than

= 2 connections. We work with

a more general rule and allow kto be any positive whole number.

Deﬁnition.

numbered k-grid

is a ﬁnite collection of nodes and connections between nodes satisfying

the following:

1. each node is given coordinates (x, y)where x, y ∈N∪ {0};

2. each node is labeled with a magnitude n∈N;

3. there are no more than kconnections between nodes;

4. connections can only exist connecting horizontally or vertically adjacent nodes.

We denote a numbered

-grid by Γ

and write

= [

x, y, n

]for a node in Γ

. Often we will denote the

magnitude of pas magn(p).

The choice of a square to represent the nodes in a numbered grid is somewhat arbitrary. One could use a

circle or other ﬁgure to represent nodes. We choose to use a square to emphasize the top, bottom, left and

right neighbors of the node. We now deﬁne what it means to be neighboring to a node p.

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AMathematicalFoundationfortheNumberlinkGameAndreaArauzaRivera,MattMcClinton,DavidSmithOctober7,2022*AndreaArauzaRivera,Ph.D.AssistantProfessor,MathematicsCalStateEastBayandrea.arauzarivera@csueastbay.eduMattMcClintonCalStateEastBaymmcclinton2@horizon.csueastbay.eduDavidSmithCalStateEastBaydavid.smit...

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A Mathematical Foundation for the Numberlink Game Andrea Arauza Rivera Matt McClinton David Smith October 7 2022

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