Polyominoes Investigated Via Partitions

Part of what makes the study of polyominoes without holes rich and intriguing is the different points of view that can be adopted in investigating their mathematical properties.

A polyomino is a collection of 1x1 squares in the (Euclidean) plane where pairs of squares, if they touch each other, share an edge and there are no "holes." Figure 1 shows all of the inequivalent polyominoes with 4 squares - the coloring of the squares is not of importance. Note two polyominoes are considered the same if they are congruent (have the same shape). I will denote the number of squares in a polyomino by m, and although the number of "different" polyominoes with m squares has been enumerated for quite large values of m there is no known "formula" for the number of polyominoes with m squares as cells. Starting with m = 1 the number of polyominoes with m cells (holes allowed) is given by this sequence: 1, 1, 3, 5, 12, 35, 108, .... The first time that the counts for when holes are allowed and the enumeration of polyominoes without holes occurs when m = 7. For m = 7 with no holes there are 107 different incongruent polyominoes.

Note: The information below was created with the assistance of AI.

Level of Mathematics

Primary Level:

• Upper High School (Grades 10–12)
• Early Undergraduate (Introductory Discrete Mathematics / Combinatorics)

Reasons

• Uses discrete mathematics concepts such as graphs, partitions, combinatorics, and geometric reasoning.
• Requires ability to interpret mathematical notation and explore open-ended investigations.
• Encourages mathematical research-style thinking rather than routine exercises.

Difficulty progression within the article

Subject Matter

The document integrates multiple mathematical domains.

Primary Mathematical Topics

Discrete Mathematics

• Polyomino structures
• Graph theory representation of shapes
• Integer partitions
• Enumeration problems

Geometry

• Polygons
• Convex vs non-convex shapes
• Angles (90°, 270°)
• Polygon side lengths

Combinatorics

• Counting polyomino configurations
• Partition structures
• Structural classification

Graph Theory

• Vertices
• Edges
• Faces
• Vertex degree (valence)

Example from the article:

• Polyominoes are analyzed as plane graphs with vertices, edges, and faces, connecting geometry and graph theory.

Key Mathematical Concepts

Core Concepts

1. Polyominoes
• Shapes formed by joining unit squares edge-to-edge.
• No holes allowed in the discussed cases.
2. Graph Representation
• Polyomino viewed as a planar graph with
• vertices (v)
• edges (e)
• faces (f)
3. Integer Partitions
• Writing a number as a sum of positive integers.
• Used to classify polyomino edge lengths.
4. Polygon Structure
• Number of sides
• Corner angles (90°, 270°)
5. Convexity
• Vertically convex
• Horizontally convex
6. Combinatorial Enumeration
• Counting distinct polyomino shapes.
7. Edge Length Partitions
• Example partition: {7,1}

Mathematical Investigations in the Article

The document proposes open-ended research questions, such as:

Investigation 1

• Relationship between:
• = number of squares
• = number of sides
• Determine minimum and maximum possible values of .

Investigation 2

Study equilateral polyominoes.

Questions include:

• What values of and occur?
• How many inequivalent shapes exist?

Investigation 3

Analyze edge partitions:

Possible types:

• all even
• all odd
• distinct even
• distinct odd

These investigations model mathematical research exploration.

Prerequisites

Students should have the following background:

Mathematical Skills

Geometry

• polygons
• angles
• convex shapes

Algebra

• integer manipulation
• sequences and sums

Discrete mathematics foundations

• basic graph theory
• combinatorics
• counting arguments

Problem-solving skills

• pattern recognition
• conjecture formation
• mathematical reasoning

Conceptual prerequisites

Students should understand:

• plane geometry
• coordinate plane concepts
• combinatorial reasoning
• mathematical notation

Application Areas

Although presented as a research exploration, the mathematics connects to several fields.

Mathematics

• combinatorics
• graph theory
• tiling theory
• enumerative geometry

Computer Science

• algorithmic tiling problems
• shape recognition
• combinatorial optimization
• grid-based modeling

Physics

• lattice models
• statistical mechanics

Chemistry / Biology

• molecular tiling structures
• protein folding models

Game and Puzzle Design

• Tetris-like puzzle systems
• tiling puzzles

Polyomino theory also underlies recreational mathematics and computational geometry.

Correlation to Mathematics Standards

Common Core State Standards (CCSS)

High School Geometry

HSG-CO

• Understand geometric definitions and constructions.

HSG-GMD

• Apply geometric concepts in modeling situations.

High School Mathematical Practices

The article strongly aligns with:

NCTM Standards

The article strongly aligns with:

Problem Solving

Students explore open-ended research questions.

Reasoning and Proof

Students formulate conjectures about polyomino properties.

Connections

Links between:

• geometry
• combinatorics
• graph theory

Representation

Uses:

• diagrams
• graphs
• partitions

Educational Value

The document is designed for mathematical exploration and student research.

Pedagogical strengths:

• inquiry-based learning
• interdisciplinary mathematics
• visual reasoning
• open-ended research tasks

It encourages students to:

• formulate conjectures
• classify structures
• explore mathematical patterns.

Summary Table