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Toward a Canonical Representation of Blocked Rectangular Grids with an Application to Finite Tiling Problems

Noah Jensen, Stephanie Treneer

TL;DR

This work develops a canonical representation framework for blocked rectangular grids under symmetry by partitioning boards into region-based board partitions and applying a main reduction theorem that yields a weighted count of all boards from a reduced subset. The method guarantees at least one representative from each symmetry class while substantially limiting redundancy, with exact counting formulas derived via Burnside’s lemma across multiple dimension parities. It then applies the framework to families of polyomino tiling problems, using backtracking and linear-system approaches to efficiently determine solvability and compute totals of solvable/unsolvable boards, supplemented by open-source code. The paper also introduces a quantitative measure of canonical quality, explores its behavior through several conjectures, and demonstrates practical impact on complex tiling datasets such as Genius Square and other tiling problems.Overall, the work provides a principled, scalable approach to symmetry-aware enumeration and tiling analysis, enabling exact class counts and near-optimal canonical representations for a broad class of blocked-grid problems.

Abstract

Given the collection of all $m\times n$ rectangular grids which have a fixed number $1\leq r\leq mn$ of blocked cells, we explicitly describe a proper subset of the collection which is guaranteed to contain at least one grid from each equivalence class under symmetry, eliminating the majority of redundant grids. We analyze the extent to which redundant grids remain in the reduced set, and give general cases in which our methods exactly produce a complete set of canonical representatives for the equivalence classes. As an application of our results, we specify collections of polyomino tiling problems and find all solvable grids in each collection.

Toward a Canonical Representation of Blocked Rectangular Grids with an Application to Finite Tiling Problems

TL;DR

This work develops a canonical representation framework for blocked rectangular grids under symmetry by partitioning boards into region-based board partitions and applying a main reduction theorem that yields a weighted count of all boards from a reduced subset. The method guarantees at least one representative from each symmetry class while substantially limiting redundancy, with exact counting formulas derived via Burnside’s lemma across multiple dimension parities. It then applies the framework to families of polyomino tiling problems, using backtracking and linear-system approaches to efficiently determine solvability and compute totals of solvable/unsolvable boards, supplemented by open-source code. The paper also introduces a quantitative measure of canonical quality, explores its behavior through several conjectures, and demonstrates practical impact on complex tiling datasets such as Genius Square and other tiling problems.Overall, the work provides a principled, scalable approach to symmetry-aware enumeration and tiling analysis, enabling exact class counts and near-optimal canonical representations for a broad class of blocked-grid problems.

Abstract

Given the collection of all rectangular grids which have a fixed number of blocked cells, we explicitly describe a proper subset of the collection which is guaranteed to contain at least one grid from each equivalence class under symmetry, eliminating the majority of redundant grids. We analyze the extent to which redundant grids remain in the reduced set, and give general cases in which our methods exactly produce a complete set of canonical representatives for the equivalence classes. As an application of our results, we specify collections of polyomino tiling problems and find all solvable grids in each collection.

Paper Structure

This paper contains 15 sections, 23 theorems, 19 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Group Actions
  4. Polyomino tiling
  5. The Main Theorem of Board Set Reduction
  6. Board Partition Restrictions
  7. Square boards with even side length
  8. Square boards with odd side length
  9. Non-square boards with even side lengths
  10. Non-square boards with odd side lengths
  11. Non-square boards with odd and even side lengths
  12. Boards with one side of length one.
  13. Counting Equivalence Classes
  14. Collections of Tiling Problems
  15. Canonical Representation

Key Result

Theorem 3.1

Let $G$ be the symmetry group acting on the set $\mathcal{B}(m,n;r)$ and suppose that a subset $\overline{\mathcal{B}}(m,n;r)\subseteq \mathcal{B}(m,n;r)$ satisfies the following conditions: For each $1\leq i\leq t$, let $K_i\leq G$ be the subgroup consisting of all symmetries that preserve the set $\pi_i$. Then

Figures (7)

  • Figure 1: A $6\times 6$ board with 7 blockers and a tiling solution.
  • Figure 2: A square board divided into four equal regions.
  • Figure 3: A square board, of odd side length, divided into nine regions.
  • Figure 4: A non-square board, of even side lengths, divided into four regions.
  • Figure 5: A non-square board, of odd side lengths, divided into nine regions.
  • ...and 2 more figures

Theorems & Definitions (54)

  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Remark
  • Theorem 4.1
  • proof
  • Corollary 4.2
  • proof
  • Theorem 4.3
  • ...and 44 more