Perfect Rectangular Tilings with Two Colors

Oswin Aichholzer; Robert Ganian; Phillip Keldenich; Maarten Löffler; Gert Meijer; Ids de Vlas; Alexandra Weinberger; Carola Wenk

arXiv · 2609.26022

Perfect Rectangular Tilings with Two Colors

Oswin Aichholzer·Robert Ganian·Phillip Keldenich·Maarten Löffler·Gert Meijer·Ids de Vlas·Alexandra Weinberger·Carola Wenk

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Abstract

We study a finite tiling problem, where tiles are unit squares whose four edges are colored with one of two colors. We ask whether a given rectangle admits a perfect rectangular tiling: every cell of the rectangle is occupied by one tile, neighboring edge colors match, and exactly $n_i$ tiles of type $i$ are used, where rotations of the tiles are allowed. Our problem is related to classical Wang tilings, more general finite tile-placement problems, and edge placement puzzles. But in our problem, the multiplicities of the tile types are part of the input and the tile alphabet is fixed and extremely small; thus the complexity of the problem arises from the interaction between the rectangle dimensions and the prescribed tile multiplicities. We provide a comprehensive study of the perfect rectangular tiling problem. For this we consider all classes of subsets of the six possible tile types for two-colored edges, and we characterize for each class whether multiplicities either always allow a perfect rectangular tiling or whether their existence can be decided efficiently.

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BibTeX RIS

Oswin Aichholzer, Robert Ganian, Phillip Keldenich, Maarten Löffler, Gert Meijer, Ids de Vlas, Alexandra Weinberger, Carola Wenk. 2026-09-22. Perfect Rectangular Tilings with Two Colors. https://arxiv.org/abs/2609.26022

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