Wang tilings problem of $m\times n$ rectangles with monochromatic edges and a specific tile set $\mathcal{T}$

Da Ronch, Mickael

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Wang tilings problem of $m\times n$ rectangles with monochromatic edges and a specific tile set $\mathcal{T}$

Formation et professionnalisation

Da Ronch, Mickael ORCID University of Teacher Education Valais

2026

Submitted to:

The Mathematical Intelligencer Journal . - 2026

Rectangles with monochromatic edges

English In this paper, we study the tiling of $m\times n$ rectangles with monochromatic edges using a particular set $\mathcal{T}$ of Wang tiles of four colors. We distinguish three types of rectangle according to the colors assigned to their opposite edges. For each type, we give necessary and sufficient conditions on $m$ and $n$ for the existence of a $\mathcal{T}$-tiling. The sufficient conditions are established through explicit constructions and an induction argument, whereas the necessary conditions follow from a parity invariant. We then use this classification to determine the minimum number of colors required to tile $m\times n$ rectangles of every type and all dimensions $m, n\geqslant 2$, and prove that this minimum is five colors. Finally, we propose several friendly problems for further investigation, ranging from determining the minimum number of tiles needed for tiling to studying analogous problems in dimensions $d\geqslant 3$.

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