Trihexagonal Magic Figures: A High-Constraint-Density Triangular Combinatorial Design and the Possibility of a Phase Transition

Abstract

We introduce a new class of magic figures defined on a finite triangular region of the trihexagonal tiling. The vertices of the region are labeled with the integers $1,2,\dots,3n(n+1)/2$, each used exactly once. The labeling is called \textbf{trihexagonal magic figure of order $n$} if all triangular faces have the same vertex-sum and all hexagonal faces have the same vertex-sum, with the latter equal to twice the former. We note three immediate structural features. First, the ratio between the number of independent constraints and the number of variables approaches $1$ as $n\to\infty$, indicating that the system is asymptotically tight. Second, a necessary parity condition arises: since the labeling uses the integers $1,2,\dots,3n(n+1)/2$, the total number of vertices must be odd, which holds if and only if $n \equiv 1$ or $2 \pmod{4}$. Third, the vertex set forms a perfectly regular triangular boundary, so the configuration has an exact polygonal symmetry without boundary irregularities.