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,…,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→∞, indicating that the system is asymptotically tight. Second, a necessary parity condition arises: since the labeling uses the integers 1,2,…,3n(n+1)/2, the total number of vertices must be odd, which holds if and only if n ≡ 1 or 2 pmod4. Third, the vertex set forms a perfectly regular triangular boundary, so the configuration has an exact polygonal symmetry without boundary irregularities.
Related
- Stretched exponential asymptotics for bases of triangular bootstrap percolation
- Triangular array progressions for population count models involving Markov-P\'{o}lya distributions and relatives
- The discontinuous Galerkin method for solving the two-dimensional transport equation with distributed-order memory kernel on triangular meshes
- Laws for triangular matrices
- Enumeration of Upper Triangular Symplectic Renner Monoids
BibTeX
@misc{park2026trihexagonal,
title = {{Trihexagonal Magic Figures: A High-Constraint-Density Triangular Combinatorial Design and the Possibility of a Phase Transition}},
author = {Donghwi Park},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05536},
eprint = {2609.05536},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05536},
}
RIS
TY - UNPB
TI - Trihexagonal Magic Figures: A High-Constraint-Density Triangular Combinatorial Design and the Possibility of a Phase Transition
AU - Donghwi Park
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.05536
UR - https://arxiv.org/abs/2609.05536
AB - 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,…,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→∞, indicating that the system is asymptotically tight. Second, a necessary parity condition arises: since the labeling uses the integers 1,2,…,3n(n+1)/2, the total number of vertices must be odd, which holds if and only if n ≡ 1 or 2 pmod4. Third, the vertex set forms a perfectly regular triangular boundary, so the configuration has an exact polygonal symmetry without boundary irregularities.
ER -