Hex-Triangles with One Interior \(H\)-Point

Abstract

It is known that triangles with vertices in the integral lattice \(\mathbb{Z}^2\) and exactly one interior lattice point can have \(3, 4, 6, 8\), and \(9\) lattice points on their boundaries. No such triangles with \(5\), nor \(7\), nor \(n \geq 10\) boundary lattice points exist. The purpose of this note is to study an analogous property for Hex-triangles, that is, triangles with vertices in the set \(H\) of corners of a tiling of \(\mathbb{R}^2\) by regular hexagons of unit edge. We show that any Hex-triangle with exactly one interior \(H\)-point can have \(3, 4, 5, 6, 7, 8,\) or \(10\), \(H\)-points on its boundary and cannot have \(9\) nor \(n \geq 11\) such points.