A pentagonal number theorem for tribone tilings

TL;DR

This paper characterizes when certain hexagonal regions can be tiled by tribones, revealing a connection to paired pentagonal numbers, thus advancing understanding of tribone tilings in hexagonal grids.

Contribution

It establishes a precise criterion for tribone tilings of hexagonal regions based on paired pentagonal numbers, linking combinatorial tiling problems to number theory.

Findings

Tiling exists if and only if parameters are paired pentagonal numbers

Provides a complete characterization of tribone tilings for a family of regions

Connects tiling problems to classical figurate numbers

Abstract

Conway and Lagarias showed that certain roughly triangular regions in the hexagonal grid cannot be tiled by shapes Thurston later dubbed tribones. Here we study a two-parameter family of roughly hexagonal regions in the hexagonal grid and show that a tiling by tribones exists if and only if the two parameters associated with the region are the paired pentagonal numbers $k(3k \pm 1)/2$.