Tiling a Badge with Two Pentahexes

Introduction

A polyhex is a plane figure formed by joining equal regular hexagons edge to edge. A pentahex is a polyhex with 5 cells. There are 22 pentahexes, not distinguishing reflections and rotations.

Let a polyhex badge be a polyhex whose cell centers form a convex hexagon whose alternate sides have equal length. Here I study the problem of arranging copies of two given pentahexes to form a badge.

A triangular polyhex is an extreme form of a badge. Many of the solutions below are triangular. See the bottom of the page for non-triangular variants.

Carl Schwenke and Johann Schwenke contributed some improvements.

Nomenclature

Table of Results

This table shows the number of tiles in the smallest known badges. If you find a smaller badge or solve an unsolved pair, please write.

	A	C	D	E	F	H	I	J	K	L	N	P	Q	R	S	T	U	V	W	X	Y	Z
A	•	18	9	2	12	33	9	15	9	12	9	9	9	15	126	123	42	12	141	9	9	9
C	18	•	5	42	15	12	51	135	18	12	15	12	15	33	—	—	—	18	12	—	11	18
D	9	5	•	5	9	12	5	12	9	5	5	5	9	5	9	12	12	9	12	9	2	12
E	2	42	5	•	12	9	5	9	18	12	9	5	9	9	60	144	12	21	9	9	9	9
F	12	15	9	12	•	12	15	27	33	12	12	5	12	18	—	—	—	54	27	15	9	18
H	33	12	12	9	12	•	27	33	—	12	9	12	—	33	—	—	15	51	—	—	9	—
I	9	51	5	5	15	27	•	15	24	9	5	9	18	21	18	2898	2124	12	51	102	5	9
J	15	135	12	9	27	33	15	•	5	12	9	12	12	12	—	66	—	12	—	18	3	15
K	9	18	9	18	33	—	24	5	•	12	12	9	—	12	—	—	30	12	—	—	9	—
L	12	12	5	12	12	12	9	12	12	•	5	5	12	12	12	12	12	12	12	12	9	12
N	9	15	5	9	12	9	5	9	12	5	•	9	12	12	15	15	18	12	12	15	9	12
P	9	12	5	5	5	12	9	12	9	5	9	•	12	11	12	12	12	12	12	9	9	12
Q	9	15	9	9	12	—	18	12	—	12	12	12	•	12	—	—	15	27	—	—	9	—
R	15	33	5	9	18	33	21	12	12	12	12	11	12	•	—	—	15	12	33	141	9	12
S	126	—	9	60	—	—	18	—	—	12	15	12	—	—	•	—	—	12	—	—	9	—
T	123	—	12	144	—	—	2898	66	—	12	15	12	—	—	—	•	—	29	—	—	15	—
U	42	—	12	12	—	15	2124	—	30	12	18	12	15	15	—	—	•	15	—	—	9	30
V	12	18	9	21	54	51	12	12	12	12	12	12	27	12	12	29	15	•	33	51	9	12
W	141	12	12	9	27	—	51	—	—	12	12	12	—	33	—	—	—	33	•	—	12	—
X	9	—	9	9	15	—	102	18	—	12	15	9	—	141	—	—	—	51	—	•	9	—
Y	9	11	2	9	9	9	5	3	9	9	9	9	9	9	9	15	9	9	12	9	•	9
Z	9	18	12	9	18	—	9	15	—	12	12	12	—	12	—	—	30	12	—	—	9	•