Tiling a Badge with a Pentahex and an Isolated Tetrahex

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.

A tetrahex is a polyhex with 4 cells. There are 7 pentahexes, not distinguishing reflections and rotations.

Call a polyhex convex if the region enclosed by joining centers of adjacent cells is convex. A badge is a convex polyhex with 3-rotary symmetry and horizontal mirror symmetry. Here I tile minimal known badges with copies of a pentahex and a tetrahex so that copies of the tetrahex do not touch.

Table of Results

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

	Pentahex
	A	C	D	E	F	H	I	J	K	L	N	P	Q	R	S	T	U	V	W	X	Y	Z
I	42	?	27	27	141	333	60	42	42	42	27	42	42	213	?	?	?	73	393	?	27	42
J	27	27	27	153	27	27	106	27	27	48	27	27	27	27	42	?	27	57	57	27	42	42
O	27	123	27	66	?	?	126	141	27	33	42	27	27	27	27	?	?	27	?	?	33	?
Q	27	144	27	28	27	153	57	27	27	42	27	28	27	60	621	?	27	57	66	27	42	42
S	306	?	63	66	87	?	330	42	?	87	87	42	?	636	?	?	27	522	?	?	73	?
U	57	?	57	42	?	27	147	57	27	73	42	33	42	126	?	?	27	477	?	27	63	201
Y	645	?	73	696	351	?	?	?	?	87	57	57	?	?	?	?	?	582	?	?	123	?

Navigation

[27 Cells] [28 Cells] [33 Cells] [42 Cells] [48 Cells]
[57 Cells] [60 Cells] [63 Cells] [66 Cells] [73 Cells] [87 Cells]
[106 Cells] [123 Cells] [126 Cells] [141 Cells] [144 Cells] [147 Cells]
[153 Cells] [201 Cells] [213 Cells] [306 Cells] [330 Cells] [333 Cells]
[351 Cells] [393 Cells] [477 Cells] [522 Cells] [582 Cells] [621 Cells]
[636 Cells] [645 Cells] [696 Cells]

Col. George Sicherman [ HOME | MAIL ]

	Pentahex
	A	C	D	E	F	H	I	J	K	L	N	P	Q	R	S	T	U	V	W	X	Y	Z
I	42	?	27	27	141	333	60	42	42	42	27	42	42	213	?	?	?	73	393	?	27	42
J	27	27	27	153	27	27	106	27	27	48	27	27	27	27	42	?	27	57	57	27	42	42
O	27	123	27	66	?	?	126	141	27	33	42	27	27	27	27	?	?	27	?	?	33	?
Q	27	144	27	28	27	153	57	27	27	42	27	28	27	60	621	?	27	57	66	27	42	42
S	306	?	63	66	87	?	330	42	?	87	87	42	?	636	?	?	27	522	?	?	73	?
U	57	?	57	42	?	27	147	57	27	73	42	33	42	126	?	?	27	477	?	27	63	201
Y	645	?	73	696	351	?	?	?	?	87	57	57	?	?	?	?	?	582	?	?	123	?

	Pentahex
	A	C	D	E	F	H	I	J	K	L	N	P	Q	R	S	T	U	V	W	X	Y	Z
I	42	?	27	27	141	333	60	42	42	42	27	42	42	213	?	?	?	73	393	?	27	42
J	27	27	27	153	27	27	106	27	27	48	27	27	27	27	42	?	27	57	57	27	42	42
O	27	123	27	66	?	?	126	141	27	33	42	27	27	27	27	?	?	27	?	?	33	?
Q	27	144	27	28	27	153	57	27	27	42	27	28	27	60	621	?	27	57	66	27	42	42
S	306	?	63	66	87	?	330	42	?	87	87	42	?	636	?	?	27	522	?	?	73	?
U	57	?	57	42	?	27	147	57	27	73	42	33	42	126	?	?	27	477	?	27	63	201
Y	645	?	73	696	351	?	?	?	?	87	57	57	?	?	?	?	?	582	?	?	123	?

	Pentahex
	A	C	D	E	F	H	I	J	K	L	N	P	Q	R	S	T	U	V	W	X	Y	Z
I	42	?	27	27	141	333	60	42	42	42	27	42	42	213	?	?	?	73	393	?	27	42
J	27	27	27	153	27	27	106	27	27	48	27	27	27	27	42	?	27	57	57	27	42	42
O	27	123	27	66	?	?	126	141	27	33	42	27	27	27	27	?	?	27	?	?	33	?
Q	27	144	27	28	27	153	57	27	27	42	27	28	27	60	621	?	27	57	66	27	42	42
S	306	?	63	66	87	?	330	42	?	87	87	42	?	636	?	?	27	522	?	?	73	?
U	57	?	57	42	?	27	147	57	27	73	42	33	42	126	?	?	27	477	?	27	63	201
Y	645	?	73	696	351	?	?	?	?	87	57	57	?	?	?	?	?	582	?	?	123	?