Author:  David Goodger <goodger@python.org> 

Date:  20150224 
Revision:  600 
Web site:  http://puzzler.sourceforge.net/ 
Copyright:  © 19982015 by David J. Goodger 
License:  GPL 2 
Polyhexes are polyforms constructed from unit hexagons joined edgetoedge on a regular hexagonal grid (a honeycomb grid).
Here is a puzzle containing all the polyhexes of order 1 through 5:
See Polyhexes: Puzzles & Solutions and Pentahexes: Puzzles & Solutions for many more puzzles.
The number and names of the various orders of polyhexes are as follows:
Order  Polyform
Name

Free
Polyhexes

OneSided
Polyhexes


1  monohex  1  1 
2  dihex  1  1 
3  trihexes  3  3 
4  tetrahexes  7  10 
5  pentahexes  22  33 
6*  hexahexes  82  147 
"*" above means that forms with enclosed holes exist.
The numbers of polyhexes can also be found in the following sequences from The OnLine Encyclopedia of Integer Sequences: A000228 (free) and A006535 (onesided).
Examples of the polyhexes from order 1 (monohex) to order 6 (hexahexes) are given in the tables below.
The polyhexes are named with a letternumber scheme, like "H1", "A3", and "I06". The initial letter is the letter of the alphabet that the polyhex most closely resembles. In some cases, that resemblance is weak, and the letters are arbitrary. The final digit of the number represents the polyform order (how many unit hexagons are in the polyhex). There are more hexahexes than letters in the alphabet, so their names have an extra middle digit (numbered from 0) to differentiate the variations.
In the tables below, "Aspects" refers to the number of unique orientations that a polyform may take (different rotations, flipped or not). This varies with the symmetry of the polyform.
The "OneSided" column identifies polyforms that are asymmetrical in reflection. Treating the flipped and unflipped versions of asymmetrical polyhexes as distinct polyforms (and disallowing further reflection or "flipping"), results in "onesided" polyhexes and puzzles.
There are 7 free tetrahexes (order4 polyhexes) and 10 onesided tetrahexes:
Name  Image  Aspects  OneSided 

I4  3  
J4  12  yes  
O4  3  
P4  12  yes  
S4  6  yes  
U4  6  
Y4  2 
There are 22 free pentahexes (order5 polyhexes) and 33 onesided pentahexes:
Name  Image  Aspects  OneSided 

A5  6  
B5  12  yes  
C5  6  
D5  6  
E5  6  
F5  12  yes  
G5  12  yes  
H5  12  yes  
I5  3  
J5  12  yes  
L5  6  
N5  12  yes  
P5  12  yes  
Q5  12  yes  
R5  12  yes  
S5  6  yes  
T5  12  yes  
U5  6  
V5  6  
W5  6  
X5  3  
Y5  6 
There are 82 free hexahexes (order6 polyhexes) and 147 onesided hexahexes:
Name  Image  Aspects  OneSided 

A06  2  
A16  12  yes  
A26  12  yes  
C06  6  
C16  6  
C26  12  yes  
C36  12  yes  
C46  12  yes  
C56  12  yes  
C66  12  yes  
C76  12  yes  
E06  6  
F06  12  yes  
F16  12  yes  
H06  12  yes  
H16  12  yes  
H26  12  yes  
I06  3  
J06  12  yes  
J16  12  yes  
J26  12  yes  
J36  12  yes  
J46  12  yes  
K06  12  yes  
L06  12  yes  
L16  12  yes  
L26  12  yes  
L36  12  yes  
M06  12  yes  
M16  12  yes  
M26  12  yes  
M36  6  yes  
M46  6  yes  
N06  6  yes  
N16  12  yes  
O06  1  
P06  12  yes  
P16  12  yes  
P26  12  yes  
P36  12  yes  
P46  12  yes  
P56  12  yes  
P66  12  yes  
P76  12  yes  
Q06  12  yes  
Q16  12  yes  
Q26  12  yes  
Q36  12  yes  
R06  12  yes  
R16  12  yes  
S06  6  yes  
S16  6  yes  
S26  6  yes  
S36  12  yes  
T06  6  
T16  6  
T26  12  yes  
T36  12  yes  
T46  12  yes  
T56  6  
T66  12  yes  
T76  12  yes  
U06  6  
U16  6  
U26  12  yes  
V06  6  
V16  12  yes  
W06  12  yes  
W16  12  yes  
W26  12  yes  
W36  12  yes  
X06  3  
X16  3  
X26  12  yes  
Y06  6  
Y16  6  
Y26  6  
Y36  12  yes  
Y46  12  yes  
Y56  12  yes  
Y66  4  yes  
Z06  6  yes 
Polyhex puzzles use a skewed 2D coordinate system, where the X and Y axes are 60° apart instead of the usual 90°. The typical representation (as seen in the Polyform Puzzler solution data files) positions the Y axis vertically with the X axis 30° counterclockwise from horizontal:
__ 3 __/ \ __/ \ / 2 __/ __/ \ __/ / __/ 1 __/ \__ \ / \ __/ __ \__/ \ / y=0 / \ / \__/ \__/ \ 3 \ / \__ __/ __/ / \__/ \ / __/ 2 \ / / \__/ 6 / \ \__/ 5 1 \ / __/ 4 / \__/ 3 y=0 \__/ 2 1 x=0
Each unit hexagon has 6 immediate neighbors. The neighbors of the hexagon at coordinates (x, y) are:
{(x+1, y), (x, y+1), (x1, y1), (x1, y), (x, y1), (x+1, y1)}