Polyforms

The number and names of the various orders of polyhexes are as follows:

"*" above means that forms with enclosed holes exist.

The numbers of polyhexes can also be found in the following sequences from The On-Line Encyclopedia of Integer Sequences: A000228 (free) and A006535 (one-sided).

Examples of the polyhexes from order 1 (monohex) to order 6 (hexahexes) are given in the tables below.

The polyhexes are named with a letter-number 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 "One-Sided" 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 "one-sided" polyhexes and puzzles.

Monohex

There is only one monohex (order-1 polyhex):

Name	Image	Aspects	One-Sided
H1		3

Dihex

There is only one dihex (order-2 polyhex):

Name	Image	Aspects	One-Sided
I2		3

Trihexes

There are 3 trihexes (order-3 polyhexes):

Name	Image	Aspects	One-Sided
A3		2
I3		3
V3		6

Tetrahexes

There are 7 free tetrahexes (order-4 polyhexes) and 10 one-sided tetrahexes:

Name	Image	Aspects	One-Sided
I4		3
J4		12	yes
O4		3
P4		12	yes
S4		6	yes
U4		6
Y4		2

Pentahexes

There are 22 free pentahexes (order-5 polyhexes) and 33 one-sided pentahexes:

Name	Image	Aspects	One-Sided
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

Hexahexes

There are 82 free hexahexes (order-6 polyhexes) and 147 one-sided hexahexes:

Name	Image	Aspects	One-Sided
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

Coordinate System

Polyhex puzzles use a skewed 2-D 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° counter-clockwise 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), (x-1, y-1), (x-1, y), (x, y-1), (x+1, y-1)}