Polyforms

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

The numbers of polyominoes can also be found in the following sequences from The On-Line Encyclopedia of Integer Sequences: A000105 (free) and A000988 (one-sided).

Examples of the polyominoes from order 1 (monomino) to order 6 (hexominoes) are given in the tables below.

The polyominoes are named with letters (like the pentomino "X") or a letter-number scheme (like the "I3" tromino or the "X06" hexomino). The traditional names for the pentominoes are used by Polyform Puzzler, without a number suffix. The names of the hexominoes below roughly correspond to the Kadon naming system for hexominoes. The rest of the polyomino names correspond loosely to Kadon's names for their Poly-4 Supplement set (exceptions noted in the tables below).

The initial letter of each name is the letter of the alphabet that the polyomino most closely resembles, or an initial. 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 squares are in the polyomino). For the hexominoes, some letters are used for multiple pieces (there are more pieces than letters in the alphabet), so a middle index digit is used to differentiate (e.g. "X06" and "X16").

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 polyominoes as distinct polyforms (and disallowing further reflection or "flipping"), results in "one-sided" polyominoes and puzzles.

Alternate names and name origins are noted in the "Name" column.

Monomino

There is only one monomino (order-1 polyomino):

Name	Image	Aspects	One-Sided
O1 ("M", from "Monomino")		1

Domino

There is only one domino (order-2 polyomino):

Name	Image	Aspects	One-Sided
I2 ("D", from "Domino")		2

Tromino

Sometimes called "triominoes", there are 2 trominoes (order-3 polyominoes):

Name	Image	Aspects	One-Sided
I3		2
V3		4

Tetromino

There are 5 free tetrominoes (order-4 polyominoes) and 7 one-sided tetrominoes:

Name	Image	Aspects	One-Sided
I4		2
L4		8	yes
O4		1
T4		4
Z4 ("S4")		4	yes

Pentominoes

There are 12 free pentominoes (order-5 polyominoes) and 18 one-sided pentominoes:

Name	Image	Aspects	One-Sided
F		8	yes
I		2
L		8	yes
N		8	yes
P		8	yes
T		4
U		4
V		4
W		4
X		1
Y		8	yes
Z		4	yes

Hexominoes

There are 35 free hexominoes (order-6 polyominoes) and 60 one-sided hexominoes:

Name	Image	Aspects	One-Sided
A06 (Kadon's "A")		4
C06		4
D06		4
E06		4
F06 ("hi F")		8	yes
F16 ("low F")		8	yes
F26 ("hi 4")		8	yes
F36 ("low 4")		8	yes
G06		8	yes
H06		8	yes
I06		2
J06		8	yes
K06		4
L06		8	yes
M06		8	yes
N06 [*] ("short N")		4	yes
N16 ("long N")		8	yes
O06		2
P06		8	yes
Q06		8	yes
R06		8	yes
S06 ("long S")		4	yes
T06 ("long T")		4
T16 ("short T")		8	yes
U06		8	yes
V06		8	yes
W06 ("Wa")		8	yes
W16 ("Wb")		4	yes
W26 ("Wc")		8	yes
X06		4
X16 ("italic X")		4	yes
Y06 ("hi Y")		8	yes
Y16 ("low Y")		4
Z06 ("long Z")		4	yes
Z16 ("short Z")		8	yes

[*]	Kadon's Sextillions set includes a second copy of N06 called "short S", also adopted by Polyform Puzzler's Hexominoes Plus as "S16".

Coordinate System

Polyominoes use a standard 2-dimensional (x,y) Cartesian coordinate system.

Each unit square has 4 immediate neighbors. The neighbors of the square at coordinates (x, y) are:

{(x+1, y), (x-1, y), (x, y+1), (x, y-1)}

See the FAQ for more details.