An Introduction to Polytwigs (Hexagonal-Grid Polysticks)

Author: David Goodger <>
Date: 2015-02-24
Revision: 600
Web site:
Copyright: © 1998-2015 by David J. Goodger
License:GPL 2


Polytwigs (a.k.a. polycules) are polyforms constructed from unit line segments (edges) joined end-to-end on a regular hexagonal grid. Unit line segments on a square grid are called "polysticks", so polytwigs could also be called "polysticks on a hexagonal grid" or "hexagonal-grid polysticks" or "hexagonal polysticks". The polytwigs are a subset of the polytrigs (triangular-grid polysticks); polytwigs would fit on a polytrig grid, but there are a lot fewer of them.

Here is a puzzle containing all the polytwigs of order 1 through 5:


And here is a puzzle containing all the hexatwigs (order 6):


See Polytwigs: Puzzles & Solutions for many more puzzles.

Most of the polytwigs of order N can be thought of as the projective duals of the polyiamonds of order N+1 (i.e., joining the centers of the triangles of a hexiamond results in a pentatwig). The exceptions are the polytwigs with loops (e.g. the "O06" hexatwig), which are duals of the same order or lower-order polyiamonds.

I invented the name "polytwigs" because "hexagonal-grid polysticks" is too unwieldy, especially when combined with order-prefixes (mono-, di-, tri-, tetra-, etc.; "hexagonal-grid tristicks"!?). The name comes from "twig" being a small stick, often bent and branching. "Polytwig" is also similar to "polytrig", the name I invented for triangular-grid polytsicks. I hope these names catch on.

Prior Research

When I began working on the hexagonal-grid polysticks, I was unable to find any references to prior research. While (to my knowledge) nothing seems to have been published, I couldn't believe that this was a completely novel idea — an intuition that proved to be true.

In February 2012 I received email from Mr. Colin F. Brown, a puzzle enthusiast from Dudley, England, who shared the results of his unpublished work on hexagonal-grid polysticks from the late 1970s and early 1980s. Mr. Brown wrote,

I did name them ‘Polycules’, from the fact that sea sponges have spicules that resemble a tri-cule (your 3-stick hexagon gridded ‘polytwig’) each segment being set at 120° apart. I also gave each pentacule a name...

See the Wikipedia article on "sponge spicules" for an image demonstrating the resemblance. In fact, Mr. Brown used the term "polycules" for sqare-grid polysticks and triangular-grid polytrigs as well (prefixed with "rectangular-", "triangular-", and "hexagonal-"), but:

... for constructional problems, I chose the hexagonal stick forms, partly because the rectangular and triangular types admitted tilings and constructions with ‘crossings’ – impossible with polytwigs.

I'll use "polycules" as a synonym for "polytwigs" here. Mr. Brown's pentacule names are listed in the pentatwigs section below. See Polytwigs: Puzzles & Solutions for examples of several of Mr. Brown's puzzle constructions.

Mr. Brown also explored the quasi-tritwigs with gaps between segments limited to length-1; see Quasi-Polyforms below.

At the Gathering for Gardner 10 (G4G10, 2012), Les Shader gave me a copy of a page from a Russian journal "Charade" ("ШАРАДА"), N6(6), December 1993. It featured a solution to the "Hexonet" puzzle ("ГЕКСОСЕТИ"), exactly the pentatwigs triangle (side length 5). Thanks to Tanya Khovanova of MIT for translating.

If you know of any other work on this type of puzzle, please let me know!


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

1 monotwig 1 1
2 ditwig 1 1
3 tritwigs 3 4
4 tetratwigs 4 6
5 pentatwigs 12 19
6 hexatwigs 27 49
7 heptatwigs 78 143

The numbers of polytwigs can also be found in the following sequences from The On-Line Encyclopedia of Integer Sequences: A197459 (free) and A197460 (one-sided).

Examples of the polytwigs from order 1 (monotwig) to order 6 (hexatwigs) are given in the tables below.

The polytwigs are named with a letter-number scheme, like "I1", "C3", and "R16". The letter part is the letter of the alphabet that the polytwig 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 line segments are in the polytwig). There are more hexatwigs 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 polytwigs as distinct polyforms (and disallowing further reflection or "flipping"), results in "one-sided" polytwigs and puzzles.


There is only one monotwig (order-1 polytwig):

Name Image Aspects One-Sided
I1 images/pieces/polytwigs/I1.png 3  


There is only one ditwig (order-2 polytwig):

Name Image Aspects One-Sided
L2 images/pieces/polytwigs/L2.png 6  


There are 3 free tritwigs (order-3 polytwigs) and 4 one-sided tritwigs:

Name Image Aspects One-Sided
C3 images/pieces/polytwigs/C3.png 6  
S3 images/pieces/polytwigs/S3.png 6 yes
Y3 images/pieces/polytwigs/Y3.png 2  


There are 4 free tetratwigs (order-4 polytwigs) and 6 one-sided tetratwigs:

Name Image Aspects One-Sided
C4 images/pieces/polytwigs/C4.png 6  
P4 images/pieces/polytwigs/P4.png 12 yes
W4 images/pieces/polytwigs/W4.png 6  
Y4 images/pieces/polytwigs/Y4.png 12 yes


There are 12 free pentatwigs (order-5 polytwigs, a.k.a. pentacules) and 19 one-sided pentatwigs. Colin F. Brown's original pieces and their evocative names are listed in the "Pentacule" and last "Name" columns.

Name Image Aspects One-Sided Pentacule Name
C5 images/pieces/polytwigs/C5.png 6   images/pieces/polytwigs/pentacules/bowl.png Bowl
H5 images/pieces/polytwigs/H5.png 12 yes images/pieces/polytwigs/pentacules/chair.png Chair
I5 images/pieces/polytwigs/I5.png 6 yes images/pieces/polytwigs/pentacules/stick.png Stick
L5 images/pieces/polytwigs/L5.png 12 yes images/pieces/polytwigs/pentacules/signpost.png Signpost
P5 images/pieces/polytwigs/P5.png 12 yes images/pieces/polytwigs/pentacules/hook.png Hook
R5 images/pieces/polytwigs/R5.png 12 yes images/pieces/polytwigs/pentacules/pipe.png Pipe
S5 images/pieces/polytwigs/S5.png 6 yes images/pieces/polytwigs/pentacules/snake.png Snake
T5 images/pieces/polytwigs/T5.png 6   images/pieces/polytwigs/pentacules/bird.png Bird
U5 images/pieces/polytwigs/U5.png 6   images/pieces/polytwigs/pentacules/bridge.png Bridge
W5 images/pieces/polytwigs/W5.png 12 yes images/pieces/polytwigs/pentacules/club.png Club
X5 images/pieces/polytwigs/X5.png 6   images/pieces/polytwigs/pentacules/trestle.png Trestle
Y5 images/pieces/polytwigs/Y5.png 6   images/pieces/polytwigs/pentacules/fork.png Fork


There are 27 free hexatwigs (order-6 polytwigs) and 49 one-sided hexatwigs:

Name Image Aspects One-Sided
C06 images/pieces/polytwigs/C06.png 12 yes
F06 images/pieces/polytwigs/F06.png 12 yes
H06 images/pieces/polytwigs/H06.png 12 yes
H16 images/pieces/polytwigs/H16.png 12 yes
I06 images/pieces/polytwigs/I06.png 6  
J06 images/pieces/polytwigs/J06.png 12 yes
L06 images/pieces/polytwigs/L06.png 12 yes
L16 images/pieces/polytwigs/L16.png 12 yes
L26 images/pieces/polytwigs/L26.png 12 yes
M06 images/pieces/polytwigs/M06.png 6  
O06 images/pieces/polytwigs/O06.png 1  
Q06 images/pieces/polytwigs/Q06.png 12 yes
Q16 images/pieces/polytwigs/Q16.png 12 yes
Q26 images/pieces/polytwigs/Q26.png 12 yes
R06 images/pieces/polytwigs/R06.png 12 yes
R16 images/pieces/polytwigs/R16.png 12 yes
S06 images/pieces/polytwigs/S06.png 12 yes
S16 images/pieces/polytwigs/S16.png 12 yes
S26 images/pieces/polytwigs/S26.png 12 yes
U06 images/pieces/polytwigs/U06.png 6  
V06 images/pieces/polytwigs/V06.png 6  
W06 images/pieces/polytwigs/W06.png 12 yes
X06 images/pieces/polytwigs/X06.png 12 yes
X16 images/pieces/polytwigs/X16.png 12 yes
Y06 images/pieces/polytwigs/Y06.png 4 yes
Y16 images/pieces/polytwigs/Y16.png 12 yes
Y26 images/pieces/polytwigs/Y26.png 12 yes


Quasi-polyforms are polyforms where the requirement that all unit shapes be connected has been removed. In other words, quasi-polyforms are polyforms where some or all unit shapes may be separate from the others.

Without limits on the distance between unit shapes there would be an infinite number of quasi-polyforms (for orders 2 and above). We will limit the quasi-polyforms we consider to those with length-1 gaps between segments.

The number and names of the various orders of quasi-polytwigs are as follows:

1 monotwig [*] 1 1
2 quasi-ditwig 3 4
3 quasi-tritwigs 17 28
4 quasi-tetratwigs 114 214
5 quasi-pentatwigs 966 1885
[*]With only 1 line segment, there can be no disconnected quasi-monotwig. The set of the regular (fully-connected) monotwigs is identical to the set of quasi-monotwigs, and consists of one piece: the "I1" monotwig.

Examples of the quasi-polytwigs of order 2 (quasi-ditwigs) and order 3 (quasi-tritwigs) are given in the tables below. See the table legend above for column descriptions.


There are 3 free quasi-ditwigs (order-2 quasi-polytwigs) and 4 one-sided quasi-ditwigs:

Name Image Aspects One-Sided
C2 images/pieces/quasi-polytwigs/C2.png 6  
L2 images/pieces/quasi-polytwigs/L2.png 6  
S2 images/pieces/quasi-polytwigs/S2.png 6 yes


In his prior research, Colin F. Brown showed that in addition to the 3 free connected tritwigs, there are 5 semi-connected (one gap) quasi-tritwigs and 9 disconnected (two gaps) quasi-tritrigs, for a total of 17 free quasi-tritwigs (order-3 quasi-polytwigs). There are 28 one-sided quasi-tritwigs.

Name Image Aspects One-Sided
C03 images/pieces/quasi-polytwigs/C03.png 6  
C13 images/pieces/quasi-polytwigs/C13.png 12 yes
C23 images/pieces/quasi-polytwigs/C23.png 2  
H13 images/pieces/quasi-polytwigs/H13.png 12 yes
I13 images/pieces/quasi-polytwigs/I13.png 6 yes
P13 images/pieces/quasi-polytwigs/P13.png 12 yes
P23 images/pieces/quasi-polytwigs/P23.png 12 yes
P33 images/pieces/quasi-polytwigs/P33.png 12 yes
S03 images/pieces/quasi-polytwigs/S03.png 6 yes
S13 images/pieces/quasi-polytwigs/S13.png 6 yes
T13 images/pieces/quasi-polytwigs/T13.png 6  
U13 images/pieces/quasi-polytwigs/U13.png 6  
W13 images/pieces/quasi-polytwigs/W13.png 12 yes
W23 images/pieces/quasi-polytwigs/W23.png 12 yes
Y03 images/pieces/quasi-polytwigs/Y03.png 2  
Y13 images/pieces/quasi-polytwigs/Y13.png 12 yes
Y23 images/pieces/quasi-polytwigs/Y23.png 6  

Coordinate System

Polytwig puzzles use a pseudo-3D skewed 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. Solution graphics are often rotated so the X axis is horizontal, and the Y axis 30° clockwise from vertical.

The (X,Y) 2-dimensional coordinate identifies the lower-left corner of the (X,Y) hexagon in a polyhex grid (a honeycomb with hexagons with horizontal tops & bottoms). The three line segments emanating from this point share the (X,Y) coordinate, and the Z dimension is used for the direction of the line segment:

z = 0: 0° horizontal, to the right;
z = 1: 120° counter-clockwise, up and to the left;
z = 2: 240° counter-clockwise, down and to the left.

Since only three line segments may emanate from each (X,Y) intersection, there is no possibility for polytwigs to cross each other and no need to keep track of intersections. This separates polytwigs from polysticks and polytrigs, where such considerations are crucial.