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 Colin F. Brown, a puzzle enthusiast from Dudley, England, who shared the results of his unpublished work on hexagonal-grid polysticks from the late 1970s and early 1980s.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!

Polyforms

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

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.

Monotwig

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

Name	Image	Aspects	One-Sided
I1		3

Ditwig

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

Name	Image	Aspects	One-Sided
L2		6

Tritwigs

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

Name	Image	Aspects	One-Sided
C3		6
S3		6	yes
Y3		2

Tetratwigs

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

Name	Image	Aspects	One-Sided
C4		6
P4		12	yes
W4		6
Y4		12	yes

Pentatwigs

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		6			Bowl
H5		12	yes		Chair
I5		6	yes		Stick
L5		12	yes		Signpost
P5		12	yes		Hook
R5		12	yes		Pipe
S5		6	yes		Snake
T5		6			Bird
U5		6			Bridge
W5		12	yes		Club
X5		6			Trestle
Y5		6			Fork

Hexatwigs

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

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

Quasi-Polyforms

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:

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.

Quasi-Ditwigs

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

Name	Image	Aspects	One-Sided
C2		6
L2		6
S2		6	yes

Quasi-Tritwigs

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		6
C13		12	yes
C23		2
H13		12	yes
I13		6	yes
P13		12	yes
P23		12	yes
P33		12	yes
S03		6	yes
S13		6	yes
T13		6
U13		6
W13		12	yes
W23		12	yes
Y03		2
Y13		12	yes
Y23		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:

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.