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 
Contents
Polysticks are polyforms constructed from unit line segments (edges) joined endtoend on a regular square grid. Polysticks were named by and seem to have been first formally expolored by Brian R. Barwell [1].
Here is a puzzle containing all the polysticks of order 1 through 4:
See Polysticks: Puzzles & Solutions for many more puzzles.
The polysticks can be thought of as the projective duals of polyominoes. Most of the polysticks of order N can be derived from the polyominoes of order N+1 (i.e., joining the centers of the squares of a pentomino results in a tetrastick). The exceptions are the polysticks with loops (e.g. the "O" tetrastick), which are duals of the same order or lowerorder polyominoes. Also, it is not a onetoone mapping. For example, the P pentomino can be mapped to four different tetrasticks, the F, H, J, and P (depending on which loop segment of the full "P" pentastick is left out).
During a visit to India in 2010, it was pointed out to me that polystick puzzles look a lot like some of the "kolam" sand/chalk drawings that can be seen on sidewalks and patios outside of homes. These drawings are thought to bestow prosperity to homes.
[1]  Brian R. Barwell, "Polysticks," Journal of Recreational Mathematics volume 22 issue 3 (1990), p.165175 
The number and names of the various orders of polysticks are as follows:
Order  Polyform
Name

Free
Polysticks

OneSided
Polysticks


1  monostick  1  1 
2  distick  2  2 
3  tristick  5  7 
4  tetrastick  16  25 
5  pentastick  55  99 
6*  hexastick  222  416 
7*  heptastick  950  1854 
"*" above means that forms with enclosed holes exist.
The numbers of polysticks can also be found in the following sequences from The OnLine Encyclopedia of Integer Sequences: A019988 (free) and A151537 (onesided).
Examples of the polysticks from order 1 (monostick) to order 4 (tetrasticks) are given in the tables below.
The polysticks (other than the tetrasticks) are named with a letternumber scheme, like "I1" and "L3". The initial letter is the letter of the alphabet that the polystick most closely resembles. In some cases, that resemblance is weak, and the letters are arbitrary. The number represents the polyform order (how many line segments are in the polystick). The tetrasticks have established letteronly names.
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 polysticks as distinct polyforms (and disallowing further reflection or "flipping"), results in "onesided" polysticks and puzzles.
The "Welded" column identifies polyforms that contain junction points or branches (they cannot be formed by simple bending).
There are 5 free tristicks (order3 polysticks) and 7 onesided tristicks:
Name  Image  Aspects  OneSided  Welded 

I3  2  
L3  8  yes  
T3  8  yes  
U3  4  
Z3  4  yes 
There are 16 free tetrasticks (order4 polysticks) and 25 onesided tetrasticks:
Name  Image  Aspects  OneSided  Welded 

F  8  yes  yes  
H  8  yes  yes  
I  2  
J  8  yes  
L  8  yes  
N  8  yes  
O  1  ?  
P  8  yes  
R  8  yes  yes  
T  4  yes  
U  4  
V  4  
W  4  
X  1  yes  
Y  8  yes  yes  
Z  4  yes 
These pieces correspond to the sevensegment display decimal digits 0 through 9, as seen on digital watches, for a total of 49 line segments. Based on the "Digigrams" puzzle (AKA "Count On Me" or "Count Me In") by Martin H. Watson. The 0/zero digit must have a gap in one side to allow the central segment to be occupied (by a 1 or a 7 only). The 2 and 5 digits are identical (through reflection), and the 6 and 9 are identical (through rotation). The pieces comprise one distick (digit 1), one tristick (digit 7), one tetrastick (digit 4), three pentasticks (digits 2, 3, & 5), three hexasticks (digits 0, 6, & 9), and one heptastick (digit 8).
Name  Image  Aspects  OneSided  Welded 

d0  2  
d1  2  
d2  4  yes  
d3  4  yes  
d4  8  yes  yes  
d5  4  yes  
d6  8  yes  yes  
d7  8  yes  
d8  2  yes  
d9  8  yes  yes 
Polystick puzzles use a pseudo3D coordinate system. The (X,Y) 2dimensional coordinate identifies the lowerleft corner of the (X,Y,0) square in a polyomino grid. The Z dimension is used for the direction of the line segment:
As there is the possibility for polysticks to cross each other at any (X,Y) intersection, the solution algorithm needs to prevent such crossings. The intersections are simple, either used (by a straightthrough piece) or available.
In the puzzle matrix, we represent contstraints on intersections via an additional column per intersection [*], in the form i(X,Y,Z) (or "X,Yi").
[*]  Intersections at the edge of a puzzle shape may be ignored. This represents a possible future optimization. 
These intersection constraints are secondary columns, meaning that at most one polyform may use or fill a column. Unlike primary columns, secondary columns may remain unused/unfilled.
If an intersection constraint is already used (or otherwise unavailable), no other polyform with the same contstraint may be placed in the puzzle solution. This prevents two polysticks from crossing each other.