Polystick

In recreational mathematics, a polystick (or polyedge) is a polyform with a line segment (a 'stick') as the basic shape. A polystick is a connected set of segments in a regular grid. A square polystick is a connected subset of a regular square grid. A triangular polystick is a connected subset of a regular triangular grid. Polysticks are classified according to how many line segments they contain.

When reflections are considered distinct we have the one-sided polysticks. When rotations and reflections are not considered to be distinct shapes, we have the free polysticks. Thus, for example, there are 7 one-sided square tristicks because two of the five shapes have left and right versions.

Square Polysticks
Sticks	Name	Free	One-Sided
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

Triangular Polysticks
Sticks	Name	Free
1	monostick	1
2	distick	3
3	tristick	12
4	tetrastick	60
5	pentastick	375
6	hexastick	2613
7	heptastick	19074

The set of n-sticks that contain no closed loops is equivalent, with some duplications, to the set of (n+1)-ominos, as each vertex at the end of every line segment can be replaced with a single square of a polyomino. In general, an n-stick with m loops is equivalent to a (n−m+1)-omino (as each loop means that one line segment does not add a vertex to the figure).

Diagram

The free square polysticks of sizes 1 through 4, including 1 monostick (red), 2 disticks (green), 5 tristicks (blue), and 16 tetrasticks (black).

External links

• Polysticks Puzzles & Solutions, at Polyforms Puzzler
• Covering the Aztec Diamond with One-sided Tetrasticks, Alfred Wassermann, University of Bayreuth, Germany
• Polypolylines, at Math Magic