polyomino-solver

Find all multihedral tilings of an arbitrary pattern using free Polyominoes.

Installation

git clone https://github.com/aaronsnoswell/polyomino-solver.git
cd polyomino-solver
pip install -e .

Requirements

Tested with Python 3.6 on Windows. Should support other OS' just fine though. Requires Matlab to be installed and accessible.

Usage

To tile the 2x3 rectangle with combinations of Polyominoes 1 and 9:

from polyomino_solver import multihedral_tile
sols = multihedral_tile([[1, 1, 1], [1, 1, 1]], polyomino_set=[1, 9])
for label, coloring in sols: print(label)

Gives the output;

[[1. 2. 3.]
 [4. 5. 6.]]
[[3. 3. 1.]
 [2. 3. 3.]]
[[1. 3. 3.]
 [3. 3. 2.]]

Keep in mind that the underlying solver optimization requires inverting a binary linear matrix system. As such, the time complexity is approximately cubic in the area of the target shape. To put this in perspective, using an Intel 8 Core i7-8650u laptop processor at 1.9GHz to solve the tiling of a shape with area 10 takes 5 minutes. Using the same laptop to solve a pattern with area 14 takes about a week.

Acknowledgements

This package is simply a thin convenience wrapper around the amazing Matlab Polyomino library provided free under GPL by John Burkardt at his website. Much thanks to him, and please see their paper describing the method at;

• Garvie, Marcus R., and John Burkardt. "A New Mathematical Model for Tiling Finite Regions of the Plane with Polyominoes".

I have made minor modifications to the library, in particular I have enabled both H1 and H2 solution verification steps to screen out all invalid solutions. Please see this patch for details.

Also thanks to the following;

• P. Todd Decker from BoardGameGeek for the Polyomino reference PDF.