3coSoKu

3coSoKu

MiniZinc and Answer Set Programming solvers for IcoSoKu and its strongly NP-complete generalization 3coSoKu (see the papers about it here and here), completed by a 3D visualisation/solver tool for IcoSoKu built with Three.js and clingo-wasm, that you can try online now. This repository also contains some scripts benchmarking the solvers and verifying that every instance of IcoSoKu can indeed be solved (see Experimental results).

The problem

IcoSoKu is a mechanical puzzle created in 2009 by Andrea Mainini and it works as follows:

1. the game is setup by placing the 12 yellow pegs (the numbers from 1 to 12) arbitrarily on the vertices;
2. the game is played by finding an arrangement of the 20 triangular tiles shipping with the game on the faces of the icosahedron, such that the number of black dots surrounding each vertex is equal to its number of the yellow peg.

The 20 tiles are shown below.

3coSoKu is the generalization of IcoSoKu where each instance is defined by:

• the playing field, a polyhedron with triangular faces (the icosahedron, for IcoSoKu);
• the capacity of each vertex, a non-negative integer (the disposition of the yellow pegs);
• the tiles, made of three non-negative integer weights (the tiles with their dots).

To stay true to IcoSoKu, we impose the number of tiles to be equal to the number of faces. 3coSoKu is strongly NP-complete, you can read all the details in the papers that Agostino Dovier, my professor at the University of Udine, and I wrote.

Getting started

To see IcoSoKu instances and to solve them using the ASP solver you can try and play around with the web application: no installation is required, thanks to Three.js and to clingo compiled to WebAssembly.

Otherwise, if you want to test the solvers, install MiniZinc and/or clingo, then download this repository.

$ git clone https://github.com/nrizzo/3coSoKu.git
$ cd 3coSoKu

The solvers, found in solvers/MiniZinc and solvers/ASP, are already configured to solve instances of IcoSoKu. On Linux/Unix systems you can use the script icosolve.sh, found in both folders: the script accepts as input the twelve capacities specifying the yellow pegs, following the convention of the figure on the right (alphabetical order).

$ cd solvers/MiniZinc
$ ./icosolve.sh 1 2 3 4 5 6 7 8 9 10 11 12

Alternatively, you can modify the array cap in file input-ico.dzn and facts cap(V,C) in input-ico.lp, respectively, also following the convention of the figure on the right, and manually execute the solvers:

$ minizinc --solver chuffed 3coSoKu.mzn input-ico.dzn

for MiniZinc (you can also use the IDE), and

$ clingo 3coSoKu.lp variants/ico.lp input-ico.lp

for ASP.

Experimental results

Folder tests contains the Bash scripts to perform some interesting tests, also described in the papers. Details and results are described in here. In particular: the solvers make it possible to solve every IcoSoKu instance, thanks to the symmetries of the game; there are billions of different solutions for each IcoSoKu instance, so a good real-life strategy for the game is to do frequent restarts and try to "get lucky".

The 3D visualisation tool

We developed a 3D application visualising IcoSoKu instances and its solutions using three.js, Tweakpane, and stats.js. Moreover, the application uses clingo-wasm to actually solve (in-browser!) the IcoSoKu instance specified by the user, thanks to clingo compiled to WebAssembly and our ASP encoding.

You can try here the web application using any modern browser, or you can launch it locally in two ways:

• you can host the folder webapp/http on your local network with any HTTP server;

    $ cd webapp/http
    $ python3 -m http.server &
    $ firefox localhost:8000
  

• you can run the offline version of the application found in webapp/offline without doing any hosting by opening the main html file.

    $ firefox webapp/offline/index.html
  

The offline version in webapp/offline does not trigger the browser's CORS rules and it was obtained with some tricks, among which compiling clingo to JavaScript instead of WebAssembly using empscripten's options -s WASM=0 --memory-init-file 0 (resulting in poorer clingo performance).

License

All of my code (the solver, the scripts and the webapp) is licensed under the terms of the GNU GPL v3 license, whereas the software and assets I am using (located in webapp/{http,offline}/vendor and in webapp/{http,offline}/assets), that is Clingo WebAssembly, three.js, Tweakpane, and stats.js, keep their original license. My images (in the images folder) are instead licensed under CC BY.

Acknowledgments

Many thanks to my professor Agostino Dovier for the proposal of this problem and for writing the paper with me, to Marzio De Biasi for his kind words, to the organizers of CILC 2020, to their reviewers and to its attendees.