<abstract xmlns="http://www.w3.org/1999/xhtml"><p> By using two different invariants for the Rubik’s Magic puzzle, one of metric type, the other of topological type, we can dramatically reduce the universe of constructible configurations of the puzzle. Finding the set of actually constructible shapes remains however a challenging task, that we tackle by first reducing the target shapes to specific configurations: the octominoid 3D shapes, with all tiles parallel to one coordinate plane; and the planar “face-up” shapes, with all tiles (considered of infinitesimal width) lying in a common plane and without superposed consecutive tiles. There are still plenty of interesting configurations that do not belong to either of these two collections. The set of constructible configurations (those that can be obtained by manipulation of the undecorated puzzle from the starting situation) is a subset of the set of configurations with vanishing invariants. We were able to actually construct all octominoid shapes with vanishing invariants and most of the planar “face-up” configurations. Particularly important is the topological invariant, of which we recently found mention in [7] by Tom Verhoeff. </p></abstract>

By using two different invariants for the Rubik’s Magic puzzle, one of metric type, the other of topological type, we can dramatically reduce the universe of constructible configurations of the puzzle. Finding the set of actually constructible shapes remains however a challenging task, that we tackle by first reducing the target shapes to specific configurations: the octominoid 3D shapes, with all tiles parallel to one coordinate plane; and the planar “face-up” shapes, with all tiles (considered of infinitesimal width) lying in a common plane and without superposed consecutive tiles. There are still plenty of interesting configurations that do not belong to either of these two collections. The set of constructible configurations (those that can be obtained by manipulation of the undecorated puzzle from the starting situation) is a subset of the set of configurations with vanishing invariants. We were able to actually construct all octominoid shapes with vanishing invariants and most of the planar “face-up” configurations. Particularly important is the topological invariant, of which we recently found mention in [7] by Tom Verhoeff.

Exploring the “Rubik's Magic” Universe

Recreational Mathematics Magazine

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"Exploring the \u201cRubik's Magic\u201d Universe"}

By using two different invariants for the Rubik’s Magic puzzle, one of metric type, the other of topological type, we can dramatically reduce the universe of...

Exploring the “Rubik's Magic” Universe

Article Category: Games and Puzzles

Published Online: Jun 09, 2017

Page range: 29 - 64

DOI: https://doi.org/10.1515/rmm-2017-0013

Keywords
Rubik's Magic puzzle, octominoid 3D shapes, topological invariants

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Exploring the “Rubik's Magic” Universe

Article Category: Games and Puzzles

Published Online: Jun 09, 2017

Page range: 29 - 64

DOI: https://doi.org/10.1515/rmm-2017-0013

KeywordsRubik's Magic puzzle, octominoid 3D shapes, topological invariants

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
Rubik's Magic puzzle, octominoid 3D shapes, topological invariants