TY - JOUR

T1 - Exploring the “Rubik's Magic” Universe

AU - Paolini, Maurizio

AU - 27408,

AU - informatiche, Area 01 - Scienze matematiche e

AU - Tartaglia', BRESCIA - Dipartimento di Matematica e fisica 'Niccolò

AU - FACOLTA' DI SCIENZE MATEMATICHE, FISICHE E NATURALI

PY - 2017

Y1 - 2017

N2 - 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 a paper by Tom Verhoeff.

AB - 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 a paper by Tom Verhoeff.

KW - Recreational mathematics

KW - invariants

KW - puzzles

KW - Recreational mathematics

KW - invariants

KW - puzzles

UR - http://hdl.handle.net/10807/102941

U2 - 10.1515/rmm-2017-0013

DO - 10.1515/rmm-2017-0013

M3 - Article

VL - 4

SP - 29

EP - 64

JO - RECREATIONAL MATHEMATICS MAGAZINE

JF - RECREATIONAL MATHEMATICS MAGAZINE

SN - 2182-1976

ER -