Exploração das configurações do cubo mágico utilizando a teoria de grupos

Abstract

This work investigates the Rubik’s Cube through the lens of Group Theory, aiming to understand its configurations and movements using concepts from abstract algebra. It begins by presenting the fundamentals of the Rubik’s Cube, including its structure, types of pieces, and standard notations. Then, Group Theory is introduced with definitions, examples, and applications focused on interpreting cube movements as elements of a group. The study explores the symmetric group, the representation of permutations by cycles, and how these tools help describe the cube’s movements. It also addresses homomorphisms, such as the sign homomorphism, which classifies permutations as even or odd, and group actions, which are essential to understanding the possible transformations of the cube. As a main result, the work provides a mathematical characterization of valid Rubik’s Cube configurations using 4-tuples (σ, τ, x, y), establishing that a configuration is achievable if and only if the permutations have the same parity, the sum of the corner orientations is congruent to 0 (mod 3), and the sum of the edge orientations is congruent to 0 (mod 2). The study also includes illustrative examples and didactic adaptations to support undergraduate students’ understanding of the concepts.