# Rubik’s Cube: The Invisible Solve

5-2-2018

Thesis

## Abstract

The Rubik’s Cube is one of the most popular and recognizable puzzles ever made. In this research, we use group theory to identify and analyze the different solutions for the Rubik’s Cube and its variations. Since they cannot be seen on a standard Rubik’s Cube, these different solutions are called invisible solves. But by putting specialized labels on each of the center pieces of a Rubik’s Cube, we are able to track each of the invisible solves and see how they are different from one another. Dependent on the size of the Rubik’s Cube, the number of distinct invisible solves varies. For example, the 3x3x3 cube has only one center piece on each side; but the 4x4x4 has four different centers on each side. This difference in centers changes the total number of invisible solves. In addition to finding the number of invisible solves in the 3x3x3 and 4x4x4 cases, we also determine, in the 3x3x3 case, how each solve can be produced using certain algorithms. In the case of the 4x4x4 cube, we can use the parity theorem (the difference between odd and even) to verify that the number of invisible solves is correct. This research provides new insight about the structure of different solutions to the Rubik’s cube and its variations.

Mathematics

Irina Seceleanu

Rachel Stahl