#### Date of Award

2014

#### First Advisor

Brian Wynne

#### Second Advisor

Courtney Thatcher

#### Abstract

Rubik's Cube, patented in 1975 by Erno Rubik, is a common household puzzle toy that many people recognize. Besides its appeal as a logic game, the Cube provides an excellent model through which to examine concepts of group theory, a major component of the field of abstract algebra. A group is a set of elements under a specified operation that satisfies certain axioms. For example,the (finite) set of all possible sticker configurations of a Rubik's Cube and the (infinite) set of all possible moves that can be applied to it can both be modeled with suitable groups. The known mathematical structures of these "Rubik" groups provide insight into the physical properties of the Cube, while the Cube also motivates a particular understanding of how the groups are constructed and operate. This symbiotic relationship between group theory and Rubik's Cube can be employed to solve a number of problems concerning the Cube. For instance, we can find the exact number of "legal" sticker configurations one can reach without breaking apart and reassembling the Cube; furthermore, we can produce mathematical criteria that can be used to determine immediately whether a given configuration is "legal." We can also examine and develop a wide range of more complex algebraic concepts such as would appear in an undergraduate group theory text. This thesis collects some of the major algebraic concepts surrounding Rubik's Cube that have been published by mathematicians in the decades after the Cube's initial distribution. It offers some alternative formulations in proofs and provides details that are omitted by many authors. In addition, two computer programs have been created that allow the reader to easily find the mathematical objects corresponding to any Rubik's Cube algorithm of interest.

#### Recommended Citation

Gearhart, Gemma, "The Algebra of Rubik's Cube" (2014). *Senior Theses*. 846.

https://digitalcommons.bard.edu/sr-theses/846

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