Document Type

Article

Peer Reviewed

1

Publication Date

4-1-2013

Scholarship Domain(s)

Scholarship of Discovery

Abstract

The research conducted was to investigate the potential connections between group theory and a puzzle set up by color cubes. The goal of the research was to investigate different sized puzzles and discover any relationships between solutions of the same sized puzzles. In this research, first, there was an extensive look into the background of Abstract Algebra and group theory, which is briefly covered in the introduction. Then, each puzzle of various sizes was explored to find all possible color combinations of the solutions. Specifically, the 2x2x2, 3x3x3, and 4x4x4 puzzles were examined to find that the 2x2x2 has 24 different color combination possibilities, the 3x3x3 puzzle has 11,612,160 color combinations, and the 4x4x4 has at least 1,339,058,552,832,000 color combinations. We cannot say exactly how many the 4x4x4 puzzle will have due to the insufficient certainty of the possible solutions of the 4x4x4 cube. After inspecting each solution for the cube, it was found that the 2x2x2 puzzle had 4 transformations (or elements, in group theory terms), and the 3x3x3 puzzle had either 9 or 27 elements. The number of elements for the 3x3x3 puzzle was dependent on its original set up. If not every cube moved in the same direction horizontally and vertically, the puzzle would have 27 elements. Since the research was not sufficient enough to find a definite number of set ups that the 4x4x4 cube could have, there was not enough information to build upon to find a collection of the elements or groups that this puzzle would be isomorphic to. However, the other two puzzles, the 2x2x2 and 3x3x3, were successfully mapped to another group, proving that these groups are isomorphic. The 2x2x2 puzzle mapped to the group Z2 ⊕ Z2. The 3x3x3 puzzle is mapped to either the group Z3 ⊕ Z3 or Z3 ⊕ Z3 ⊕ Z3, depending on which group the original set up belonged to.

Comments

Honors Capstone Project completed in 2013 for Olivet Nazarene University.

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