Date of Award

5-2016

Document Type

Thesis

Degree Name

Master of Science (MS)

College/School

College of Science and Mathematics

Department/Program

Mathematical Sciences

Thesis Sponsor/Dissertation Chair/Project Chair

Diana Thomas

Committee Member

Michael A. Jones

Committee Member

Jonathan Cutler

Abstract

In a recent article, Jones, Shelton and Weaverdyck and Aim, Gramelspacher, and Rice provided two analyses of the Rubik’s Slide game on a board of dimensions 3x3. This paper extends the work of Jones, Shelton, and Weaverdyck to a board of dimensions 4x4. Concepts from abstract algebra and graph theory are used to calculate the God’s number of many classes of puzzles, which is the least number of moves necessary to reach any end configuration from any starting configuration of game play. It turns out that God’s number is equivalent to the diameter of a graph of the group formed by the Rubik’s Slide. Group structures are identified along with the Cayley graphs of multiple classes of puzzles and a colorbased adjacency matrix is used to explore the overall group structure of the puzzle. A Hamiltonian cycle is identified for the group with supporting graphs that isolate a subgroup and its cosets. A more challenging version (hard mode) of the puzzle that is represented by a group of order 1,625,702,400 is also explored with some interesting preliminary results that will help provide further insight into the structure of this large group. Programming code (Python) used to research this puzzle is also provided, along with GAP group definitions.

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