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Abstract

The Name Tag Problem is a thought experiment that, when formalized, serves as an introduction to the concept of an orthomorphism of $\Zn$. Orthomorphisms are a type of group permutation and their graphs are used to construct mutually orthogonal Latin squares, affine planes and other objects. This paper walks through the formalization of the Name Tag Problem and its linear solutions, which center around modular arithmetic. The characterization of which linear mappings give rise to these solutions developed in this paper can be used to calculate the exact number of linear orthomorphisms for any additive group Z/nZ, which is demonstrated in the third section. The final section establishes the equivalence between solutions to the Name Tag Problem and orthomorphisms of Z/nZ.

Author Bio

“The Name Tag Problem” (NTP) is the paper that served as Christian Carley's senior thesis in mathematics at Boise State University, Spring 2019. The ground work for this paper was laid Spring 2018 through Fall 2018, after approaching a professor about undergraduate research opportunities. Then, in Spring 2018 it was revised and served as his senior thesis. Christian likes to note that, because he began the research for this paper in his Junior year and did not take number theory until his Senior year, the original contents of this paper were drastically more complicated and lengthy. Lacking knowledge of some relevant theorems, he resorted to brute force in all of the theorems he developed. It was not until taking number theory and seeing that there was an existing framework for the ideas in this paper that he was able to put it into its current format. Christian lives in Boise, ID and hopes to pursue research in applied analysis, dynamical systems, and mathematical physics. He is an avid reader, particularly history and biographies, a metalhead, and loves to cook.

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