Date of Award

Spring 2019

Degree Name

Bachelor of Science in Mathematics

First Advisor

Samuel Coskey, Ph.D.

Abstract

A group of n people sit around a table, according to an assignment of name tags in which only one person is paired with the correct name tag. Curious to see if it will improve the number of correct pairings, everybody passes their name tag to the person on their left. Oddly, a new person, and only that person, receives the correct name tag. Indeed, every rotation provides the correct name tag to exactly one new person, until the nth rotation, whereby every person has received the correct name tag one at a time. Given any group of people, how can we assign them name tags so that this situation is reproduced? That is the name tag problem, or the NTP for short.

This paper develops a mathematical description of the NTP and walks through the construction of linear solutions, highlighting unique features with theorems and diagrams. While it was motivated as an exercise based on a fun problem, we found that the results were interesting in their own right and had connections to several areas of mathematics, including number theory, combinatorics and a particular type of group permutations called orthomorphisms, which in turn have applications in various other areas of mathematics. Section 1.1 offers a preliminary, qualitative description of the mathematics involved, detailing some of the conceptual difficulties that accompanied its formulation. Section 1.2 proceeds to work out the corresponding mathematical notation of the description from section 1.1. Once supported with a mathematical theory, section 2 constructs linear solutions to the NTP and highlights their number-theoretic properties with examples and diagrams. Section 3 is dedicated to counting linear solutions to the NTP for any given n. Finally, section 4 offers a brief introduction to the definition and applications of orthomorphic functions, establishes the equivalence of the orthomorphisms of ℤ/nℤ and solutions to the NTP, and provides resources for further reading.

Comments

A paper related to this work has been submitted to the Rose-Hulman Undergraduate Mathematics Journal.

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