Mathematics
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Continued Fraction Approximations Demonstrated Through The Musical Chromatic ScaleIn this paper, we approximate an irrational number using continued fractions through an example of a musical problem. We first define the chromatic scale. To delve into why the chromatic scale only has twelve notes, we discuss the topic of Pythagorean Tuning and how it utilizes mathematics to create scales. Since using Pythagorean Tuning to approximate the length of a scale results in an irrational number, we introduce the notion of continued fractions. These can be calculated by either using the Euclidean Algorithm or the Continued Fraction Algorithm. We define the term best approximation and finally, we use these components to solve our musical question.

Leibniz, Calculus, and The Hyperreal NumbersOur ideas revolving around Calculus, Philosophy, Law, and Theology are often so clouded that we forget to acknowledge the people behind these ideas. Through this way of thinking, we forget to look at the foundations that took countless years and even lifetimes to construct out of what we believe to be nothingness. What if I were to say everything mentioned in the first sentence was revolutionized by a German mathematician, philosopher, and logician's name is Gottfried Wilhelm Leibniz. The first part of this paper will focus on outlining his contributions to the foundations and invention of Calculus, disagreements between him and Newton, Leibniz's notation for Calculus, and other works in other areas such as law, metaphysics, and theology. This paper does not cover details including birth, death, spouses, etc. as those take away from the goals of this paper. The second part focuses on Abraham Robertson's construction of the Hyperreal Numbers and their applications proving that Leibniz's intuition of infinitesimals and Calculus correct. Accurate recognition of one's work is critical in maintaining not only credibility over future pieces of work but also recognizing the accomplishments of one's work. Understanding Leibniz's work and the instrumental construction of Calculus and infinitesimals allows us to also focus on the foundations of our modern societies and trace where many of our common ideas and innovations stem from. Ultimately, by the end of this paper, one should have a better understanding of the impact Leibniz, infinitesimals, and the foundational understandings of Calculus.

Teaching Higher Level Vector Concepts To Elementary Age Students By Exploring The Use Of GamificationMathematics has been known by many students to be their least favorite subject due to the complexity of the concepts, and the way in which this subject has historically been taught. However, with how practical mathematical skills are in so many diverse fields, helping foster a student's positive relationship with math is extremely beneficial for their academic and career growth. This literature review explores how educators can make math education more engaging starting from the elementary school level while also providing students with a good math foundation that will support them in higher grades and beyond their schooling. This goal can be accomplished by introducing elementary age students to the higherlevel math concept of vectors. With the application of vectors in careers ranging from STEM fields to animation, students will see how practical and useful math can be in the outside world. Combining this concept with researchedbased math engagement strategies, such as gamification and handson activities, will help support students' intrinsic motivation with math. This literature review concludes with lesson plan resources that educators can use to effectively teach students this practical, higher level math concept in an engaging way.

Math in OrigamiGeometry is not only used for math but also to create art! Geometry is used to create Origami, the ancient Japanese art of folding paper. There are seven origami axioms that can be used to solve general cubic equations through the Beloch fold (corresponds to an origami axiom). Origami is also being used to revolutionize technology, from space, to the least explored environment on Earth, the ocean! NASA James Webb Space Telescope and the ocean robotic device RAD, are just a few of the many new origamiinspired technologies.

Guaranteed To Win: Optimal Strategies For Discrete Bidding GamesMany of us are familiar with two player games, such as TicTacToe or chess, where each player alternates taking turns. Players compete against each other, strategically making a move once it’s their turn. The goal of the game is simply to “win”, depending on the rules of the game. We can add an extra layer to these games that creates some mathematical questions. Instead of alternating turns, players are now “bidding” to make a move. Not only does this add more competition, strategy, and excitement to the game, but it also adds mathematical intricacies. We call these Richman games, studied by David Richman in the 1980s. In Richman games, players make a bid (or auction)[1] of a nonnegative number of chips to make a move. The player that bids the most plays their turn, and then “pays” their chips to the other player. By studying Richman games, this paper will explore the optimal bidding strategies to maximize game play. The goal of each player is to win the game  not have the most amount of chips. In order to win the game, players need to have bidding strategies to ensure they are making moves. The proportion of chips a player has in their possession at a certain point, or critical threshold, is crucial within bidding games. We will explore how to find the critical threshold for games, and how it optimizes a player’s chance of winning (also referred to as winning strategies). We will also dissect the use of the tiebreaking advantage when two players bid the game amount of chips. Through these strategies, we will explore a game of bidding Tug O’ War and applications to more extensive games, such as bidding TicTacToe.

Pebbling And Cover Pebbling Numbers Of GraphsCover pebbling is a method in graph theory that was first brought about by Largias and Saks. This topic in graph theory was used to come up with a way to calculate how much of a consumable resource would being needed to begin transportation of said consumable resource. This paper will tackle the basics on cover pebbling, brush upon weight pebbling, and problem 9 of the open problems that can be found in Betsy Crull's paper The cover pebbling number of graphs. Problem 9: What are the cover pebbling number for other graphs G, for example cubes, complete rpartite graphs, etc.

Confidence Levels of SSU Education Students in Writing AssessmentsSalem State University has a newly accredited 4+1 master’s in education program but has been teaching education for decades. Given the newness of the 4+1 program, are the students getting a good idea of what assessments to write, what they look like, and when to do which type of assessment? This study investigates the confidence levels of SSU Education students in writing assessments based on their experiences and classes that they have taken in the School of Education. In order to examine this question, a survey was completed by students in the education program in which they specified their knowledge on each type of assessment (summative and formative), which classes they have taken, what experiences they have had, and how comfortable they are writing both. It is then analyzed by the classes they have taken the program, their comfortability, and knowledge of each assessment.

Exploring The Thirteen Colorful Variations Of Guthrie's FourColor ConjectureColoring is an important part of graph theory. Historically, it was thought that only four colors could be the minimal number of colors. This paper discusses the Four Color Theorem and how the Four Color Theorem is applied to graphs. This paper gives an overview of several different definitions involved with graphs and shows how to create a dual graph. This paper also discusses how a graph of 12 regions has at least one region bounded by less than five edges. The paper includes several figures which include graphs, dual graphs, and different colorings. The paper also provides a proof which shows mathematically why a graph of 12 regions has at least one region bounded by less than five edges.

Ranking College Football Teams Independent of Victory MarginsThis paper addresses David Mease's formula for ranking college football teams. It is just one of the numerous formulas that can be used by the Bowl Championship Series in ranking the top teams in the country. Mease uses his formula to rank teams independent of victory margins, something not all formulas take into consideration, Winning margin may help teams gain higher ranking in some formulas, so this formula ignores that statistic.

NQueens ProblemUsing combinatorics in this paper, we will discuss three different methods in solving the nqueens problem. We will find the maximum and minimum number of queens we can place on an n x n chessboard. Also, we will use latin squares, latin rectangles and circulant matrices as another method of placing the queens on a chessboard.

Mathematical Understandings Of A Rubik's CubeMany people are familiar with the 3x3x3 Rubik’s Cube as a puzzle or a toy. But, what most people do not realize is that the cube is a great physical visual of a group. The goal of this paper is to discuss the Rubik’s Cube as a group and dive into a specific subgroup of the cube. Through this discussion, we will also explore homomorphisms in the slice group. This paper will also give insight on permutations, commutators, and conjugates in terms of the cube, as well as “God’s number”.

Using Matrices And Hungarian Method To Solve The Traveling Salesman ProblemIn this paper, we introduce the Traveling Salesman Problem (TSP) and solve for the most efficient route of the problem using the steps of the Hungarian method. Specifically, this paper discusses the properties of a TSP matrix, provides the steps for the Hungarian method, and presents examples that apply these concepts to a Traveling Salesman Problem. We do not consider any constraints on the order in which the localities are visited, nor do we take into account possible traffic at differing times. We use examples to show how the Hungarian method is used and why it is an efficient way to solve the Traveling Salesman Problem.

Investigation Of Plane Symmetry In Lattice DesignsThe purpose of this research project is to analyze the scholarly article The Plane Symmetry Groups: Their Recognition and Notation by Doris Schattschneider. In this article, Schattschneider discusses an application of abstract algebra which is useful in art as well as crystallography: frieze groups and wallpaper groups. I was interested in pursuing this topic because it combines mathematics with its applications, particularly with my own interest in chemistry. The article provides a compiled resource of terminology and rules of these groups, but not one which was easily accessible to undergraduate students. In my research, I elaborated on the descriptions of certain types of periodic patterns to add to the accessibility, and analyzed a few designs to prove their classification based on the rules from Schattschneider's article. I found that this resource provided a good source of rules for which mathematical proofs could be based, and proved the classification of two different periodic plane designs.

A Combinatorial Method To Producing PortfoliosThere is a lack of research on portfolios and combinatorial methods in finance. In this paper, we outline a new method for producing longterm portfolios of stock using a combinatorial approach. A retrospective data analysis shows that this method produces protable longterm portfolios.

Topspin: OvalTrack Puzzle, Taking Apart The Topspin One Tile At A TimeA Topspin “Oval Track” puzzle consists of 20 numbered tiles in an ovalshaped track and a flipping window that reverses the 4 tiles in the window. The solvability of the puzzle uses permutations which are combinations where the order matters. A puzzle is considered solvable if each permutation in can be mapped to a spot in the original position through the three different moves the puzzle can make; a left shift, a right shift, and the flip which reverses the order of the 4 tiles in the window. I wanted to find out what math was involved in solving this puzzle. I had certain topics that I wanted to find out more information about, but the major question I had was “what is the fewest number of moves it takes to solve a puzzle”. Other topics I had were what made a puzzle unsolvable and what other types of puzzles use the same kind of math to solve them. To construct this research, I had read different scholarly articles that talked about the Topspin as well as physically looked at the puzzle and see how it works. I had found that there was no way to determine the fewest number of moves it takes to solve a puzzle since it’s impossible to decide what a “more scrambled” puzzle is compared to another. One scrambled puzzle might look completely different from another, and still have the same number of moves to solve it. In addition to this, I was also able to find that the Rubik’s Cube is solved like the Topspin.