# Projects

## 2021

## MathFest Virtual

## Survey Labyrinth Probabilities

## Nathan LeRoy with mentor Lindsey Bosko-Dunbar

Labyrinth is a board game that was manufactured by Ravensburger in 1986. It is for 2-4 players with an age range of 7-99. The goal of the game is to slide the moveable columns and rows with an extra piece to maneuver your character to the treasure icons scattered about the board. We will be exploring mathematical elements such as graph theory, game theory, and probability in relation to the configurations of the standard 7×7 board for this game. We will be starting our research investigating a 3×3 board with similar ratios of pieces to the actual board. From this, we hope to scale up our findings to the larger board and perhaps even generalize to a board of arbitrary size. Once configuration numbers can be confirmed, we can break up or research with my direction being more probability and graph theory-based. I will explore a connection to the google maps path optimization algorithms that can help choose the best moves to get anywhere on the board in a game.

## An Analysis of Strategic Thinking in Ravensburger Labyrinth

## Jax Mader with mentor Lindsey Bosko-Dunbar

Labyrinth is a board game for two to four players by Ravensburger in 1986. The object of the game is to collect all of your treasures and return to your starting corner before your opponents. Turns are sequential and consist of two parts–1) shift the board by inserting an extra game tile into a moveable row or column to change the configuration of the current board and 2) take the option of either moving your playing token along an open path made by connecting the game tiles or staying at the current coordinate on the board. Turns are completed in sequential order by players in a clockwise direction until one player obtains all their respective treasures and returns to their starting corner. The board consists of three basic types of game pieces: T, L, and I. The standard gameboard is a 7×7 square, with even rows and columns movable. We explore several factors that would optimize success during gameplay. Some of these factors considered are the configuration of the board, the location of the intended target, and the least amount of moves to a target. The analysis builds from a 3×3 reduced version to a 5×5 reduced version to analyze possible strategies for the full 7×7 board.

## 2020

## MathFest Cancelled

## Expanding Lanchester’s Combat Laws to Three Armies

## Will Ulrich with mentor Samuel Hokamp

In 1916, Frederick Lanchester formulated a series of differential equations to describe the power relationships between opposing forces. This model has since been used to study a variety of aggressive interactions, including some among chimpanzees and fire ants. In this project, we showed that the only solution to the original model is the pair hyperbolic sine and hyperbolic cosine. We also analyzed a suitable expansion of the model to three forces and devised victory conditions for fighting effectiveness and initial army size.

## A New Measure for Partisan Bias in Redistricting

## Megan Struble with mentor Jonathan Dunbar

Each decade, the U.S. Census Bureau supplies states with population data so they may draw new state maps with new political district boundaries. When this redistricting process is done in a way that is intentionally biased in favor of a political party, we call it partisan gerrymandering. We propose a method for measuring partisan bias in redistricting, and apply this method to election data from the state of Kansas.

## 2019

## MathFest @ Cincinnati, OH

## How Not to be Lazy: Reducing Procrastination Through Two-Player Task Modification

## Jessica Pomplun with mentor Seth Meyer

In 2014, Kleinberg and Oren introduced a graph-theoretic model of a time-inconsistent agent planning a strategy to complete a task. This model is able to predict several real world behaviors including procrastination and abandonment of long-range tasks. In this project, I investigate an extension to the model which allows two actors to modify the task graph during the agents progress towards task completion.

## Variation in Procrastinating over Repeated Tasks

## Joshua Schill with mentor Seth Meyer

In 2014, Kleinberg and Oren introduced a graph-theoretic model of a time-inconsistent agent planning a strategy to complete a task. This model is able to predict several real world behaviors including procrastination and abandonment of long-range tasks. In this project, I investigate the situation where an agent repeats the same task multiple times and updates its willingness to procrastinate based on how poorly it performed the task in the past.

## Modeling the Population Dynamics of Diseased Fish

## Bao Van with mentor Terry Jo Leiterman

Columnaris Disease is a fatal disease affecting most freshwater fish and fish-farming communities. Flavobacterium columnare is the disease-causing agent. Fish become infected through contact with this agent in water. Flavobacterium columnare forms into biofilms on the body of fish, affecting respiratory process and motility. As the biofilm grows, it sheds off the host into the water, which results in continued transmission of the disease. Flavobacterium columnare is saprophytic, meaning that it does not need a host to survive. Moreover, the biofilm continues to grow on deceased fish. Columnaris disease is unconventional because the deceased members of the population infect the population’s healthy members. The conventional SIR models do not capture this phenomenon. In this project, I present a population model in which the deceased population cannot be removed, or ignored, when predicting infectious transmission. In this project, I intersect biology, mathematical modeling, and scientific computation to build accurate predictions of the population dynamics of fish infected by Columnaris Disease and compare the results to field data from a fish-farming community

## 2018

## MathFest @ DENver, CO

## Columnaris Disease and the Population Dynamics of Infected Fish

## Allison Gerk with mentor TERRY JO Leiterman

Flavobacterium columnare is a bacterial pathogen that forms biofilms on the surface of freshwater fish. As the biofilm grows, the resulting infection causes a fatal disease to fish known as columnaris. F. columnare grows not only on the surface of healthy fish but also on deceased fish. During this biofilm growth, F. columnare sheds from the host fish and enters the aquatic ecosystem where it resides, remaining viable, until it infects another fish. Consequently, shedding is a threat to the healthy fish population. In this project, I present a model for the population dynamics of freshwater fish after F. columnare is introduced into the healthy population. We explore the interconnected relationship between healthy fish, infected fish, and deceased fish as a result of the shedding of F. columnare due to biofilm growth.

## Examples of Graphs that Admit No Normal Nonabelian Sylow p-Subgroup

## Mark Nichols with mentor Jacob Laubaucher

When studying a given graph, we frequently wish to determine whether or not the graph occurs as the prime character degree graph, denoted ∆(G), for some finite solvable group G. En route to showing a graph does not occur as ∆(G), one known strategy is to show there are no normal nonabelian Sylow p-subgroups. One way to ensure this result is to show ∆(G) satisfies specific technical conditions, of which the graphs I created exhibit

## Building Low Rank Matroids

## Bao Van with mentor Simon Pfeil

We explore low rank matroids constructed by an iterative process of adding elements to circuits of other low rank matroids. Throughout this process we can ask, for example, how likely a resulting matroid is to be graphic, representable, or connected

## 2017

## MathFest @ Chicago, IL

## Properties of the Easter Cycle

## Dayle Duffek with mentor Anders Hendrickson

The date of Easter is mathematically determined by approximations to the solar, lunar, and hebdomadal cycles. We study the function which maps each year to the date of Easter in that year, according to the Gregorian calendar, and describe patterns and properties of this function.

## Generalizing the Tower of Hanoi

## Alexander Grover with mentor Anders Hendrickson

The smallest number of moves to solve the 3-peg Tower of Hanoi is well-known, but when more pegs are available, the problem becomes quite difficult. We observe several patterns for arbitrary numbers of disks and pegs, and present proofs of some of those patterns

## Legendre’s Equation

## Colleen Mandell with mentor TERRY JO Leiterman

In this project, I solve Legendre’s equation, which is a linear, homogeneous second-order, ordinary differential equation. We construct the Legendre polynomials through a power series solution approach. The Sturm-Liouville theory is introduced to demonstrate how Legendre polynomials are used to solve the Laplace equation in spherical coordinates.

## Constructing the Airy’s Function, and Other Solutions within a Special Class of ODEs

## Brittany Sheahan with mentor TERRY JO Leiterman

We examine a class of homogenous, second order, linear ordinary differential equations that take the form (d/dx)(dy/dx) + f(x)y = 0 where the coefficient on the y term is the power function f(x) = y^n. The case where n = 0 provides the familiar constant coefficient solution. The case where n = 1 provides a solution called the Airys function that has been named after the founder, George Biddell Airy (1801-1892). In this project, I research some brief historical remarks about George Airy. We derive the Airys function by imposing a power series solution in the corresponding f(x) = x differential equation. We further this work by organizing solutions of the ODE for all integers n > 1. Interesting and predictable patterns emerge. We introduce some challenges for constructing solutions of the ODE for integers n < 0.

## Equivalency of Easter Algorithms

## Emily Simon with mentor Anders Hendrickson

The algorithm used to determine Easter in the Gregorian calendar was formulated in the 1500’s in Latin, using ancient mathematical tools such as epacts and golden numbers. The algorithm has been reformulated and modernized several times over the centuries, most famously by Gauss. We exhibit several algorithms for determining the date of Easter and show why they are equivalent.

## An Exploration in Electrical Circuits and Mass-Spring Systems

## Hanna Strohm and Joseph Mohr with mentor TERRY JO Leiterman

In this project, I derive the mathematical models for both the displacement on a mass induced by a forced, damped spring motion and for the charge in an RLC electrical circuit. Both models are linear, nonhomogeneous, second-order, ordinary differential equations. We compare and contrast selected solutions with a focus on applications. We introduce the conditions under which this differential equation relates to the motion of a pendulum

## 2016

## MathFest @ Columbus, OH

## Non-Existence of Uniformly Most Reliable Two-Terminal Networks

## Hayley Bertrand with mentor Seth Meyer

A 2-terminal network is a network in which resources are assumed to flow from one node, called the source, to another, called the sink. We represent these networks as graphs, where the 2-terminal reliability of such a graph is the probability that there exists a path from the source vertex to the sink vertex when each edge is included with probability p. Given a fixed number of vertices n and a fixed number of edges m, we look for the graph that is most reliable for all p over [0, 1]. In this project, I present specific values of n and m for which a most optimal graph does not exist as well as values of n and m for which there does exist an optimal graph.

## Generalizations of the Josephus Problem

## Chance Browning with mentor John Frohliger

In the classic Josephus Problem, you start with m people in a circle and go around, eliminating every kth person until only one person remains. Where should someone be in the original circle in order to be the last one remaining? We will answer this question and discuss variations of the problem.

## 2015

## MathFest @ Washington, DC

## Minimum Rank for Disconnected Circulant Graphs

## Hayley Bertrand with mentor Seth Meyer

The minimum rank problem is to determine, for a given simple graph, the smallest rank of a Hermitian matrix whose off-diagonal zero-nonzero pattern is that of the adjacency matrix of the graph. When the graph in question is a circulant graph, there is a connection between polynomials with few terms, orthogonal representations of the graph in few dimensions, and positive semidefinite matrices with small rank. In this project, I explore various connections between circulant graphs with multiple components, as well as their associated polynomials

## Minimum Rank of Circulant Graphs

## Sam Potier with mentor Seth Meyer

The minimum rank problem is to determine, for a given simple graph, the smallest rank of a Hermitian matrix whose off-diagonal zero-nonzero pattern is that of the adjacency matrix of the graph. When the graph in question is a circulant graph, there is a connection between polynomials with few terms, orthogonal representations of the graph in few dimensions, and positive semidefinite matrices with small rank. In this project, I discuss how the cyclic orders of complex roots of unity give equality in the minimum positive semidefinite circulant rank of certain related circulant graphs.

## 2014

## MathFest @ Portland, OR

## A Particular Polarity, Part I

## Marissa Hartzheim with mentor John Frohliger

In projective geometry, a polarity is a type of correlation between points and lines that preserves incidence. One such correlation associates points (a, b) in the Cartesian plane with non-vertical lines y = ax − b. We will discuss this polarity and explore some of its properties.

## A Particular Polarity, Part II

## Taylor Miller with mentor John Frohliger

A special polarity in the Cartesian plane associates points (a, b) with non-vertical lines described by the equations y = ax − b. In this project, I will look into the possibility of extending this correlation to a three-dimensional version.

## 2013

## MathFest @ Hartford, CT

## Atomisticity of Supercharacter Theory Lattices

## Alex Leitheiser with mentor Anders Hendrickson

The study of finite groups dates to the early 1800’s, and at the turn of the twentieth century the German mathematician Frobenius developed the powerful tool of character theory to study them. Quite recently, in 2006 P. Diaconis (Stanford) and I.M. Isaacs (Wisconsin) defined certain generalizations of character theory called supercharacter theories. Every finite group has several supercharacter theories, which form a mathematical structure called a lattice; the properties of this lattice depend on the group from which it was constructed. Necessary and sufficient conditions on the size of a cyclic group are already known to tell exactly when its supercharacter theory lattice is upper semimodular, lower semimodular, or coatomistic, but no one has yet proved when the lattice is atomistic. Alex and Professor Hendrickson worked towards a proof of a 2012 conjecture that would determine exactly which cyclic groups are atomistic.

## Cayley Groups

## Laura Staver with mentor Seth Meyer

A graph is consistent if when the vertex at (1,1) is adjacent to the vertex at (2,2), then every vertex is adjacent to the vertex above and to the right of it. The graph whose vertices form a rectangular lattice and whose edges are consistent is a Cayley graph for the cartesian product of the integers mod m and the integers mod n. We studied the associated class of matrices and computed the zero forcing numbers for these graphs based on which edges are present in the graph. This combinatorial question has applications in quantum physics and electrical systems.

## Helping the Environment through Strategic Placing of Power Plants

## Sarah Stiemke with mentors John Frohliger and TERRY JO Leiterman

This work is inspired by an article by Christopher Schaufele and Nancy Zumoff in the MAA publication Environmental Mathematics in the Classroom. This presentation examines the effects a coal burning power plant has on water and the organisms that use the water. This project attempts to find the height and location of the smokestacks that would result in the smallest effect on the environment.

## An Economic Approach to Environmental Sustainability of Public Beaches, Oceans and Waterways

## Doug Wickingson with mentor TERRY JO Leiterman

This project tackles two environmental considerations by employing the economics associated with both cost-benefit and marginal analysis in simple ways. It will discuss the reduction of pollution on public beaches with a focus on costs, benefits, and efficiency. It will also discuss the conservation of water resources that have free public access. The work is inspired by an article by Ginger Holmes Rowell in the MAA publication, Environmental Mathematics in the Classroom. This project attempts to quantify the sustainable use of free access beaches, ocean, and waterways from both environmental and economic vantage points. It uses the analysis of hypothetical data sets to better understand the complex, interconnected relationship that exists between the public and our natural resources.

## 2012

## MathFest @ Madison, WI

## Neither Rain, Nor Sleet, Nor Snow. What About the Internet?

## HanQin (Caesar) Cai

The internet is having a profound impact on the United States Postal Service. The volume of “snail mail” sent in this country decreased by 42.36% from 2008-2009. In 2010, I participated in a group competition studying this effect and this project will revisit and update our results.

## An Evaporation Investigation

## Jaci Kulow and Laura Sommerfeld with mentor John Frohliger

In this project, the amount of water lost due to evaporation by the use of a decorative water fountain was calculated. The goal was to reduce water waste and a build a convincing case for implementation of a new water use plan at a residential community in Florida.

## An Exploration in Differential Equations for Modeling Population Growth

## Jeff LaJeunesse with mentor TERRY JO Leiterman

We investigate the population dynamics of phytoplankton, which form the base of aquatic ecosystems. Predicting phytoplankton growth contributes to a better understanding of climate change. We focus on how light availability, particle geometry, and fluid mixing affect changes in the population by using field data, laboratory data, and scientific computation.

## From Golf Balls to Airplanes; What are the Powers of Dimples?

## Erik Miller with mentor Rick Poss

Dimples are known to improve the performance of golf balls by extending the boundary layer over the surface. Could this principle be applied to airplane design? With the addition of dimpled airfoils, the performance of aircraft could be greatly improved. We’ll investigate the aeronautical theory behind the dimples and see how it could affect the phases of flight

## A Scale, Some Coins, A Problem

## Sarah Stiemke with mentor.John Frohliger

This is a variation of the classic counterfeit coin problem. Given a collection of n coins of weights 1, 2, or 3 grams and a balance scale, we prove the minimum number of weighings needed to determine the weight of each coin is n

## 2011

## MathFest @ Lexington, KY

## Parameterizing the Koch Curve

## Brian Pietsch with mentors John Frohliger and Kevin Murphy

One of the most famous fractals is the self-similar Koch curve. It is known for having infinite length, and it is generated by infinite iterations of four affine transformations. We use these transformations to create a continuous parameterization of the Koch curve.

## Fractals and Their Dimensions

## HanQin (Caesar) Cai with mentors John Frohliger and Kevin Murphy

Fractals such as the Koch curve and Sierpinski Triangle are self-similar geometric shapes. During the summer of 2011, I participated in a research program at St. Norbert College focusing on fractals. We will begin by introducing fractal dimension which describes how densely the shape occupies space, and use it to compare fractals. Then, I will present some of my findings.

## Mathematics Involved in the Game “Shut Box”

## Ryan Hallberg

In this project, I research the game “Shut Box”. The game involves two dice and numbered boxes. The dice are rolled simultaneously, and the player must choose a combination of boxes to shut based on the sum of the dice. The rules of the game are simple, but the math involved is challenging due to the randomness of dice rolls and player’s choice. To solve these problems we reduce the game to a smaller variant and propose a fixed strategy. The primary math used are probability trees and Markov chains.

## An Extension of the Hiring Problem

## Nicole Harp with mentors John Frohliger and Kevin Murphy

AbstractA classic probability problem, the Hiring Problem, looks at the optimal strategy for hiring the best-qualified candidate from a pool of applicants. It is assumed that an applicant cannot be revisited once interviewed. If the best applicant is hired, it is considered a success. This problem was extended from hiring the best applicant to hiring one of the top, pre-determined applicants. The probability of success cannot be maximized using basic techniques so numerical methods were implemented.

## Assessing the Risk of an Oil Spill in the Arctic

## Abigail Wendricks with Mentor Teena Carroll

Drilling in the arctic region is a topic which has been discussed at length in the American legislature. Current legislation proposes the placement of 20 oils rigs in the Arctic Ocean. We are interested in estimating the probability of a major oil spill if this plan is enacted. We use data on drilling productivity, the frequency and severity of past spills from oil rigs and oil transport to estimate the probability and impact of an oil spill in the Arctic ecosystem.

## 2010

## MathFest @ Pittsburgh, PA

## Balanced Sequences and Egyptian Fractions

## Haoqi Chen with mentor Teena Carroll

Balanced sequences are sequences whose first k terms have the same sum and product. The first balanced sequence found was related to the well-known Sylvester sequence, whose reciprocals sum to one. Investigating this connection leads to the ancient idea of an Egyptian fraction representation of a rational number a/b. Enumeration problems were a primary focus of this work, including finding how many Egyptian fraction representations of one there are which use only even denominators.

## Variations of the Counterfeit Coin Problem

## Kayla Pope with mentor John Frohliger

AbsThe original counterfeit coin problem involves using a balance scale to determine a single counterfeit from among a collection of coins. This project generalized the problem to the case in which the number of counterfeit coins is unknown. A weighing scheme is a tree diagram in which the edges leaving a node are determined by the results of that weighing (and the preceding weighings). The most unexpected result involved the size of a minimal weighing tree.tract

## A Movement Algorithm for a Zero Turn Machine

## Corey Vorland

Is it possible to develop a zero-turn-lawnmower simulator? We will present the concept of zero turn movement from a mathematical perspective. Using only basic trigonometry, we will show how the simulator can calculate the new coordinates of a zero turn machine using user input

## 2009

## MathFest @ Portland, OR

## Genome Exploration

## Kathleen Miller with mentor TERRY JO Leiterman

Due to high-throughput genomics, massive amounts of data on DNA protein structure and protein sequences are becoming rapidly available – at a faster rate than we can keep up! This data is only as useful as long as it is interpreted. Based on summer laboratory research in bioinformatics, we explore a biological question using statistical and computational models to create algorithms. These algorithms allow for comparison between databases allowing for further interpretation and exploration within genomes

## Generating Sudoku Puzzles

## Stephanie Schauer with mentor John Frohliger

A (solved) Sudoku puzzle can be viewed as a function f : (Z^4) mod 3 → Z mod 9 with certain near one-to-one properties. In this case, for fixed a and b, f(x, y, a, b), f(a, x, y, b), and f(a, b, x, y) are injective functions of (x, y). Functions of the form f(x, y, z, w) = g(x, y) + h(z, w) mod 9 were studied. This form produces filled Sudoku puzzles if and only if two criteria are satisfied. An unsolved puzzle gives you some of the values of f; you need to find the rest. Sudoku puzzles were generated using this function

## Modeling Diatom Growth in Trout Lake, Part 2

## Corey Vorland and Stephanie Schauer with mentor TERRY JO Leiterman

Aulacoseira is a freshwater diatom which forms string-like colonies. Aulacoseiras growth is determined by a complex, interconnected relationship between mixing and light availability in the lake. Mixing, generated by turbulent convection, alters the location of Aulacoseira within the depth of the lake, consequently altering its ability to obtain light for growth. Aulacoseiras abundance and colony size have been measured at varying depths in Trout Lake in Northern Wisconsin. In previous work, we built a mathematical model which accounted for growth and sinking of the diatom. However, sinking was only qualitatively included. In this work, the model takes a more quantitative approach to including the diatoms sinking velocity which is not well known in the biological community. This work is in collaboration with Stephanie Schauer, an undergraduate student at St. Norbert College.

## 2008

## MathFest @ Madison, WI

## How We Roll: The Theory and Construction of the Square Wheel Bicycle

## Alicia Brinkman with mentors John Frohliger and TERRY JO Leiterman

In this project, I will research the history of the square wheel bicycle and develop a differential equation that describes the road required for the smooth motion of the wheel. The resulting solution for the road is given by a catenary curve. As part of the Math Modeling course at St. Norbert College, twelve students collectively built a square wheel bicycle and the required road. The result is that it works.

## Teaching Math with Pokemon

## Samantha Goeben

Children are living in a media-centered world and teachers are on a constant search for relevant and interesting techniques. One such technique uses a website titled “Pokmon Learning League”. It is an interactive, narrative-based website that uses the well-known cartoons to help teach mathematics to elementary students. Research has shown that this instructional strategy increases the attitude and aptitude in both individual and group case studies, regardless of opinion towards math or Pokemon

## Genetic Modeling of the White Buffalo

## Kathleen Miller with mentor TERRY JO Leiterman

In August 1994, a female white buffalo was born in Wisconsin. In this project, I will research a mathematical model for predicting the frequency of such an event. The model is built around the theory of genetics, including mutations and albinism. We then attempt to answer: when will it happen again?

## Introduction to Sudoku for the Algorithmically-Minded

## Ryan Pavlik with mentor Rick Poss

Whether or not you know about the popular sudoku logic puzzles, you can enjoy this discussion of basic sudoku strategy. Given the technically-minded audience, we will explore the basic strategies to solve a sudoku without guessing, and reflect on the algorithmic techniques we are using and learning as we go.

## Rolling Smoothly on a Saw-Tooth Road: The Theory for a Wheel

## Stephanie Schauer with mentors John Frohliger and TERRY JO Leiterman

Given a road constructed from a periodic pattern of isosceles triangles, how does one build a wheel that will traverse the road smoothly? Through a change of variables to polar coordinates, a “messy problem reveals itself as a simple spiral. In this project, I will research the details behind the solution and discuss some interesting discoveries.

## Modeling Diatom Growth in Trout Lake

## Corey Vorland and Stephanie Schauer with mentor TERRY JO Leiterman

Aulacoseira is a freshwater diatom whose abundance and colony size has been measured at varying depths in Trout Lake in Northern Wisconsin. Its population growth patterns are influenced by temperature, light availability, and nutrients. In this project, the vertical distribution of Aulacoseira is investigated through modeling, which incorporates natural characteristics of the lake as well as effects of the diatom’s buoyancy. Predicted outcomes are compared to measured observations. This work is joint with Stephanie Schauer, an undergraduate student also at St. Norbert College.