Undergraduate Research in Mathematics

# Senior Projects 2012

Luke Giberson - A Simple Learning Heuristic Demonstrated in the NxN Pawn Game, Advisor Dr. Bernd Rossa

In his book, The Colossal Book of Mathematics, Martin Gardner constructs a hexapawn learning machine using matchboxes and beads. In this study we implement his learning heuristic on the Maple software in an attempt to simulate a series of matches of the NxN Pawn Game. Specifically, we are interested in the distribution of Critical Points, that is, the number of games played in a series before one side learns enough information to win all subsequent games. Results of the simulations show that these critical points are dependent on the pseudo-random choices made each game and the relative strength of the opposition.

Wan-Yuan Hung - The 10.5 Game, Advisor Dr. Max Buot

10.5 is a card game that is similar to Blackjack. The game originated in Taiwan and is typically just played for fun. Most people have not tried to figure out the probabilities of winning in the game because it is not popular around the world. For this reason, we were interested to study the probabilities of winning given certain card outcomes. After calculating the probabilities by hand, we programmed in R to check if the manual answers calculated matched the simulated results of the R program. In this talk, we discuss some of the interesting cases observed. For example, hands that have different cards but whose sum is the same, will also result in different probabilities of winning.

Jessie Sherman - Combinatorial Geometry and its Applications to Perspective Drawing, Advisor Dr. Danny Otero

Is it possible to create a faithful planar drawing of a three-dimensional figure? In this talk, we will explore how properties of combinatorial geometry can be used to create and analyze these drawings. We will also examine the effects of using different viewpoints to create the perspective renditions of these figures. We will seek to determine if there are certain viewpoints that produce drawings with no loss of information and thus most closely reflect the real world.

Luke Wasserman - Concrete Euclidean and Non-Euclidean Geometry, Advisor Dr. Danny Otero

This project presents concrete models for the three classical plane geometries, Euclidean, Spherical and Hyperbolic. It will discuss how inner products for 2- and 3-dimensional real vector spaces allow for the definition of the Euclidean and Lorentz metrics, and how coordinatizations of space provide environments in which faithful models of the three geometries can be realized. In each of these models the identification of lines (geodesics), isometries, and the notions of parallelism are the building blocks of triangle geometry that mirror classical synthetic approaches to plane geometry. At the end of this presentation, one, if interested, will be able to continue research of Geometry in a concrete way.