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Lights Out is a puzzle game where the goal is to turn off all the lights on a nxn board starting from a random configuration. In order to find the solution of a configuration, the game is constructed using a matrix basis in the span of the field Z mod 2.This the game can be modeled by the system Ap=s which will be the center of the investigation when determining the solvability for any n×n board since A is not always invertable leading to some interesting cases. The goal of this thesis was to construct a model that will allow the player to solve for the pushes to attain the zero-state for an nxn system. Constructing the model gave a procedure that will allow to solve the puzzle game. The procedure presented here first uses a simple clearing technique (valid for any board size) to turn off all the lights except in the last row, which we call the standard-clear. The heart of the technique, is to give a way to use the information about which lights remain lit in the last row to determine which switches in the first row need to be pushed before the standard-clear. This part of the solution algorithm we call the first row adjustment, and it depends heavily on the specific board size n of the problem. Finally, after these first row pushes are made, the standard clear will now turn off all the lights including (seemingly magically) the last row. Thus the solution to the Lights Out puzzle of a given size is reduced to finding a first row adjustment for that size. (Please refer to the actual thesis for the full abstract)
ContributorsBarraza, Jay Romeo (Author) / Vaz, Paul (Thesis director) / Paupert, Julien (Committee member) / Civil, Environmental and Sustainable Engineering Programs (Contributor) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor)
Created2015-05
Description
The Jordan curve theorem states that any homeomorphic copy of a circle into R2 divides the plane into two distinct regions. This paper reconstructs one proof of the Jordan curve theorem before turning its attention toward generalizations of the theorem and their proofs and counterexamples. We begin with an introduction to elementary topology and the different notions of the connectedness of a space before constructing the first proof of the Jordan curve theorem. We then turn our attention to algebraic topology which we utilize in our discussion of the Jordan curve theorem’s generalizations. We end with a proof of the Jordan-Brouwer theorems, extensions of the Jordan curve theorem to higher dimensions.
ContributorsClark, Kacey (Author) / Kawski, Matthias (Thesis director) / Paupert, Julien (Committee member) / Department of Physics (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2020-05
Description
Applying a classical theorem due to Macbeath applied to a suitably sized horoball, we calculate novel group presentations for singly-cusped Bianchi groups. We find new presentations for Bianchi groups with d = -43, -67, -163. With previously known presentations for d = -1, -2, -3, -7, -11, -19, this constitutes a complete set of presentations for singly-cusped Bianchi groups.
ContributorsReese, Tanner (Author) / Paupert, Julien (Thesis director) / Childress, Nancy (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Computer Science and Engineering Program (Contributor)
Created2022-05