%This is a sample thesis file for all CPP Math Grad Students to follow.
% After downloading this file to your computer, save it with whatever name you wish.
% Then download the file CPP.cls and be sure to save it in the same folder where
% you saved this file. Do NOT change the name of CPP.cls
\documentclass[senior]{CPP}
\usepackage{amsthm}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{latexsym}
\usepackage{verbatim}
\newtheorem{thm}{Theorem}
\newtheorem{quest}{Question}
\newtheorem{prop}{Proposition}
\newtheorem{lem}{Lemma}
\newtheorem{cor}{Corollary}
\newtheorem{claim}{Claim}
\newtheorem{conj}{Conjecture}
%Enter the information indicated inside the curly brackets below.
\begin{document}
\titleone{Equal Relabelings}
\titletwo{for $PQ$-sided dice}
\doctype{Thesis}
\doctypeUp{Thesis}
\degree{Master of Science}
\field{Mathematics}
\Author{Alec Lewald}
\Advisor{Amber Rosin}
\MemberA{Berit Givens}
\MemberB{John Rock}
\Year{2017}
\quarter{Winter}
%Type your abstract here.
\Abstract{\addcontentsline{toc}{chapter}{Abstract}
For $m$ $n$-sided dice, we will call the sums $m, m+1, m+2, \ldots, mn$ the \emph{standard sums}, and we will say that $m$ $n$-sided dice have an \emph{equal relabeling} if the dice can be labeled with positive integers in such a way that the standard sums are equally likely to occur. We will consider the case when $n=pq$ for distinct primes $pa_1$. Then we know that $k+m-1$ appears at least $a_k* b_{m-1}$ times since $m-1$ appears $b_{m-1}$ times on die $2$. Next consider that the only values we can use that sum to $m$ are $1\in D_1$ and ${m-1}\in D_2$ since these are the minimum values on each die. Therefore the number of times that the sum $m$ appears is exactly $a_1*b_{m-1}$. Since $a_1*b_{m-1}