\documentclass{beamer}

\mode<presentation>
{
  \usetheme{AnnArbor}
}


\usepackage[english]{babel}
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\usepackage[utf8]{inputenc}
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\usepackage{times}
\usepackage[T1]{fontenc}
% Or whatever. Note that the encoding and the font should match. If T1
% does not look nice, try deleting the line with the fontenc.
\usepackage{amsmath,amsxtra,amssymb,latexsym, amscd,amsthm}
\usepackage{graphicx,graphpap}
\usepackage{floatflt}
\usepackage{epic}
\usepackage{curves}
\usepackage[utf8]{vietnam}
\beamerdefaultoverlayspecification{<+->}
\setbeamercolor{alerted text} {fg = magenta}
\newtheorem{thm}{Theorem}
\newtheorem{cor}{Corolary}
\newtheorem{lem}{Lemma}
\newtheorem{disc}{Discussion}
\newtheorem{defn}{Definition}
\newtheorem{rem}{Remark}
\newtheorem{conj}{Conjeture}
\newtheorem{exam}{Example}
\newtheorem{ex}{Exercise}
%\newtheorem{ans}{To see the answer hit PgDn}
\newtheorem{hint}{Rule}
%\newtheorem{rule}{Rule}
\newtheorem{ask}{Question}
\newtheorem{com}{Comment}
\newtheorem{ans}{Answer}
\newtheorem{drill}{Drill}
\newtheorem{blank}{}
\newcommand\RR{\mathbb{R}}
\title{\textbf{Discrete Mathematics 2012\\ Lecture 1}}
\begin{document}
\titlepage
\begin{frame}
\frametitle{1. Introduction}
\begin{ask}
\Large
What is mathematics?
\end{ask}
One of my favorite 
\begin{defn}
{\Large \color{green} \bf  Mathematics is the study of numbers, shapes and patterns}
\end{defn}
\begin{ask}
\Large How do you study mathematics?
\end{ask}
\begin{ans}
\Large \color{red} \bf With love and enhtusiasm.
\end{ans}
\end{frame}
%***************************************************************************************
\begin{frame}
\frametitle{Introduction}
\begin{ask}
\large \bf \color{magenta}
\Large How do you really study mathematics?
\end{ask}
\begin{ans}
\Large
\begin{enumerate}[<+-|alert@+>]
\item Observe
\item Hypothesize
\item Verify   
\item  Modify (if needed)
\item  Prove
\end{enumerate}
\end{ans}
\pause
\Large \bf \color{purple} Let us start with a small sample of examples.
\end{frame}
%****************************************************************************
\begin{frame}
\frametitle{\huge Examples}
\begin{enumerate}[<+-|alert@+>]
\item 
\begin{itemize}[<+-|alert@+>]
\Large
\item  Consider the sequence $1,2,4,$ \pause $7,$ \pause $11,$ \pause $16,$ \pause $22 \dots$ \pause
\item Can you express your observation in a ``mathematical'' sentence?
\item Does this sequence represent anything?
\end{itemize}
\item
\begin{itemize}[<+-|alert@+>]
\Large
\item Consider the sequence: $13, 29, 53, \pause 149, \pause 173, \pause 269, \pause 293, \pause 317, \dots $
\item What is the next number?
\item Can you express it in a ``mathematical'' sentence?
\item Can you formulate an hypothesis?
\item Can you prove it?
\end{itemize}
\end{enumerate}
\end{frame}
%*******************************************************************************
\begin{frame}
\frametitle{\bf \Large Shapes}
\setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(60,60)
\thicklines
\drawline(-10,-10)(90,70)
\pause
\drawline(90,0)(-30,60)
\pause
\drawline(-30,10)(100,10)
\pause
\drawline(50,-10)(40,65)
\pause
\put(-3,4){\bf \color{red} 1}
\pause
\put(-3,24){\bf \color{green} 2}
\pause
\put(20,44){\bf \color{blue} 3}
\pause
\put(50,44){\bf \color{magenta} 4}
\put(55,24){\bf \color{brown} 5}
\put(41,26){\bf \color{blue} 6}
\put(33,12){\bf  7}
\put(49,12){\bf  8}
\put(33,4){\bf \color{red} 9}
\put(53,4){\bf \color{red} 10}
\put(87,4){\bf \color{red} 11}
\end{picture}
\end{center}
\end{frame}
%**********************************************************************************
\begin{frame}
\frametitle{\bf \Large Shapes}
\begin{enumerate}[<+-|alert@+>]
\Large
\item How many regions can be generated by 5 lines?  
\item By 6 lines? 
\item Have you seen this sequence of numbers before?
\item So can you tell how many regions can be generated by 17 lines?
\item Can you explain why?
\item Can you find a ``formula'' for $f(n)$ the number of regions generated by $n$ lines?
\item Can you prove that it is $\binom{n+1}{2} + 1$?
\end{enumerate} 
\end{frame}
%**********************************************************************************
\begin{frame}
\frametitle{\Large \bf Curiosity led research}
\Large  You have four points on a line, the distance between any two consecutive points is {\bf 1}. \pause
\vskip 10pt
\hskip 100pt {\bf \color{red} o---o---o---o}
\vskip 10pt
\begin{enumerate}[<+-|alert@+>]
\large
\item How many times does the distance {\bf 1} occur?
\item  How many times does the distance {\bf 2} occur?
\item  How many times does the distance {\bf 3} occur?
\item Can you place $n$ points in the plane so one distance will occur $n-1$ times, a second distance will occur $n-2$ times, a third $n-3$ times etc.
\item This was easy,  can you do it so no three points are on a line?
\item This was not too difficult,
\item but can you do it so the points wil be in {\bf general position} (no three on a line, no four on a circle)?

\end{enumerate}
\end{frame}
\end{document}

