\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,epsfig}
\newcommand{\ds}{\displaystyle}
\newcommand{\vp}{$\vphantom{\big|}$}

\newcommand{\R}{\textbf{R}}
\newcommand{\Item}{\item{$\Box$}\quad}

\newcommand{\Answer}[1]{$\phantom{.}$\hfill
        #1\quad\fboxsep 20pt \fbox{$\phantom{\Big|Answer}$}} 

\newcommand{\MyTallAnswer}[2]{$\phantom{.}$\hfill
        #1\quad\fboxsep 20pt\fbox{$\vphantom{\Big|}#2$}}

\newcommand{\MyAnswer}[2]{$\phantom{.}$\hfill
        #1\quad\fboxsep 5pt\fbox{$\vphantom{\Big|}#2$}}

\newcommand{\soln}{\paragraph{Solution}}
\newcommand{\sep}{{\medskip\hrule\bigskip}}
\newcommand{\blank}{\underline{$\phantom{A blank}$}}

\setlength{\textwidth}{6.5in}
\setlength{\topmargin}{-0.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\textheight}{9.0in}
\begin{document}
\begin{center}

\textbf{College of the Holy Cross} \\
\textbf{MONT 104Q -- Mathematical Journeys}\\
\textbf{Final Exam -- December 18, 2015}
\end{center}
\vskip 10pt
\vfil

\noindent\textbf{Your Name}: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$


\vskip 30pt
\noindent\textbf{Instructions}:  Do all work on the test pages.
Place your answers in the space provided for the problem.  Use the 
back of the preceding page if you need more space for scratch work.  
The numbers next to each part of the questions are their point values.  
There are 100 total points on this exam.

\vfil

\noindent\textbf{Please do not write in the space below}

\begin{center}
\begin{tabular}{|l||r|}
\hline
Problem & Points/Poss \\
\hline\hline
I  & \hfil/ 10\  \hfil \\
\hline
II  & \hfil/ 15\  \hfil \\
\hline
III  & \hfil/ 10\  \hfil \\
\hline
IV  & \hfil/ 10\  \hfil \\
\hline
V  & \hfil/ 20\  \hfil \\
\hline
VI  & \hfil/ 35\  \hfil \\
\hline\hline
Total   & \hfil/ 100\hfil \\
\hline
\end{tabular}
\end{center}

\begin{center}
$$\mathcal{HAVE\ \ A\ \ PEACEFUL\ \ AND\ \ ENJOYABLE\ \ HOLIDAY\ \ SEASON}!$$
\end{center}
\vfil\eject
\noindent
I.  G. H. Hardy included two proofs from Euclid's {\it Elements} as
prime examples of beautiful, serious mathematics in his book 
{\it A Mathematician's Apology}.  The first was the proof that there are infinitely many
prime numbers.  
\begin{enumerate}
\item[A)]  (5) Prove this result using Euclid's method.
\vskip 3.5truein
\item[B)]  (5) What is the name of the method of proof that Euclid (and you)
used here?  Describe briefly how that method works.  Hardy say this is ``a 
far finer gambit than any chess gambit:  a chess player may offer 
the sacrifice of a pawn or even a piece, but a mathematician 
offers {\it the game}?''  What does he mean by that?
\end{enumerate} 
\vfil\eject
\noindent
II.  
\begin{enumerate}
\item[(A)] (10)  Give the statement and proof of Proposition 29 in Book I of 
Euclid's {\it Elements}.  
\vskip 5.0truein
\item[(B)] (5) How does this proposition relate to Proposition 27?  
What is special about the place of Proposition 29 in Book I?  Explain briefly. 
\end{enumerate}
\vfil\eject
\noindent
III. (10)  What does it mean to say that a polyhedron is \emph{convex}?  What is
true about the number of vertices $V$, the number of edges $E$, and the number
of faces $F$ in that case?  
\vfil\eject
\noindent
IV.  In our proof of the ``power theorem'' about the remainders $r^{m-1}\ R\ m$, where
$m$ is prime and $1 \le r \le m - 1$,
\begin{enumerate}
\item[(A)] (5)  The first step was to show that for each fixed $r$ as above, the remainders
$r\ R\ m$, $(2\cdot r)\ R\ m$, ... , $((m - 1)\cdot r)\ R\ m$ were all distinct.  Show this 
is true.  You may assume the statement of Euclid's lemma about primes: if a prime
$m$ divides a product of integers $c \cdot d$, then $m$ divides $c$ or $m$ divides $d$.    
\vskip 4.0truein
\item[(B)] (5)  What does the fact from part A show about the remainders $r^{m-1}\ R\ m$?
Explain.
\end{enumerate}
\vfil\eject
\noindent
V.  Proposition 47 in Book I of the {\it Elements} is a form of the Pythagorean theorem,
illustrated by the figure above.  Use the labeling here in your answers to all parts.
\begin{figure}
\begin{center}
\includegraphics[width=.5\linewidth]{EuclidI47.jpg}
\caption{Figure for Proposition 47, Book I}
\end{center}
\end{figure}
Euclid's form of the statement is that the area of the square on the hypotenuse of the 
right triangle $\Delta ABC$ is equal to the sum of the areas of the squares on the sides.
\begin{enumerate}
\item[(A)] (5)  How is the dotted line $AM$ in the figure constructed?
\vglue 1.0truein
\item[(B)] (5)  In the first part of the proof, Euclid shows that $\Delta GBF$ has the same
area as what other triangle in the figure?  Why does that follow?
\vfil\eject
\item[(C)] (5)  The second part of the proof consists of showing that $\Delta FBC$ and 
$\Delta ABD$ are congruent.  How does that follow?  (Show that is true using one of the 
triangle congruence results proved before in Book I.)  
\vglue 3.0truein
\item[(D)] (5)  How does Euclid conclude that $ABFG$ and $BLMD$ have the same area?  And
how does he conclude the proof?
\end{enumerate}
\vfil
\eject
\noindent
VI.  Essay.  (35)
\begin{enumerate}
\item[] {\bf Option A:} Oliver Heaviside, 1850-1925, an English engineer, applied mathematician, 
and physicist, once wrote the following about the role of Euclid in mathematical education in 
his time in England:  ``As to the need of improvement there can be no question whilst the reign 
of Euclid continues.  My own idea of a useful course is to begin with arithmetic, 
and then not Euclid but algebra.  Next, not Euclid, but practical geometry, solid 
as well as plane; not demonstration, but to make acquaintance. Then not Euclid, 
but elementary vectors, conjoined with algebra, and applied to geometry ... Elementary 
calculus should go on simultaneously ... . Euclid might be an extra course for 
learned men, like Homer.  But Euclid for children is barbarous.''  On the other hand, 
about 5 years ago, Peter Rudman, a contemporary physicist, wrote this: ``High school 
mathematics education today, ... , 
all too often neglects the derivations where mathematics is learned and emphasizes 
memorizing the equations that provide quick solutions in the standardized tests but 
that are then rapidly forgotten ... .''  What aspects of mathematics does each of 
these authors seem to value most highly and think students should learn?  How does 
what each of them says relate to the ideas of G. H. Hardy in {\it A Mathematician's 
Apology}?  Why might Heaviside say that teaching Euclid to children is ``barbarous?''  
Was your high school mathematics more or less like what Heaviside is recommending?  
Was your experience like that Rudman describes?  Do you think that emphasizing proofs 
more would make mathematics more interesting for more people?  Or is that too much 
to hope for?
\item[] {\bf Option B:}  (If you are choosing this option, ask Prof. Little for
a copy of the poem to consult while you are writing.) 
The poem {\it Ithaka} by C. Cavafy clearly draws on themes from the {\it Odyssey}, but 
does it just retell parts of Homer's story, or does it end up making something quite 
different of them?  In particular, is the return of Odysseus the main point here?  If not, 
what is the main point?
Why doesn't Cavafy mention Telemachus or Penelope? Finally, how do you think what 
Cavafy is saying here relates to the CHQ theme (especially the ``how then shall we
live?'' part)?
\end{enumerate}
\vfill\eject
$\qquad$
\vfill\eject
$\qquad$
\end{document}
