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\begin{center}

\textbf{College of the Holy Cross, Fall 2016} \\
\textbf{MONT 106Q -- Mathematical Thinking} \\
\textbf{Final Exam, December 13, 2016}
\end{center}
\vskip 50pt
\leftline{{\bf Your Name:}  $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$ }
\vskip 70pt
\noindent
{\bf Directions}
\vskip 10pt
Do all work on the sheets provided (if you use the back of a sheet,
please place a note telling me to look there).
There is an extra blank sheet of paper at the end that you can
use either as scratch paper or as extra space for your essay.  You may
detach that if you like, but please put your name on it and hand it in with your exam
if you do detach it.
The numbers in parentheses next to the questions are there point
values (100 points total).

\vskip 140pt
\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/ 10\  \hfil \\
\hline
III  & \hfil/ 10\  \hfil \\
\hline
IV   & \hfil/ 10\ \hfil\\
\hline
V    & \hfil/ 10\ \hfil\\
\hline
VI   & \hfil/ 10\ \hfil\\
\hline
Essay & \hfil/ 40\ \hfil\\
\hline\hline
Total   & \hfil/100\hfil \\
\hline
\end{tabular}
\end{center}
\vskip 10pt
\centerline{\it Have a peaceful and joyous holiday season!}
\vfill\eject
\noindent
I.
\noindent
A) (5) Express the base 10 number $457$ in base 2.
\vglue 4.0truein
\noindent
B) (5) What is the next integer after $111111$ in base 2?
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\noindent
II.  Compute ``the Egyptian way"
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\noindent
A) (5) $53 \times 123$
\vglue 4.0truein
\noindent
B) (5) $301 \div 21$ (that is, ``calculate with 21 to yield 301''; for the fractional part, get a
sum of distinct Egyptian fractions any way you can, using the $2/n$ table provided on the accompanying sheet.)
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\noindent
III.  Let $m,n$ be two integers $\ge 1$.  The graph $K_{m,n}$ has two groups of vertices, one containing
$m$ vertices and one containing $n$ vertices.  There are edges containing each of the vertices in the first
group to each of the vertices in the second, and no other edges.  For instance one of these graphs is shown
in the figure above.
\begin{figure}
\begin{center}
\includegraphics[height=1.5in]{K23.png}
\caption{The graph $K_{2,3}$}
\end{center}
\end{figure}
\vskip 10pt
\noindent
A. (5)  Does the graph in the figure have Eulerian circuits?  Why or why not?
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\noindent
B. (5)  For which pairs $m,n$ does $K_{m,n}$ have an Eulerian circuit?  Explain your reasoning.
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\noindent
IV.  (10) Suppose you are playing \emph{mancala} and it is your turn.  Your pits contain numbers of
stones left to right as follows:  $6,4,2,2,0,0$.  Is it possible to get all 14 stones into your
store before the other player gets to play?  If so, how would you do it, step by step?  If not, why not?
\vfill\eject
\begin{figure}
\begin{center}
\includegraphics{StLouis.jpg}
\caption{Figure for question V, part C}
\end{center}
\end{figure}
\noindent
V.  Multiple Choice.  Circle the correct arabic-number option for each question.
\begin{enumerate}
\item[A.] (2) In this course we saw examples of the mathematical structure of a \emph{group}
in
\begin{enumerate}
\item[I.] analyzing the Warlpiri kinship system
\item[II.] considering the game of \emph{mancala}
\item[III.] studying symmetries of strip patterns
\end{enumerate}
\begin{enumerate}
\item[(1)]  I only
\item[(2)]  III only
\item[(3)]  I and III only
\item[(4)]  I,II,and III
\end{enumerate}
\item[B.] (2) Which Mayan number symbols represent the base-10 number $383$?
\begin{enumerate}
\item[(1)]
\item[(2)]
\item[(3)]
\item[(4)]
\end{enumerate}
\item[C.] (2) What is the type of the symmetric strip pattern in Figure 2?
\begin{enumerate}
\item[(1)] $p111$
\item[(2)] $pm11$
\item[(3)] $pma2$
\item[(4)] $pmm2$
\end{enumerate}
\item[D.] (2) Which of the following combinations of the mother $m$ and father $f$ relations
is the same as $fm$ in the Warlpiri kinship system?
\begin{enumerate}
\item[(1)]  $mf$
\item[(2)]  $m^2f$
\item[(3)]  $m^3f$
\item[(4)]  $mfm$
\end{enumerate}
\item[E.] (2) In {\it Flatland}, who was the originator of the
phrase ``configuration makes the man.''
\begin{enumerate}
\item[(1)]  the Sphere
\item[(2)]  Pantocyclus
\item[(3)]  the Square
\item[(4)]  Chromatistes
\end{enumerate}
\end{enumerate}
\vskip 10pt
\noindent
VI.  Short answer.  Answer \emph{any five} of the following.  If you answer
more than five, only the best five will be used.
\vskip 10pt
\begin{enumerate}
\item[A)] (2) Draw a diagram representing which sections are allowed to marry in the Warlpiri
kinship system and how the section of the children is determined.
\vglue 1.5truein
\item[B)] (2)  Why is the Maya number system called a ``mixed-base'' system?  Explain.
\vglue 1.5truein
\item[C)] (2)  Apart from as entertainment, how was the game of {\it mancala} used in some traditional
African societies?
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\item[E)] (2)  What is the main difference between $pmm2$ and $pma2$ symmetric strip patterns?
\vglue 2.0truein
\item[F)] (2)  From what part of the world do the textiles we saw in the Cantor Art Gallery come?
\vglue 2.0truein
\item[G)] (2)  Is it possible for a symmetric strip pattern to have reflection symmetry across the
main translation axis (usually horizontal) and no glide reflection symmetries?  Explain.
\end{enumerate}
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\noindent
IV.  Essay.  (40)  Pick one of the two following topics.  Say clearly which one you are addressing.
\begin{enumerate}
\item[] {\it Option 1:}  What are some of the borders/boundaries that Shakespeare uses to structure
his plot and characters in {\it Othello}?  For instance, what is a ``Moor?''  What does the word mean,
and how does the fact that Othello is a Moor create a boundary between him and Venetian society?  Where
does the action of the play take place, and how does that include a political or physical border?  Who
besides Othello has transcended borders and how?  Does transcending those borders end well or result
in tragedy?
\item[] {\it Option 2:}  A key episode in {\it Flatland} involves the interaction between the Sphere
from Spaceland and the Square from Flatland where the border of the two-dimensional Flatland is
literally transcended.  What does the Sphere do to demonstrate the existence of a third dimension?  Why
does the Sphere say, ``Listen, no stranger must witness what you have witnessed"?  What is the Square's
reaction to being taken into space?  Find three words in the text that have the prefix \emph{omni-}
and explain what they mean and how they relate to this episode.  Why does the Sphere say that
omnividence does not make a person a God?  What qualities does he say make one more divine?  What
is the Square's reaction?
\end{enumerate}
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$\qquad$
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