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\begin{center}

\textbf{College of the Holy Cross, Fall 2016} \\
\textbf{MONT 106Q -- Mathematical Thinking} \\
\textbf{Solutions for Final Exam Questions, December 13, 2016}
\end{center}
\vskip 10pt
\noindent
I.
\noindent
A) (5) Express the base 10 number $457$ in base 2.
\vskip 10pt
\noindent
{\it Solution:} The number 457 = $256 + 128 + 64 + 8 + 1$, so the base 2 form is $111001001$.
\vskip 10pt
\noindent
B) (5) What is the next integer after $111111$ in base 2?
\vskip 10pt
\noindent
{\it Solution:}  It's $1000000$.  (You can see this either by adding 1 as described in the 
solutions for the practice problems or converting to base 10:  $111111 = 63$, so the next number is 64, which
is $1000000$.)
\vskip 10pt
\noindent
II.  Compute ``the Egyptian way"
\vskip 10pt
\noindent
A) (5) $53 \times 123$
\vskip 10pt
\noindent
{\it Solution:}  We double repeatedly
\begin{eqnarray*}
\Rightarrow 1 \times 123 &=& 123\\
2 \times 123 &=& 246\\
\Rightarrow 4 \times 123 &=& 492\\
8 \times 123 &=& 984\\
\Rightarrow 16\times 123 &=& 1968\\
\Rightarrow 32\times 123 &=& 3936
\end{eqnarray*}
Since $53 = 32 + 16 + 4 + 1$, the product is $3936 + 1968 + 492 + 123 = 6519$.
\vskip 10pt
\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.)
\vskip 10pt
\noindent
{\it Solution:} We double $21$ repeatedly until the next doubling takes us past $301$:
\begin{eqnarray*}
1\times 21 &=& 21\\
\Rightarrow 2\times 21 &=& 42\\
\Rightarrow 4\times 21 &=& 84\\
\Rightarrow 8\times 21 &=& 168.
\end{eqnarray*}
So $301 = 14 \times 21 + 7$.  The remaining part of the computation is to write $\frac{7}{21}$ in 
terms of Egyptian fractions.  But that is easy: $301 \div 21 = 14 + \frac{1}{3}$. 
\vskip 10pt
\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?  
\vskip 10pt
\noindent
{\it Solution:}
There is an Eulerian path starting at one of the top vertices and ending at the other, but
that is not an Eulerian circuit.  There are not any of those because not all of the vertices
have even degree. 
\vskip 10pt
\noindent
B. (5)  For which pairs $m,n$ does $K_{m,n}$ have an Eulerian circuit?  Explain your reasoning.
\vskip 10pt
\noindent
{\it Solution:}  An Eulerian circuit exists if and only if $m,n$ are \emph{both even}.  The reason
is that each of the $m$ vertices in the first group has degree $n$ and each of the $n$ vertices
in the second group has degree $m$.  Both of those must be even for the $K_{m,n}$ graph to 
have an Eulerian circuit.
\vskip 10pt
\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?
\vskip 10pt
\noindent
{\it Solution:} Playing from the rightmost harvestable pit (underlined) each time we would proceed as follows:
\begin{eqnarray*}
\underline{6},4,2,2,0,0&&\\
0,5,3,3,1,\underline{1}&&\\
0,\underline{5},3,3,1,0&&\\
0,0,4,4,2,\underline{1}&&\\
0,0,4,4,\underline{2},0&&\\
0,0,4,4,0,\underline{1}&&\\
0,0,\underline{4},4,0,0&&\\
0,0,0,5,1,\underline{1}&&\\
0,0,0,5,1,0\\
\end{eqnarray*}
and we are stuck.  This is not one of the ``good positions.''  $6,4,2,0,0,0$ is the ``closest'' one that does
work, in a way.

\vskip 10pt
\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[(3)]  I and III only
\end{enumerate}
\item[B.] (2) Which Mayan number symbols represent the base-10 number $383$?
\begin{enumerate}
\item[(1)]  
\end{enumerate}
\item[C.] (2) What is the type of the symmetric strip pattern in Figure 2?
\begin{enumerate}
\item[(2)] $pm11$  (because it has no rotation symmetries, but it does have vertical reflection symmetry)
\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[(3)]  $m^3f$
\end{enumerate}
\item[E.] (2) In {\it Flatland}, who was the originator of the 
phrase ``configuration makes the man.''
\begin{enumerate}
\item[(4)]  Pantocyclus
\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. 
\item[] {\it Solution:}  See the diagram on page 71 of {\it Ethnomathematics} or your class notes.
\item[B)] (2)  Why is the Maya number system called a ``mixed-base'' system?  Explain.
\item[] {\it Solution:}  It is a mixed-base system because while it is mostly base-20, the third
digits of numbers represent numbers of $18 \times 20 = 360$'s rather than numbers of $20^2 = 400$'s.
The following digits continue from there:  $18 \times 20^2, 18 \times 20^3$, etc.
\item[C)] (2)  Apart from as entertainment, how was the game of {\it mancala} used in some traditional 
African societies?
\item[] {\it Solution:}  It was reportedly used as a test of strategic thinking for prospective chiefs.
\item[E)] (2)  What is the main difference between $pmm2$ and $pma2$ symmetric strip patterns?   
\item[] {\it Solution:}  The main difference is that $pmm2$ patterns have horizontal reflection symmetry, 
while $pma2$ patterns do not.
\item[F)] (2)  From what part of the world do the textiles we saw in the Cantor Art Gallery come?
\item[] {\it Solution:}  From the island of Sarawak in Indonesia (south Asia).
\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.  
\item[] {\it Solution:}  No, this is not possible.  The transformation obtained by translating
by one unit along the main axis, followed by reflection across that axis must also be in the 
symmetry group, and that is a glide reflection.
\end{enumerate}
\vskip 10pt
\noindent
VII.  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}
\end{document}
