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\textbf{College of the Holy Cross, Fall 2016} \\
\textbf{MONT 106Q -- Mathematical Thinking} \\
\textbf{Solutions for Midterm Exam, November 4, 2016}
\end{center}
\vskip 10pt
\noindent
I. 
\noindent
A) (10) Express the base 2 number $(1010110)_2$ in base 10.
\vskip 10pt
\noindent
{\it Answer:}
This is the number 
$$1 \times 2^6 + 0 \times 2^5 + 1 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 86$$
in base 10.
\vskip 10pt
\noindent
B) (10) Express the base 10 number $137$ in base 2.
\vskip 10pt
\noindent
{\it Answer:}
This is 
$$1 \times 128 + 0\times 64 + 0\times 32 + 0 \times 16 + 1 \times 8 + 0 \times 4 + 0\times 2 + 1\times 1 
= 10001001$$
in base 2.  
\vskip 10pt
\noindent 
II.  Compute ``the Egyptian way"
\vskip 10pt
\noindent
A) (10) $54 \times 763$.
\vskip 10pt
\noindent
{\it Answer:}  We double repeatedly starting from $763$:
\begin{eqnarray*}
1 \times 763 &=& 763\\
*2 \times 763 &=& 1526\\
*4 \times 763 &=& 3052\\
8 \times 763 &=& 6104\\
*16 \times 763 &=& 12208\\
*32 \times 763 &=& 24416
\end{eqnarray*}
Since the next power of 2 is $64$, which is bigger than $54$ we stop here.  Then 
$54 = 32 + 16 + 4 + 2$ (these are the numbers marked with asterisks above).  So the 
product 
$$54 \times 763 = 24416 + 12208 + 3052 + 1526 = 41202.$$
\noindent
B) (10) $1332 \div 12$ (that is, ``calculate with 12 to yield 1332'')
\vskip 10pt
\noindent
{\it Answer:}  We repeatedly double starting from 12:
\begin{eqnarray*}
*1 \times 12 &=& 12\\
*2 \times 12 &=& 24\\
*4 \times 12 &=& 48\\
*8 \times 12 &=& 96\\
16 \times 12 &=& 192\\
*32 \times 12 &=& 384\\
*64 \times 12 &=& 768
\end{eqnarray*}
We stop there since the next doubling would take us past $1332$.  Then 
$$1332 = 764 + 384 + 96 + 48 + 24 + 12$$
So the quotient comes from the marked numbers in the table
$$1332 \div 12 = 64 + 32 + 8 + 4 + 2 + 1 = 111$$
\vskip 10pt
\noindent
III.  Short answer.  Answer any four of the following.  If you answer
more than four, the best four will be used.
\vskip 10pt
\begin{enumerate}
\item[A)] (5)  What base 10 number is represented by these Egyptian
hieroglyphs? 
\vskip 10pt
\noindent
{\it Answer:} The number is $347$
\vskip 10pt
\item[B)] (5)  What base did the Maya use in their number system?
How would a Maya scribe have represented the number $151$?  
\vskip 10pt
\noindent
{\it Answer:}  They used a ``mixed'' base-20 system (meaning that numbers were expressed
as sums of multiples of $1, 20, 20\times 18, 20^2 \times 18, \ldots$, rather than purely
in powers of $20$).  The number $151 = 7 \times 20 + 11$,
so the Mayan form is:
\vskip 10pt
\item[C)] (5)  What are the principal surviving records of Egyptian mathematics and what are
their approximate dates?
\vskip 10pt
\noindent
{\it Answer:}  There are two surviving papyri of mathematical problems and their solutions
known as the Moscow and Rhind mathematical papyri.  The Moscow papyrus is thought to 
come from about 1800BCE.  The Rhind Papyrus is slightly later, about 1650BCE, although
it is thought to be copied from earlier work.  
\vskip 10pt
\item[D)] (5)  What was special about the Egyptian way of 
dealing with fractions? 
\vskip 10pt
\noindent 
{\it Answer:} They dealt almost entirely with unit fractions -- that is, fractions
of the form $\frac{1}{n}$.  They employed tables of sums of unit fractions 
equal to other fractions like $\frac{2}{n}$ to perform arithmetic with more general
fractions. 
\vskip 10pt
\noindent
\item[E)] (5)  What was the \emph{meaning} of the numerical information in the
section of the Maya \emph{Dresden Codex} that we studied in Discussion 1?
\vskip 10pt
\noindent
{\it Answer:}  This was a table of number of days between successive lunar eclipses.
It apparently served as both a record of a particular sequence of eclipses and as a
way to predict patterns of lunar eclipses in the future.
\vskip 10pt
\item[F)] (5) The ``months'' in the Maya {\it tzol'kin} calendar are, in order, 
\[\begin{tabular}{|c|l|c|l}
1. & Imix  &   11. & Chuen\\
2. & Ik    &   12. & Eb \\
3. & Akbal &   13. & Ben\\
4. & Kan   &   14. & Ix\\
5. & Chicchan & 15.& Men\\
6. & Cimi  &   16. & Cib\\
7. & Manik &   17. & Caban \\
8. & Lamat &   18. & Etznab \\
9. & Muluc &  19. & Ahab \\
10.& Oc  & 20. & Ahau\\  
\end{tabular}\]
What are the next 3 days after the day 11 Ahab in this system?
\vskip 10pt
\noindent
{\it Answer:}  They are 12 Ahau, 13 Imix, 1 Ik.
\end{enumerate}
\vskip 10pt
\noindent
IV.  Essay.  (40)  Explain in detail two examples of mathematical problems that 
Christopher Boone discusses in  {\it the curious incident of the dog in the night-time}.  
What do those problems mean for him?  More generally, what is the role of mathematics in 
his life?  Is it fair to say that he relates to the world in mathematical terms?  Why 
or why not?
\vskip 10pt
{\it Model Response:} (Note:  any two examples are OK, these are just the two that
stood out the most for me.)  Two examples of mathematical problems discussed by Christopher
Boone in {\it the curious incident of the dog in the night-time} are the ``Monty Hall''
problem and the problem about Pythagorean triples from the A-level examination he discusses 
in the Appendix.  The Monty Hall problem deals with a situation from the TV show ``Let's Make a
Deal."  The host, Monty Hall, would play a game with contestants on the show where three
doors concealed a car  and two worthless prizes like goats.   The contestant 
would select one of three doors, then Monty Hall would reveal one of the goats and
ask the contestant if they wanted to change their original choice.  The question was
whether it was better strategy to always change or to always stay with the original choice.
This question and a proposed solution that it was better to change appeared in the column
by Marilyn Vos Savant in the 1980's.  A number of people, including many professional mathematicians
disagreed.  But Christopher is able to reason correctly that switching doors is the better
strategy.  He explains that he likes this problem because it shows that mathematics is \emph{not}
just ``cut and dried'' and ``safe'' as his teacher Mr. Jeavons had said.  There are genuinely
surprising and difficult things in the subject.  But his (Christopher's) insight and skill allow
him to reason through to correct answers, and he feels powerful for that reason.

The second problem asks one to show that any triangle with sides of the form $(n^2 - 1, 2n, n^2 + 1)$
for some positive integer $n \ge 2$ must be a right triangle.  Christopher solves this very carefully
and completely by showing that the Pythagorean theorem equation follows:
$$(n^2 + 1)^2 = n^4 + 2n^2 + 1 = (n^2 - 1)^2 + (2n)^2$$
Since the relation $c^2 = a^2 + b^2$ implies the triangle has a right angle opposite the 
side of length $c$, this shows the triangle has a right angle.  This problem shows
how well Christopher understands basic secondary-school mathematics and that he 
really is ready to go on to study more advanced subjects.  His biggest success in
this novel is taking and passing this examination.  It again shows his comfort
and skill in dealing with mathematical ideas.

In both of these examples and in general, we can see that Christopher is most at home
thinking in mathematical terms--that is where his brain works best, where he is at
his most comfortable, and where he is most himself.  His first inclination is often to 
try to figure out situations where he finds himself by thinking in mathematical or 
logical terms.  So yes, it is pretty fair to say he relates to the world mostly in 
mathematical terms.  Part of what makes his life difficult is that he cannot do this
all of the time--situations like the impasse between his parents, or his inability 
to handle enviroments with too many stimuli don't allow him to use his strengths
in the same way.  
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