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\leftline{MONT 109N -- Mathematics Across Cultures}
\leftline{Midterm Exam}
\leftline{March 25, 2011}
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\leftline{Your Name: $\underline{\hbox{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}}$}
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\noindent
{\bf Directions:}  Do all work on these sheets.  You may use the backs of the page if you need more space.  Note the table of information provided on the accompanying sheet.   There are 100 total points.  
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\noindent
I.  In the Mayan calendar system, today is the day {\it 4 Akbal 16 Cumku} in the Calendar Round 
and 12.19.18.4.3 (in baktuns, katuns, tuns, uinals, days) in the Long Count.
\begin{enumerate}
\item[A.] (15) What day will it be three of our weeks from today (on Friday, April 15)?  
Show all calculations needed to determine this. 
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\item[B.] (10)  What will the Long Count date be three weeks from today?
\end{enumerate}
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\noindent
II.  (15) Suppose you are playing {\it mancala} and it is your turn.  
Your 6 pits contain numbers of stones left to right 
as follows:  0,5,3,0,0,0.  Is it possible to
get all 8 stones to your store before the other player gets to play?  If so,
show how to do it, step by step.  If not, explain why not.
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\noindent
IV. Short Answer (choose any 6 for 30 total points -- if you answer more than 6, only 
the best 6 scores will be used).
\begin{enumerate}
\item[A.] (5)  What base did the Maya use in their number system?
How would a Classic Maya scribe have represented the number $151$?  
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\item[B.] (5) The role of {\it mancala} games in African cultures is often 
compared to the role of what game in European cultures?  
In what ways could it be said that these games are similar?  
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\item[C.] (5)  What was the actual information represented in the {\it mettang}
``stick charts'' used by the Marshall Islanders in their culture?  
Were they {\it maps} in our sense?
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\item[D.] (5)  What is the name of the medieval palace in Spain 
whose decorations stimulated M.C. Escher's interest in periodic patterns and near 
which city is it located?
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\item[E.] (5)  In what modern-day countries are most of the sites of Classic Maya 
cities located?  Approximately when did the collapse of the Classic Maya civilization occur?
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\item[F.] (5)  What are some of the common features shared by the Classic Maya collapse 
and the collapse of the Easter Island culture?
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\item[G.] (5)  What was the meaning of the numerical information in the
section of the Maya {\it Dresden Codex} that we studied in Problem Set 1?  
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\item[H.] (5)  Give two examples of transformations
that are symmetries of the pattern in the Escher drawing reproduced on 
the next page.  (Imagine the pattern extended to fill out the whole plane 
of that sheet.)  Are you taking the shading (coloring) into account?  Explain.  
\end{enumerate}
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\noindent
IV.  Essay.  (30)  What is the mathematical meaning of {\it symmetry}?
What examples of symmetry transformations have we used in discussing examples from
visual art in this course?  What is the mathematical concept of a {\it group}
and why is it important for the study of symmetries?  Describe how
symmetry groups can be used to classify plane periodic patterns.  In what 
sense are two patterns of the same type similar?  In what ways can they
be different?
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\noindent
{\it The ``Months'' in the Maya Sacred Round}
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\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}                   
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\noindent
{\it The ``Months'' in the Maya Vague Year}
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\begin{tabular}{|c|l|c|l}
  1. & Pop  & 10. & Yax\\
  2. & Uo   & 11. & Zak\\
  3. & Zip  & 12. & Ceh\\
  4. & Zotz & 13. & Mac\\
  5. & Zec  & 14. & Kankin\\
  6. & Xul  & 15. & Muan\\
  7. & Yaxkin & 16. & Pax\\
  8. & Mol  & 17. & Kayab\\
  9. & Chen & 18. & Cumku\\
     &      & 19. & Uayeb\\
\end{tabular}
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