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\centerline{MONT 109N -- Mathematics Across Cultures}
\centerline{Discussion/Problem Set 4 -- Chinese Mathematics}
\centerline{March 30, 2011}
\vskip 10pt
\noindent
{\it Background and Assignment}
\vskip 10pt
In this assignment, we want to look at a sample of Chinese
mathematics starting from the time of the {\it Zhou Bi} 
(which was perhaps collected in written form as early as 500 B.C.E.) 
almost to the beginning of the Ming dynasty.  Individual writeups are
due in class on Monday, April 4.  
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\noindent
{\it Questions}
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\noindent
A.  The {\it Zhou Bi} contained a diagram like the following one, including four
right triangles arranged around a small square to make a larger square, and 
illustrating a relationship the Chinese knew as the {\it gou gu} theorem.

\begin{center}
\begin{picture}(100,100)
\put(45,55){\line(1,0){55}}
\put(55,55){\line(0,-1){55}}
\put(55,45){\line(-1,0){55}}
\put(45,45){\line(0,1){55}}
\put(0,45){\line(9,11){45}}
\put(45,100){\line(11,-9){55}}
\put(100,55){\line(-9,-11){45}}
\put(55,0){\line(-11,9){55}}
\put(13,72){$c$}
\put(21,49){$a$}
\put(34,64){$b$}
\end{picture}
\end{center}
\noindent 
(The labels $a,b,c$ are in our Roman alphabet, of course.  The original Chinese
diagram had Chinese characters, and a square grid overlaid on this picture.)
\begin{enumerate}
\item[1)]  What do {\it we} call the {\it gou gu} theorem?
\item[2)]  How does this diagram essentially furnish a {\it proof} of that theorem?
(Note: It is thought by many historians that the mathematics presented in the
{\it Zhou Bi} goes back to the time of the Greek ``Dark Age'' between the 
end of the Bronze Age and the start of the Archaic Period.)  
\end{enumerate}
\bigskip
\noindent
B.  The {\it Jiu Zhang} (The ``Nine Chapters in Mathematical Arts'')
from the Han period (200 B.C.E. to 222 C.E.)
contained problems similar to the following one:
{\it Three sheaves of a good crop and 2 sheaves of a bad crop are sold
for 39 {\it dou} (a unit of money).  Two sheaves of a good crop and one sheaf
of a bad crop are sold for 25 {\it dou}.  What are the prices for a sheaf
of the good crop and the bad crop?}
\begin{enumerate}
\item[1)]  Write the given information as a pair of equations for $x = $ price
per sheaf of the good crop and $y = $ price for a sheaf of the bad crop.
Solve your equations by modern algebraic methods.
\item[2)]  The solution method for this type of problem 
described in the {\it Jiu Zhang}  worked like this.  (They used the {\it heng/zong}
counting rod numerals, but we will use our Hindu-Arabic numerals for simplicity.)
Lay out a rectangular array of numbers representing the
given information like this:
\[
\begin{array}{ccc}
3 & 2 & 39\\
2 & 1 & 25
\end{array}  
\]
Then, multiply every element on the second row by $2$ and subtract from the corresponding
entry on the first row to get a new array:
\[
\begin{array}{ccc}
-1 & 0 & -11\\
2 & 1 & 25
\end{array}  
\]
Multiply every entry on the first row by 2 and add to the corresponding entry on 
the second row:
\[
\begin{array}{ccc}
-1 & 0 & -11\\
0 & 1 & 3
\end{array}  
\]
Explain how and why this process gives the solution you found in part (1).
{\it Note:}  Over time, by the Song and Ming dynasties, the Chinese 
were able to apply similar methods to much larger systems of 
equations (examples exist of up to 10 equations in 10 unknowns).  
The algebra is more tedious but the idea is essentially the same -- represent
the equations in a rectangular array like this, and  
combine rows in a systematic way to make enough zeroes 
in the table so that you can ``read off'' the solutions.  
(This was only rediscovered in Europe around the beginning of the 
19th century!)
\item[3)]  Have you seen an equivalent way to solve systems
of linear equations like this in an algebra class?   Note: I believe 
the latest standards for 10th grade mathematics in Massachusetts include
this topic (!)  
\end{enumerate}
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\noindent
C.  Another problem from the {\it Jiu Zhang} (and repeated later in a book
of Yang Hui, 1261 C.E.). -- {\it There is a bamboo 10 {\it chang} (a unit of length) high.
The bamboo is broken so the upper end upper end reaches the ground 3 {\it chang}
from the stem.  Find the height of the break.}  See the top illustration on the last page
for the way the problem appeared in Yang Hui.
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\noindent
D.  (Magic squares)  The Chinese were very taken with {\it magic squares} and
similar arrangements of numbers.  The following method was learned by a French envoy
in present-day Thailand in 1687 C.E., but it is thought that the method was developed
first in China.  A {\it normal $n$th order magic square} is an $n\times n$
table consisting of the numbers $1,2,3,\ldots,n^2$ arranged so that 
the sum of the elements in each row and each column of the table is 
the same number (which must be $n(n^2 +1)/2$ in this case).  Here is an example
of the method using $n = 5$.  We first draw a square array of $25$ boxes in 
$5$ rows and $5$ columns.  An additional row is placed along the top and the 
right side.  The square at the top right of the extended table is not used.
Starting with $n = 1$ in the middle square on the top row of the 5, put the 
numbers $1,2,3,\ldots,25$ in order going one step to the right and one up 
each time.

\begin{itemize}
\item If that ``move'' takes you to the row on the top, ``wrap back'' to the 
bottom row and continue from there.
\item If the ``move'' takes you to the column on the right, ``wrap back''
to the left-most column.
\item If the move runs into a previously filled square (or past the upper
right corner), drop down one square instead and continue from there.  (The $X$
on the top row is a reminder not to fill that space.)
\end{itemize}

This gives the following table:
\[
\begin{tabular}{c|c|c|c|c|c}
   & 18 & 25 & 2 & 9 &  X \\ \hline
17 & 24 & 1 & 8 & 15 & 17\\ \hline
23 & 5 & 7 & 14 & 16 & 23\\ \hline
4 & 6 & 13 & 20 & 22 & 4\\ \hline
10 & 12 & 19 & 21 & 3 & 10\\ \hline
11 & 18 & 25 & 2 & 9
\end{tabular}
\]
The magic square is then obtained by deleting the ``extra'' top and right 
entries:
\[
\begin{tabular}{c|c|c|c|c}
17 & 24 & 1 & 8 & 15 \\ \hline
23 & 5 & 7 & 14 & 16 \\ \hline
4 & 6 & 13 & 20 & 22 \\ \hline
10 & 12 & 19 & 21 & 3\\ \hline
11 & 18 & 25 & 2 & 9
\end{tabular}
\]
\begin{enumerate}
\item[1.]  Check that that this is indeed a normal magic square of order $5$.
\item[2.]  The same method can be used to produce a normal magic square of
order $n$ for any odd $n$.  Use it to produce a normal magic square of order
$7$.
\item[3.]  (Extra Credit)  Can you {\it prove} why this method works? 
\end{enumerate}
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\noindent
E.  The diagram on the bottom of the last page appeared in the {\it Si Yuan Yu Jian}
of Zhu Shijie (from 1303 C.E.) Each circle contains a number written in {\it heng/zong}
form.  The small circles are zeroes (a late development in Chinese mathematics
that can be traced to contacts with India).  
\begin{enumerate}
\item[1)] Transcribe all of the numbers into our
Hindu-Arabic numerals.   (Note:  Some of the orientations of the {\it hong/zeng} symbols,
and the alternation between {\it hongs} and {\it zengs},
are slightly different from the description in Joseph's book; try to figure them out
from ``internal evidence'')
\item[2)]  What is this table (what are the numbers)?  
\item[3)]  What European mathematician/philosopher is credited with finding this pattern
in histories of mathematics in Europe and when did he live?  Also, do you know why was he 
interested in these numbers?  
\end{enumerate}
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