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\noindent
Hope Garcia and Margot Reed -- Applications of Latin Squares
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This is generally a good discussion of Latin squares and their
applications.  However, there are a few things that are either
not explained very clearly, or where I'm not sure you completely
understood what the point of the source discussion you were
looking at was (see in particular the comments below).  Writing about
 mathematics is different (and harder) than other sorts of writing 
because you have to be careful to state things so precisely.

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\noindent
{\it Specific Comments}
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\item{1)}  Pages 1, 2, 7, 8--The distinction between magic squares and Latin
squares is not very clear from what you write in the first historical discussion
(you fixed this in the presentation and later on page 6, but not here).  They're not
the same thing because {\it all} the entries in a magic square are distinct (usually
the integers $1,2,\ldots,n^2$).  In Latin squares, on the other hand, the
same $n$ entries are reused on every row.   As you discuss later, though, Latin squares and
magic squares are very closely related (as in the paper by Euler that
I sent you after the run-through of your talk).   You discuss one way he mentioned on
your page 8.  But there are other ways too that do not require the Latin squares
used in the construction to be ``diagonal'' Latin squares.  Here's another construction:
For instance, if $n = 5$ and 
$$L = \pmatrix{ a & b & c & d & e\cr
                b & c & d & e & a\cr
                c & d & e & a & b\cr
                d & e & a & b & c\cr
                e & a & b & c & d\cr}
\qquad {\rm and} \qquad  L' = 
\pmatrix{ E & D & C & B & A\cr
          A & E & D & C & B\cr
          B & A & E & D & C\cr
          C & B & A & E & D\cr
          D & C & B & A & E\cr}$$
are any Latin square matrices with $a + b + c + d + e = 5e$ and $A + B + C + D + E = 5E$, 
and $\{a,b,c,d,e\} = \{1,2,3,4,5\}$, $\{A,B,C,D,E\} = \{0,1,2,3,4\}$, then $L$ and $L'$
are mutually orthogonal Latin squares ($L$ is of the cyclic type and $L'$ is 
cyclic but in reverse order on each row), 
and moreover the linear combination
$L + 5L'$ (using arithmetic in ${\bf Z}$)  is a {\it magic square}.  
Something similar works for any odd $n$.  (This is a nice exercise -- try it!)  
Euler also discussed yet other, different
constructions.
\item{2)}  Page 3--It would be better to say that every row (and every column)
of a Latin square is a permutation of the set $S$ of symbols.  The permutations
corresponding to different rows can be different, so your statement "Latin Squares 
are examples of permutations of the n elements contained in S" is not exactly
right.   
\item{3)}  Page 3 -- I don't quite understand what you are saying about
the generalized pigeonhole principle and Latin squares toward the bottom of the page.  
It's true that $k = m$ means there don't have to be any duplicates (not 
necessarily any pigeonholes getting more than one pigeon).  But it's not 
actually that complicated.  If you have $|S| = n$, then each element of $S$ has to 
appear exactly once in each row and column of an $n\times n$ Latin square 
(same comment about the corresponding place in the presentation -- it was not
clear there either).
\item{4)}  Page 4 -- In the definition of mutually orthogonal Latin squares,
what you mean by ``the $n^2$ pairs'' isn't too clear because you haven't
described where the pairs come from.  The idea, of course, is that you 
want to ``overlay'' or take the table of pairs of elements in corresponding
positions.  For instance with the Latin squares $L$ and $L'$ from comment
1 above, you would have 
$$L\times L' = \pmatrix{ (a,E) & (b,D) & (c,C) & (d,B) & (e,A)\cr
                         (b,A) & (c,E) & (d,D) & (e,C) & (a,B)\cr
                         (c,B) & (d,A) & (e,E) & (a,D) & (b,C)\cr
                         (d,C) & (e,B) & (a,A) & (b,E) & (c,D)\cr
                         (e,D) & (a,C) & (b,B) & (c,A) & (d,E)\cr}$$
\item{5)}  Page 5 -- the ``banquet'' example you give is OK, but it 
presupposes that the department is evenly split into four groups 
of equal sizes in both the breadth area and the length of service
components.  There is also another aspect
that isn't really addressed by the Latin square structure and that's who
sits with whom.  The orthogonal Latin squares just deal with the combinations
of breadth area and years of experience, not with the actual seating arrangements
for the banquet.  I'm not seeing how you are making up the tables for the 
banquet, or am I misunderstanding what you meant?  
\item{6)}  Page 6 -- the $xx$ in Euler's notation is what we would 
write as $x^2$ :)  This was a hold-over from the days before the introduction
of numerical exponents to represent powers of numbers. 
\item{7)}  Page 11 -- the Sudoku X, Even-Odd Sudoku, and Colored Sudoko variations
are interesting.  I think they are essentially different ways of presenting the
known information to the puzzle solver that probably allow fewer of the squares
to be given at the start.  For the puzzle designer, they represent additional
constraints on the construction of the puzzle.   
\item{8)}  Page 12 -- I really don't understand what you are saying here
about Sudoku solution algorithms.   
A "rejection algorithm" based on randomly generating permutations seems much too
brute-force to be practicable as a way of solving Sudoku puzzles.  That's not
what you described in your presentation either, so I'm not sure whether you 
really understand what this algorithm is saying.  Note that in the presentation,
you were analyzing the constraints on the unknown squares in a row or column 
based on the squares already known in a row or column, which is a much smarter 
approach than randomly sampling permutations to see if the Sudoku or Latin
square condition is satisfied.  Especially
after the puzzle is partially filled in, there is a lot of extra information
about excluded potential solutions that can be exploited.  
\item{9)}  The definition of quasi-complete Latin squares you give is
confusing.  I think the better way to say it is that every unordered
pair ${p,q}$ of symbols appear in {\it adjacent} squares either as $p,q$
or as $q,p$ twice in the rows and twice in the columns.  You said 
``quasi-complete Latin squares are defined as pairs of adjacent entries, 
$L_{i,j}$ and $L_{i,j+1}$, and $L_{i, j}$ and $L{i+1,j}$,
that can appear in either order, twice in rows and twice in columns."  That
does not mean the same thing even though it includes (most of) the same ``ingredients.''
The $L_{i,j}$ and $L_{i,j+1}$, and $L_{i, j}$ and $L{i+1,j}$ are just what
we mean by adjacent elements in a row or a column.  It's not that "quasi-complete 
Latin squares are defined as pairs of adjacent entries ... that can appear in either
order ... ." It's that a Latin square is quasi-complete if every unordered pair
does appear twice in the rows and twice in the columns  
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\noindent
Final Project Grade Computation
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   Bibliography:  10/10
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   Paper:         54/60
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   Presentation:  27/30
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   Total:         91/100


