
diff -r 70306f6c65b1 -r fdc2c6fb7a24 cws/cw02.tex
--- a/cws/cw02.tex	Tue Nov 15 23:04:52 2016 +0000
+++ b/cws/cw02.tex	Tue Nov 15 23:08:09 2016 +0000
@@ -78,7 +78,7 @@
 knight's tour is \underline{not} closed (it is open) because the last
 step on field $(0, 4)$ is not within the reach of the first step on
 $(4, 4)$. It turns out there is no closed knight's tour on a $5\times
-5$ board. But there are on a $6\times 6$ board, for example
+5$ board. But there are on a $6\times 6$ board and bigger, for example
 
 \chessboard[maxfield=e5, 
             pgfstyle= {[base,at={\pgfpoint{0pt}{-0.5ex}}]text},
@@ -130,9 +130,9 @@
 \noindent
 where the 35th move can join up again with the 0th move.
 
-If you cannot remember how a knight moved in chess, or never played
+If you cannot remember how a knight moves in chess, or never played
 chess, below are all potential moves indicated for two knights, one on
-field $(2, 2)$ (blue) and another on $(7, 7)$ (red):
+field $(2, 2)$ (blue moves) and another on $(7, 7)$ (red moves):
 
 
 \chessboard[maxfield=g7,
@@ -148,23 +148,29 @@
 
 \subsection*{Part 1 (6 Marks)}
 
-We will implement the knight's tour problem such that we can change
-quickly the dimension of the chessboard. The fun with this problem is
-that even for small chessbord dimensions it has already an incredably
-large search space---finding a tour is like finding a needle in a
-haystack. In the first part we want to see far we get with
-exhaustively exploring the complete search space for small dimensions.
+You are asked to implement the knight's tour problem such that the
+dimension of the board can be changed.  Therefore most functions will
+take the dimension as an argument.  The fun with this problem is that
+even for small chessbord dimensions it has already an incredably large
+search space---finding a tour is like finding a needle in a
+haystack. In the first task we want to see far we get with
+exhaustively exploring the complete search space for small
+chessboards.\medskip
 
-Let us first fix the basic datastructures for the implementation.  A
-\emph{position} (or field) on the chessboard is a pair of integers. A
-\emph{path} is a list of positions. The first (or 0th move) in a path
-should be the last element in this list; and the last move is the
-first element. For example the path for the $5\times 5$ chessboard
-above is represented by
+\noindent
+Let us first fix the basic datastructures for the implementation.  The
+board dimension is an integer (we will never go boyond board sizes of
+$100 \times 100$).  A \emph{position} (or field) on the chessboard is
+a pair of integers, like $(0, 0)$. A \emph{path} is a list of
+positions. The first (or 0th move) in a path is the last element in
+this list; and the last move in the path is the first element. For
+example the path for the $5\times 5$ chessboard above is represented
+by
 
 \[
 \texttt{List($\underbrace{\texttt{(0, 4)}}_{24}$,
-  $\underbrace{\texttt{(2, 3)}}_{23}$, ..., (3, 2), $\underbrace{\texttt{(4, 4)}}_0$)}
+  $\underbrace{\texttt{(2, 3)}}_{23}$, ...,
+  $\underbrace{\texttt{(3, 2)}}_1$, $\underbrace{\texttt{(4, 4)}}_0$)}
 \]
 
 \noindent
@@ -179,51 +185,46 @@
 \begin{itemize}
 \item[(1a)] Implement a is-legal-move function that takes a dimension, a path
 and a position as argument and tests whether the position is inside
-the board and not yet element in the path.
+the board and not yet element in the path. \hfill[1 Mark]
 
 \item[(1b)] Implement a legal-moves function that calculates for a
-  position all legal follow-on moves. If the follow-on moves are
+  position all legal onward moves. If the onward moves are
   placed on a circle, you should produce them starting from
   ``12-oclock'' following in clockwise order.  For example on an
   $8\times 8$ board for a knight on position $(2, 2)$ and otherwise
-  empty board, the legal-moves function should produce the follow-on
+  empty board, the legal-moves function should produce the onward
   positions
 
   \begin{center}
   \texttt{List((3,4), (4,3), (4,1), (3,0), (1,0), (0,1), (0,3), (1,4))}
   \end{center}
 
-  If the board is not empty, then maybe some of the moves need to be filtered out from this list.
-  For a knight on field $(7, 7)$ and an empty board, the legal moves
-  are
+  in this order.  If the board is not empty, then maybe some of the
+  moves need to be filtered out from this list.  For a knight on field
+  $(7, 7)$ and an empty board, the legal moves are
 
   \begin{center}
   \texttt{List((6,5), (5,6))}
-  \end{center}  
+  \end{center}
+  \mbox{}\hfill[1 Mark]
 
 \item[(1c)] Implement two recursive functions (count-tours and
   enum-tours). They each take a dimension and a path as
-  arguments. They exhaustively search for \underline{open} tours
+  arguments. They exhaustively search for \underline{\bf open} tours
   starting from the given path. The first function counts all possible
   open tours (there can be none for certain board sizes) and the second
-  collects all open tours in a list of paths.
+  collects all open tours in a list of paths.\hfill[2 Marks]
 \end{itemize}
 
-\noindent \textbf{Test data:} For the marking, these functions will be
-called with board sizes up to $5 \times 5$. If you only search for
-open tours starting from field $(0, 0)$, there are 304 of them. If you
-try out every field of a $5 \times 5$-board as a starting field and
-add up all open tours, you obtain 1728. A $6\times 6$ board is already
-too large to search exhaustively: the number of open tours on
-$6\times 6$, $7\times 7$ and $8\times 8$ are
-
-\begin{center}
-\begin{tabular}{ll}  
-  $6\times 6$ & 6637920\\
-  $7\times 7$ & 165575218320\\
-  $8\times 8$ & 19591828170979904
-\end{tabular}
-\end{center}
+\noindent \textbf{Test data:} For the marking, the functions in (1c)
+will be called with board sizes up to $5 \times 5$. If you only search
+for open tours on $5 \times 5$ board starting from field $(0, 0)$,
+there are 304 of them. If you try out every field of a $5 \times
+5$-board as a starting field and add up all open tours, you obtain
+1728. A $6\times 6$ board is already too large to be searched
+exhaustively.\footnote{For your interest, the number of open tours on
+  $6\times 6$, $7\times 7$ and $8\times 8$ are 6637920, 165575218320,
+  19591828170979904, respectively.}
 
 \subsubsection*{Tasks (file knight2.scala)}
 
@@ -235,31 +236,38 @@
 
   \[
   \begin{array}{lcl}
-  first(\texttt{Nil}, f) & \dn & \texttt{None}\\  
-  first(x\!::\!xs, f) & \dn & \begin{cases}
+  \textit{first}(\texttt{Nil}, f) & \dn & \texttt{None}\\  
+  \textit{first}(x\!::\!xs, f) & \dn & \begin{cases}
     f(x) & \textit{if}\;f(x) \not=\texttt{None}\\
-    first(xs, f) & \textit{otherwise}\\
+    \textit{first}(xs, f) & \textit{otherwise}\\
                               \end{cases}
   \end{array}
   \]
 
+  \noindent That is, we want to find the first position where the
+  result of $f$ is not \texttt{None}.\newline\mbox{}\hfill[1 Mark]
+  
 \item[(2b)] Implement a first-tour function. Using the first-function
-  from (2a), search recursively for an open tour. Only use the field
-  $(0, 0)$ as a starting field of the tour. As there might not be such
-  a tour at all, the first-tour function needs to return an
-  \texttt{Option[Path]}. For the marking, this function will be called
-  with board sizes up to $8 \times 8$.
-\end{itemize}  
+  from (2a), search recursively for an open tour.  As there might not
+  be such a tour at all, the first-tour function needs to return an
+  \texttt{Option[Path]}.\hfill[2 Marks]
+\end{itemize}
+
+\noindent
+\textbf{Testing} The first tour function will be called with board
+sizes of up to $8 \times 8$. 
 
 
 \subsection*{Part 2 (4 Marks)}
 
-For open tours beyond $8 \times 8$ boards and also for searching for
-closed tours, an heuristic (called Warnsdorf's rule) needs to be
-implemented. This rule states that a knight is moved so that it
-always proceeds to the square from which the knight will have the
-fewest onward moves.  For example for a knight on field $(1, 3)$,
-the field $(0, 1)$ has the fewest possible onward moves.
+As you should have seen in Part 1, a naive search for open tours
+beyond $8 \times 8$ boards and also for searching for closed tours
+takes too much time. There is a heuristic (called Warnsdorf's rule)
+that can speed up finding a tour. This heuristice states that a knight
+is moved so that it always proceeds to the square from which the
+knight will have the \underline{fewest} onward moves.  For example for
+a knight on field $(1, 3)$, the field $(0, 1)$ has the fewest possible
+onward moves, namely 2.
 
 \chessboard[maxfield=g7,
             pgfstyle= {[base,at={\pgfpoint{0pt}{-0.5ex}}]text},
@@ -272,8 +280,8 @@
             setpieces={Na3}]
 
 \noindent
-Warnsdorf's rule states that the moves sould be tried out in the
-order
+Warnsdorf's rule states that the moves on the board above sould be
+tried out in the order
 
 \[
 (0, 1), (0, 5), (2, 1), (2, 5), (3, 4), (3, 2)




