# HG changeset patch
# User Christian Urban <urbanc@in.tum.de>
# Date 1479307038 0
# Node ID 9891c9fac37eaa6d62aca12b6d1bcc9fd21fbdba
# Parent  fdc2c6fb7a2469ee6fa66f8a03c6b0eeb9c9e4ea
updated

diff -r fdc2c6fb7a24 -r 9891c9fac37e cws/cw01.pdf
Binary file cws/cw01.pdf has changed
diff -r fdc2c6fb7a24 -r 9891c9fac37e cws/cw02.pdf
Binary file cws/cw02.pdf has changed
diff -r fdc2c6fb7a24 -r 9891c9fac37e cws/cw02.tex
--- a/cws/cw02.tex	Tue Nov 15 23:08:09 2016 +0000
+++ b/cws/cw02.tex	Wed Nov 16 14:37:18 2016 +0000
@@ -16,13 +16,23 @@
 
 \section*{Coursework 7 (Scala, Knight's Tour)}
 
-This coursework is about searching and backtracking, and worth
-10\%. The first part is due on 23 November at 11pm; the second, more
-advanced part, is due on 30 November at 11pm. You are asked to
-implement Scala programs that solve various versions of the
+This coursework is worth 10\%. It is about searching and
+backtracking. The first part is due on 23 November at 11pm; the
+second, more advanced part, is due on 30 November at 11pm. You are
+asked to implement Scala programs that solve various versions of the
 \textit{Knight's Tour Problem} on a chessboard. Make sure the files
 you submit can be processed by just calling \texttt{scala
-  <<filename.scala>>}.
+  <<filename.scala>>}.\bigskip
+
+\noindent
+\textbf{Important:} Do not use any mutable data structures in your
+submissions! They are not needed. This excluded the use of
+\texttt{ListBuffer}s, for example. Do not use \texttt{return} in your
+code! It has a different meaning in Scala, than in Java.  Feel free to
+copy any code you need from files \texttt{knight1.scala},
+\texttt{knight2.scala} and \texttt{knight3.scala}. Make sure the
+functions you submit are defined on the ``top-level'' of Scala, not
+inside a class or object.
  
 \subsection*{Disclaimer}
 
@@ -78,7 +88,8 @@
 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 and bigger, for example
+5$ board. But there are on a $6\times 6$ board and on bigger ones, for
+example
 
 \chessboard[maxfield=e5, 
             pgfstyle= {[base,at={\pgfpoint{0pt}{-0.5ex}}]text},
@@ -146,15 +157,15 @@
             markfields={f5, e6},
             setpieces={Ng7, Nb2}]
 
-\subsection*{Part 1 (6 Marks)}
+\subsection*{Part 1 (7 Marks)}
 
 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
+take the dimension of the board 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 how far we get
+with exhaustively exploring the complete search space for small
 chessboards.\medskip
 
 \noindent
@@ -183,9 +194,10 @@
 \subsubsection*{Tasks (file knight1.scala)}
 
 \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. \hfill[1 Mark]
+\item[(1a)] Implement an 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. \hfill[1 Mark]
 
 \item[(1b)] Implement a legal-moves function that calculates for a
   position all legal onward moves. If the onward moves are
@@ -193,15 +205,15 @@
   ``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 onward
-  positions
+  positions in this order:
 
   \begin{center}
   \texttt{List((3,4), (4,3), (4,1), (3,0), (1,0), (0,1), (0,3), (1,4))}
   \end{center}
 
-  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
+  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))}
@@ -210,16 +222,16 @@
 
 \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{\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
+  arguments. They exhaustively search for {\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.\hfill[2 Marks]
 \end{itemize}
 
 \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
+will be called with board sizes up to $5 \times 5$. If you search
+for open tours on a $5 \times 5$ board starting only from field $(0, 0)$,
+there are 304 of tours. 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
@@ -231,8 +243,9 @@
 \begin{itemize}
 \item[(2a)] Implement a first-function. This function takes a list of
   positions and a function $f$ as arguments. The function $f$ takes a
-  position as argument and produces an optional path. The idea behind
-  the first-function is as follows:
+  position as argument and produces an optional path. So its type is
+  \texttt{Pos => Option[Path]}. The idea behind the first-function is
+  as follows:
 
   \[
   \begin{array}{lcl}
@@ -245,12 +258,12 @@
   \]
 
   \noindent That is, we want to find the first position where the
-  result of $f$ is not \texttt{None}.\newline\mbox{}\hfill[1 Mark]
+  result of $f$ is not \texttt{None}, if there is one.\mbox{}\hfill[1 Mark]
   
-\item[(2b)] Implement a first-tour function. Using the first-function
-  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]
+\item[(2b)] Implement a first-tour function that uses the
+  first-function from (2a), and searches 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
@@ -258,13 +271,13 @@
 sizes of up to $8 \times 8$. 
 
 
-\subsection*{Part 2 (4 Marks)}
+\subsection*{Part 2 (3 Marks)}
 
 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
+beyond $8 \times 8$ boards and also 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
+that can speed up finding a tour. This heuristic states that a knight
+is moved so that it always proceeds to the field 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.
@@ -281,7 +294,7 @@
 
 \noindent
 Warnsdorf's rule states that the moves on the board above sould be
-tried out in the order
+tried in the order
 
 \[
 (0, 1), (0, 5), (2, 1), (2, 5), (3, 4), (3, 2)
@@ -295,11 +308,20 @@
 \subsubsection*{Tasks (file knight3.scala)}
 
 \begin{itemize}
-\item[(3a)] orderered-moves
+\item[(3a)] Write a function ordered-moves that calculates a list of
+  onward moves like in (1b) but orders them according to the
+  Warnsdorf’s rule. That means moves with the fewest legal onward moves
+  should come first (in order to be tried out first).
+  
+\item[(3b)] Implement a first-closed-tour-heuristic function that searches for a
+  \textbf{closed} tour on a $6\times 6$ board. It should use the
+  first-function from (2a) and tries out onwards moves according to
+  the ordered-moves function from (3a). It is more likely to find
+  a solution when started in the middle of the board (that is
+  position $(dimension / 2, dimension / 2)$).
 
-\item[(3b)] first-closed tour heuristics; up to $6\times 6$
-
-\item[(3c)] first tour heuristics; up to $50\times 50$
+\item[(3c)] Implement a first-tour-heuristic function for boards up to $50\times 50$.
+  It is the same function  as in (3b) but searches for \textbf{open} tours.
 \end{itemize}  
 
 \end{document}
diff -r fdc2c6fb7a24 -r 9891c9fac37e progs/collatz_sol2.scala
--- a/progs/collatz_sol2.scala	Tue Nov 15 23:08:09 2016 +0000
+++ b/progs/collatz_sol2.scala	Wed Nov 16 14:37:18 2016 +0000
@@ -15,7 +15,7 @@
 
 
 // an alternative that calculates the steps directly
-def collatz1(n: Long): Int =
+def collatz1(n: Long): Long =
   if (n == 1) 1 else
     if (n % 2 == 0) (1 + collatz1(n / 2)) else 
       (1 + collatz1(3 * n + 1))
@@ -37,13 +37,15 @@
 //     upto 1 million. 
 
 def collatz_max(bnd: Long): (Long, Long) = {
-  (1L to bnd).view.map((i) => (collatz2(i, 1), i)).maxBy(_._1)
+  (1L to bnd).view.map((i) => (collatz1(i), i)).maxBy(_._1)
 }
 
 
 // some testing harness
 //val bnds = List(10, 100, 1000, 10000, 100000, 1000000)
-val bnds = List(10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000)
+val bnds = List(10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 2000000000)
+
+
 
 for (bnd <- bnds) {
   val (steps, max) = collatz_max(bnd)
diff -r fdc2c6fb7a24 -r 9891c9fac37e progs/knight1.scala
--- a/progs/knight1.scala	Tue Nov 15 23:08:09 2016 +0000
+++ b/progs/knight1.scala	Wed Nov 16 14:37:18 2016 +0000
@@ -1,76 +1,36 @@
-import scala.util._
+// Part 1 about finding and counting Knight's tours
+//==================================================
 
-class Computation[A,B](value: A, function: A => B) {
-  lazy val result = function(value)
-}
+type Pos = (Int, Int)    // a position on a chessboard 
+type Path = List[Pos]    // a path...a list of positions
+
+//(1a) Complete the function that tests whether the position 
+// is inside the board and not yet element in the path.
+
+def is_legal(dim: Int, path: Path)(x: Pos): Boolean = ...
 
 
-def print_board(n: Int)(steps: List[(Int, Int)]): Unit = {
-  println
-  for (i <- 0 until n) {
-    for (j <- 0 until n) {
-      print(f"${steps.indexOf((i, j))}%3.0f ")
-    }
-    println
-  } 
-}
-
-def add_pair(x: (Int, Int))(y: (Int, Int)) = 
-  (x._1 + y._1, x._2 + y._2)
-
-def is_legal(n: Int)(x: (Int, Int)) = 
-  0 <= x._1 && 0 <= x._2 && x._1 < n && x._2 < n
-
-def moves(n: Int)(steps: List[(Int, Int)])(x: (Int, Int)): List[(Int, Int)] = {
-  List((1, 2),(2, 1),(2, -1),(1, -2),
-       (-1, -2),(-2, -1),(-2, 1),(-1, 2)).map(add_pair(x)).filter(is_legal(n)).filterNot(steps.contains(_))
-}
-
-def ordered_moves(n: Int)(steps: List[(Int, Int)])(x : (Int, Int)): List[(Int, Int)] = 
-  moves(n)(steps)(x).sortBy(moves(n)(steps)(_).length)
-
-moves(8)(Nil)(1,3)
-ordered_moves(8)(Nil)(1,3)
-ordered_moves(8)(List((2, 4), (2, 6)))(1,3)
+//(1b) Complete the function that calculates for a position 
+// all legal onward moves that are not already in the path. 
+// The moves should be ordered in a "clockwise" order.
+ 
+def legal_moves(dim: Int, path: Path, x: Pos): List[Pos] = ...
 
-def first[A, B](xs: List[A], f: A => Set[B]): Set[B] = xs match {
-  case Nil => Set()
-  case x::xs => {
-    val result = f(x)
-    if (result == Set()) first(xs, f) else result
-  }
-}
-
-// non-circular tour
-def tour(n: Int)(steps: List[(Int, Int)]): Option[List[(Int, Int)]] = {
-  if (steps.length ==  n * n) Some(steps) 
-  else 
-    { val list = moves(n)(steps)(steps.head) map (x => new Computation(x, ((x:(Int, Int)) => tour(n)(x::steps))))
-      val found = list.par find (_.result.isDefined)
-      found map (_.result.get)
-    }
-}
-
-val n = 6
-println(s"simple tour: n = $n")
-
-val starts = for (i <- (0 until n).toList; 
-                  j <- (0 until n).toList) yield new Computation ((i, j), ((x:(Int, Int)) => tour(n)(x::Nil)))
-
-val found = starts.par find (_.result.isDefined)
-print_board(n)((found map (_.result.get)).get)
-
-//for measuring time
-def time_needed[T](i: Int, code: => T) = {
-  val start = System.nanoTime()
-  for (j <- 1 to i) code
-  val end = System.nanoTime()
-  (end - start)/(i * 1.0e9)
-}
-
-//for (i <- 1 to 20) {
-//  println(i + ": " + "%.5f".format(time_needed(2, matches(EVIL1(i), "a" * i))))
-//}
+//assert(legal_moves(8, Nil, (2,2)) == 
+//  List((3,4), (4,3), (4,1), (3,0), (1,0), (0,1), (0,3), (1,4)))
+//assert(legal_moves(8, Nil, (7,7)) == List((6,5), (5,6)))
+//assert(legal_moves(8, List((4,1), (1,0)), (2,2)) == 
+//  List((3,4), (4,3), (3,0), (0,1), (0,3), (1,4)))
+//assert(legal_moves(8, List((6,6)), (7,7)) == List((6,5), (5,6)))
 
 
+//(1c) Complement the two recursive functions below. 
+// They exhaustively search for open tours starting from the 
+// given path. The first function counts all possible open tours, 
+// and the second collects all open tours in a list of paths.
 
+def count_tours(dim: Int, path: Path): Int = ...
+
+def enum_tours(dim: Int, path: Path): List[Path] = ...
+
+
diff -r fdc2c6fb7a24 -r 9891c9fac37e progs/knight1_sol.scala
--- a/progs/knight1_sol.scala	Tue Nov 15 23:08:09 2016 +0000
+++ b/progs/knight1_sol.scala	Wed Nov 16 14:37:18 2016 +0000
@@ -27,9 +27,12 @@
 def legal_moves(dim: Int, path: Path, x: Pos): List[Pos] = 
   moves(x).filter(is_legal(dim, path))
 
-legal_moves(8, Nil, (2,2))
-legal_moves(8, Nil, (7,7))
-
+assert(legal_moves(8, Nil, (2,2)) == 
+  List((3,4), (4,3), (4,1), (3,0), (1,0), (0,1), (0,3), (1,4)))
+assert(legal_moves(8, Nil, (7,7)) == List((6,5), (5,6)))
+assert(legal_moves(8, List((4,1), (1,0)), (2,2)) == 
+  List((3,4), (4,3), (3,0), (0,1), (0,3), (1,4)))
+assert(legal_moves(8, List((6,6)), (7,7)) == List((6,5), (5,6)))
 
 def count_tours(dim: Int, path: Path): Int = {
   if (path.length == dim * dim) 1
diff -r fdc2c6fb7a24 -r 9891c9fac37e progs/knight2.scala
--- a/progs/knight2.scala	Tue Nov 15 23:08:09 2016 +0000
+++ b/progs/knight2.scala	Wed Nov 16 14:37:18 2016 +0000
@@ -1,78 +1,21 @@
-
-type Pos = (Int, Int)
-type Path = List[Pos]
-
-def print_board(dim: Int, path: Path): Unit = {
-  println
-  for (i <- 0 until dim) {
-    for (j <- 0 until dim) {
-      print(f"${path.indexOf((i, j))}%3.0f ")
-    }
-    println
-  } 
-}
-
+// Part 2 about finding a single tour for a board
+//================================================
 
-def add_pair(x: Pos)(y: Pos): Pos = 
-  (x._1 + y._1, x._2 + y._2)
-
-def is_legal(dim: Int, path: Path)(x: Pos): Boolean = 
-  0 <= x._1 && 0 <= x._2 && x._1 < dim && x._2 < dim && !path.contains(x)
+// copy any function you need from file knight1.scala
 
-def moves(x: Pos): List[Pos] = {
-  List(( 1,  2),( 2,  1),( 2, -1),( 1, -2),
-       (-1, -2),(-2, -1),(-2,  1),(-1,  2)).map(add_pair(x))
-}
-
-def legal_moves(dim: Int, path: Path, x: Pos): List[Pos] = 
-  moves(x).filter(is_legal(dim, path))
-
+type Pos = (Int, Int)    // a position on a chessboard 
+type Path = List[Pos]    // a path...a list of positions
 
 
-// non-circle tours
-/*
-def tour(dim: Int, path: List[Pos]): List[List[Pos]] = {
-  if (path.length ==  dim * dim) // && moves(n)(path.head).contains(path.last)) 
-    List(path)
-  else 
-    (for (x <- legal_moves(dim, path, path.head)) yield tour(dim, x::path)).flatten
-}
-*/
+//(2a) Implement a first-function that finds the first 
+// element, say x, in the list xs where f is not None. 
+// In that case return f(x), otherwise none.
 
-def tour(dim: Int, path: Path): Int = {
-  if (path.length == dim * dim) 1
-  else 
-    (for (x <- legal_moves(dim, path, path.head) yield tour(dim, x::path))).sum
-}
-
-
-def dtour(dim: Int): List[List[Pos]] = {
-  var counter = 100000000
+def first(xs: List[Pos], f: Pos => Option[Path]): Option[Path] = ...
 
-  def etour(dim: Int, path: List[Pos]): List[List[Pos]] = {
-    counter = counter - 1
-    if (counter <= 0) List() else
-      if (path.length == dim * dim) List(path)
-      else 
-        (for (x <- legal_moves(dim, path, path.head)) yield etour(dim, x::path)).flatten
-  }
-
-  (for (i <- (0 until dim).toList; 
-        j <- (0 until dim).toList) yield etour(dim, List((i, j)))).flatten
-}
-
-
+//(2b) Implement a function that uses the first-function for
+// trying out onward moves, and searches recursively for an 
+// *open* tour on a dim * dim-board.
 
-//val n = 8
-val n = 5
-println(s"number simple tours: n = $n")
-
-//println(etour(n, List((0, 0))).size)
-
-
-
-for (d <- 9 to 9) {
-  println(s"${d} x ${d} " + dtour(d).length)
-}
-
-
+def first_tour(dim: Int, path: Path): Option[Path] = ...
+ 
diff -r fdc2c6fb7a24 -r 9891c9fac37e progs/knight2_sol.scala
--- a/progs/knight2_sol.scala	Tue Nov 15 23:08:09 2016 +0000
+++ b/progs/knight2_sol.scala	Wed Nov 16 14:37:18 2016 +0000
@@ -1,9 +1,8 @@
 // Part 2 about finding a single tour for a board
 //================================================
 
-
-type Pos = (Int, Int)
-type Path = List[Pos]
+type Pos = (Int, Int)    // a position on a chessboard 
+type Path = List[Pos]    // a path...a list of positions
 
 def print_board(dim: Int, path: Path): Unit = {
   println
diff -r fdc2c6fb7a24 -r 9891c9fac37e progs/knight3.scala
--- a/progs/knight3.scala	Tue Nov 15 23:08:09 2016 +0000
+++ b/progs/knight3.scala	Wed Nov 16 14:37:18 2016 +0000
@@ -1,45 +1,65 @@
-import scala.util._
+// Part 3 about finding a single tour using the Warnsdorf Rule
+//=============================================================
+
 
-def print_board(n: Int)(steps: List[(Int, Int)]): Unit = {
-  for (i <- 0 until n) {
-    for (j <- 0 until n) {
-      print(f"${steps.indexOf((i, j))}%3.0f ")
+type Pos = (Int, Int)
+type Path = List[Pos]
+
+def print_board(dim: Int, path: Path): Unit = {
+  println
+  for (i <- 0 until dim) {
+    for (j <- 0 until dim) {
+      print(f"${path.reverse.indexOf((i, j))}%3.0f ")
     }
     println
   } 
-  //readLine()
-  System.exit(0)
-}
-
-def add_pair(x: (Int, Int))(y: (Int, Int)) = 
-  (x._1 + y._1, x._2 + y._2)
-
-def is_legal(n: Int)(x: (Int, Int)) = 
-  0 <= x._1 && 0 <= x._2 && x._1 < n && x._2 < n
-
-def moves(n: Int)(x: (Int, Int)): List[(Int, Int)] = {
-  List((1, 2),(2, 1),(2, -1),(1, -2),
-       (-1, -2),(-2, -1),(-2, 1),(-1, 2)).map(add_pair(x)).filter(is_legal(n))
 }
 
-def ordered_moves(n: Int)(steps: List[(Int, Int)])(x : (Int, Int)): List[(Int, Int)] = 
-  moves(n)(x).sortBy((x: (Int, Int)) => moves(n)(x).filterNot(steps.contains(_)).length)
+def add_pair(x: Pos)(y: Pos): Pos = 
+  (x._1 + y._1, x._2 + y._2)
 
-moves(8)(1,3)
-ordered_moves(8)(Nil)(1,3)
-ordered_moves(8)(List((2, 4), (2, 6)))(1,3)
+def is_legal(dim: Int, path: Path)(x: Pos): Boolean = 
+  0 <= x._1 && 0 <= x._2 && x._1 < dim && x._2 < dim && !path.contains(x)
+
+def moves(x: Pos): List[Pos] = 
+  List(( 1,  2),( 2,  1),( 2, -1),( 1, -2),
+       (-1, -2),(-2, -1),(-2,  1),(-1,  2)).map(add_pair(x))
 
-// non-circle tour parallel
-def tour(n: Int)(steps: List[(Int, Int)]): Unit = {
-  if (steps.length ==  n * n && moves(n)(steps.head).contains(steps.last))
-    print_board(n)(steps)
-  else 
-    for (x <- moves(n)(steps.head).par; if (!steps.contains(x))) tour(n)(x :: steps)
+def legal_moves(dim: Int, path: Path, x: Pos): List[Pos] = 
+  moves(x).filter(is_legal(dim, path))
+
+def ordered_moves(dim: Int, path: Path, x: Pos): List[Pos] = 
+  legal_moves(dim, path, x).sortBy((x) => legal_moves(dim, path, x).length)
+
+
+def first(xs: List[Pos], f: Pos => Option[Path]): Option[Path] = xs match {
+  case Nil => None
+  case x::xs => {
+    val result = f(x)
+    if (result.isDefined) result else first(xs, f)
+  }
 }
 
-val n = 7
-println(s"circle tour parallel: n = $n")
+
+def first_closed_tour_heuristics(dim: Int, path: Path): Option[Path] = {
+  if (path.length == dim * dim && moves(path.head).contains(path.last)) Some(path)
+  else
+    first(ordered_moves(dim, path, path.head), (x: Pos) => first_closed_tour_heuristics(dim, x::path))
+}
+
+for (dim <- 1 to 6) {
+  val t = first_closed_tour_heuristics(dim, List((dim / 2, dim / 2)))
+  println(s"${dim} x ${dim} closed: " + (if (t == None) "" else { print_board(dim, t.get) ; "" }))
+}
 
-val starts = for (i <- 0 until n; j <- 0 until n) yield (i, j)
 
-starts.par.foreach((x:(Int, Int)) => tour(n)(List(x))) 
+def first_tour_heuristics(dim: Int, path: Path): Option[Path] = {
+  if (path.length == dim * dim) Some(path)
+  else
+    first(ordered_moves(dim, path, path.head), (x: Pos) => first_tour_heuristics(dim, x::path))
+}
+
+for (dim <- 1 to 50) {
+  val t = first_tour_heuristics(dim, List((dim / 2, dim / 2)))
+  println(s"${dim} x ${dim}: " + (if (t == None) "" else { print_board(dim, t.get) ; "" }))
+}
diff -r fdc2c6fb7a24 -r 9891c9fac37e progs/knight3_sol.scala
--- a/progs/knight3_sol.scala	Tue Nov 15 23:08:09 2016 +0000
+++ b/progs/knight3_sol.scala	Wed Nov 16 14:37:18 2016 +0000
@@ -1,65 +1,24 @@
-// Part 2 about finding a single tour for a board
-//================================================
-
-
-type Pos = (Int, Int)
-type Path = List[Pos]
+// Part 3 about finding a single tour using the Warnsdorf Rule
+//=============================================================
 
-def print_board(dim: Int, path: Path): Unit = {
-  println
-  for (i <- 0 until dim) {
-    for (j <- 0 until dim) {
-      print(f"${path.reverse.indexOf((i, j))}%3.0f ")
-    }
-    println
-  } 
-}
-
-def add_pair(x: Pos)(y: Pos): Pos = 
-  (x._1 + y._1, x._2 + y._2)
+// copy any function you need from files knight1.scala and
+// knight2.scala
 
-def is_legal(dim: Int, path: Path)(x: Pos): Boolean = 
-  0 <= x._1 && 0 <= x._2 && x._1 < dim && x._2 < dim && !path.contains(x)
-
-def moves(x: Pos): List[Pos] = 
-  List(( 1,  2),( 2,  1),( 2, -1),( 1, -2),
-       (-1, -2),(-2, -1),(-2,  1),(-1,  2)).map(add_pair(x))
-
-def legal_moves(dim: Int, path: Path, x: Pos): List[Pos] = 
-  moves(x).filter(is_legal(dim, path))
-
-def ordered_moves(dim: Int, path: Path, x: Pos): List[Pos] = 
-  legal_moves(dim, path, x).sortBy((x) => legal_moves(dim, path, x).length)
+type Pos = (Int, Int)    // a position on a chessboard 
+type Path = List[Pos]    // a path...a list of positions
 
+//(3a) Complete the function that calculates a list of onward
+// moves like in (1b) but orders them according to the Warnsdorf’s 
+// rule. That means moves with the fewest legal onward moves 
+// should come first.
 
-def first(xs: List[Pos], f: Pos => Option[Path]): Option[Path] = xs match {
-  case Nil => None
-  case x::xs => {
-    val result = f(x)
-    if (result.isDefined) result else first(xs, f)
-  }
-}
-
-
-def first_closed_tour_heuristics(dim: Int, path: Path): Option[Path] = {
-  if (path.length == dim * dim && moves(path.head).contains(path.last)) Some(path)
-  else
-    first(ordered_moves(dim, path, path.head), (x: Pos) => first_closed_tour_heuristics(dim, x::path))
-}
+def ordered_moves(dim: Int, path: Path, x: Pos): List[Pos] = ..
 
-for (dim <- 1 to 7) {
-  val t = first_closed_tour_heuristics(dim, List((dim / 2, dim / 2)))
-  println(s"${dim} x ${dim} closed: " + (if (t == None) "" else { print_board(dim, t.get) ; "" }))
-}
-
+//(3b) Complete the function that searches for a single *closed* 
+// tour using the ordered moves function.
 
-def first_tour_heuristics(dim: Int, path: Path): Option[Path] = {
-  if (path.length == dim * dim) Some(path)
-  else
-    first(ordered_moves(dim, path, path.head), (x: Pos) => first_tour_heuristics(dim, x::path))
-}
+def first_closed_tour_heuristic(dim: Int, path: Path): Option[Path] = ...
 
-for (dim <- 1 to 50) {
-  val t = first_tour_heuristics(dim, List((dim / 2, dim / 2)))
-  println(s"${dim} x ${dim}: " + (if (t == None) "" else { print_board(dim, t.get) ; "" }))
-}
+//(3c) Sama as (3b) but searches for *open* tours.
+
+def first_tour_heuristic(dim: Int, path: Path): Option[Path] = ...