updated
authorChristian Urban <urbanc@in.tum.de>
Thu, 07 Dec 2017 12:04:31 +0000
changeset 163 84917f2e16cd
parent 162 6d25ccbb3cf2
child 164 7ada03135b2e
updated
LINKS
cws/cw02.tex
cws/cw03.pdf
cws/cw03.tex
progs/catastrophic.java
template3/bf.scala
template3/re.scala
testing2/knight1.scala
--- a/LINKS	Tue Dec 05 00:34:14 2017 +0000
+++ b/LINKS	Thu Dec 07 12:04:31 2017 +0000
@@ -4,6 +4,7 @@
 why functional programming matters
 http://queue.acm.org/detail.cfm?id=2038036
 
+=============
 
 
 BufferOverflow
--- a/cws/cw02.tex	Tue Dec 05 00:34:14 2017 +0000
+++ b/cws/cw02.tex	Thu Dec 07 12:04:31 2017 +0000
@@ -56,7 +56,7 @@
 \subsection*{Disclaimer}
 
 It should be understood that the work you submit represents
-your own effort. You have not copied from anyone else. An
+your \textbf{own} effort. You have not copied from anyone else. An
 exception is the Scala code I showed during the lectures or
 uploaded to KEATS, which you can freely use.
 
Binary file cws/cw03.pdf has changed
--- a/cws/cw03.tex	Tue Dec 05 00:34:14 2017 +0000
+++ b/cws/cw03.tex	Thu Dec 07 12:04:31 2017 +0000
@@ -45,6 +45,33 @@
 28  29.81185
 \end{filecontents}
 
+\begin{filecontents}{re-java9.data}
+1000  0.01410
+2000  0.04882
+3000  0.10609
+4000  0.17456
+5000  0.27530
+6000  0.41116
+7000  0.53741
+8000  0.70261
+9000  0.93981
+10000 0.97419
+11000 1.28697
+12000 1.51387
+14000 2.07079
+16000 2.69846
+20000 4.41823
+24000 6.46077
+26000 7.64373
+30000 9.99446
+34000 12.966885
+38000 16.281621
+42000 19.180228
+46000 21.984721
+50000 26.950203
+60000 43.0327746
+\end{filecontents}
+
 
 \begin{document}
 
@@ -92,7 +119,7 @@
 \subsection*{Disclaimer}
 
 It should be understood that the work you submit represents
-your own effort! You have not copied from anyone else. An
+your \textbf{own} effort! You have not copied from anyone else. An
 exception is the Scala code I showed during the lectures or
 uploaded to KEATS, which you can freely use.\bigskip
 
@@ -359,28 +386,53 @@
 30 seconds time limit?
 
 \begin{center}
+\begin{tabular}{@{}cc@{}}
+\multicolumn{2}{c}{Graph: $(a^*)^*\cdot b$ and strings 
+           $\underbrace{a\ldots a}_{n}$}\bigskip\\
+  
 \begin{tikzpicture}
 \begin{axis}[
-    title={Graph: $(a^*)^*\cdot b$ and strings 
-           $\underbrace{a\ldots a}_{n}$},
     xlabel={$n$},
     x label style={at={(1.05,0.0)}},
     ylabel={time in secs},
+    y label style={at={(0.06,0.5)}},
     enlargelimits=false,
     xtick={0,5,...,30},
     xmax=33,
-    ymax=35,
-    ytick={0,5,...,30},
+    ymax=45,
+    ytick={0,5,...,40},
     scaled ticks=false,
     axis lines=left,
     width=6cm,
     height=5.5cm, 
     legend entries={Python, Java 8},  
-    legend pos=outer north east]
+    legend pos=north west]
 \addplot[blue,mark=*, mark options={fill=white}] table {re-python2.data};
 \addplot[cyan,mark=*, mark options={fill=white}] table {re-java.data};
 \end{axis}
 \end{tikzpicture}
+  & 
+\begin{tikzpicture}
+\begin{axis}[
+    xlabel={$n$},
+    x label style={at={(1.05,0.0)}},
+    ylabel={time in secs},
+    y label style={at={(0.06,0.5)}},
+    %enlargelimits=false,
+    %xtick={0,5000,...,30000},
+    xmax=65000,
+    ymax=45,
+    ytick={0,5,...,40},
+    scaled ticks=false,
+    axis lines=left,
+    width=6cm,
+    height=5.5cm, 
+    legend entries={Java 9},  
+    legend pos=north west]
+\addplot[cyan,mark=*, mark options={fill=white}] table {re-java9.data};
+\end{axis}
+\end{tikzpicture}
+\end{tabular}  
 \end{center}
 \newpage
 
--- a/progs/catastrophic.java	Tue Dec 05 00:34:14 2017 +0000
+++ b/progs/catastrophic.java	Thu Dec 07 12:04:31 2017 +0000
@@ -9,12 +9,12 @@
     public static void main(String[] args) {
 
         //we always run all the tests twice -> warmup of the JVM
-        for (int runs = 0; runs < 2; runs++) {
+        for (int runs = 0; runs < 3; runs++) {
             
             Pattern pattern = Pattern.compile("(a*)*b");
             
             // Run from 5 to 28 characters
-            for (int length = 5; length < 28; length++) {
+            for (int length = 70000; length < 70001; length++) {
                 
                 // Build input of specified length
                 String input = "";
@@ -27,7 +27,7 @@
                 }
                 
                 System.out.println(length + " " + input + ": " 
-                       + ((System.nanoTime() - start) / 2000000000d) 
+                       + ((System.nanoTime() - start) / 3000000000d) 
                        + "s");
             }
         }
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/template3/bf.scala	Thu Dec 07 12:04:31 2017 +0000
@@ -0,0 +1,151 @@
+// Part 2 about an Interpreter for the Brainf*** language
+//========================================================
+
+object CW8b {
+
+type Mem = Map[Int, Int]
+
+// (2a) Complete the functions for safely reading  
+// and writing brainf*** memory. Safely read should
+// Return the value stored in the Map for a given memory
+// pointer, if it exists; otherwise it Returns 0. The
+// writing function generates a new Map with the
+// same data, except at the given memory pointer the
+// value v is stored.
+
+
+//def sread(mem: Mem, mp: Int) : Int = ...
+
+//def write(mem: Mem, mp: Int, v: Int) : Mem = ...
+
+
+// (2b) Implement the two jumping instructions in the 
+// brainf*** language. In jumpRight, given a program and 
+// a program counter move the counter to the right 
+// until the command after the *matching* ]-command. Similarly, 
+// jumpLeft implements the move to the left to just after
+// the *matching* [-command. The levels are used to find the
+// *matching* bracket.
+
+//def jumpRight(prog: String, pc: Int, level: Int) : Int = ...
+
+//def jumpLeft(prog: String, pc: Int, level: Int) : Int = ...
+
+
+
+// (2c) Complete the run function that interprets (runs) a brainf***
+// program: the arguments are a program, a program counter,
+// a memory counter and a brainf*** memory. It Returns the
+// memory at the stage when the execution of the brainf*** program
+// finishes. The interpretation finishes once the program counter
+// pc is pointing to something outside the program string.
+// If the pc points to a character inside the program, the pc, 
+// memory pointer and memory need to be updated according to 
+// rules of the brainf*** language. Then, recursively, the run 
+// function continues with the command at the new program
+// counter. 
+//
+// Implement the start function that calls run with the program
+// counter and memory counter set to 0.
+
+//def run(prog: String, pc: Int, mp: Int, mem: Mem) : Mem = ...
+
+//def start(prog: String, mem: Mem) = ...
+
+
+
+
+
+
+// some sample bf programs collected from the Internet
+//==================================================
+
+
+/*
+// first some contrived (small) programs
+
+// clears the 0-cell
+start("[-]", Map(0 -> 100)) 
+
+// copies content of the 0-cell to 1-cell
+start("[->+<]", Map(0 -> 10))
+
+// copies content of the 0-cell to 2-cell and 4-cell
+start("[>>+>>+<<<<-]", Map(0 -> 42))
+
+start("+++[>+++++<-]", Map(0 -> 10))
+
+
+// prints out numbers 0 to 9
+start("""+++++[->++++++++++<]>--<+++[->>++++++++++<<]>>++<<----------[+>.>.<+<]""", Map())
+
+
+// some more "useful" programs
+
+// hello world program 1
+start("""++++++++[>++++[>++>+++>+++>+<<<<-]>+>+>->>+[<]<-]>>.>---.+++++++
+       ..+++.>>.<-.<.+++.------.--------.>>+.>++.""", Map())
+
+// hello world program 2
+start("""++++++++++[>+++++++>++++++++++>+++>+<<<<-]>++.>+.+++++++..+++.>+
+      +.<<+++++++++++++++.>.+++.------.--------.>+.>.""", Map())
+
+
+// draws the Sierpinski triangle
+start("""++++++++[>+>++++<<-]>++>>+<[-[>>+<<-]+>>]>+[-<<<[
+      ->[+[-]+>++>>>-<<]<[<]>>++++++[<<+++++>>-]+<<++.[-]<<
+      ]>.>+[>>]>+]""", Map())
+
+//Fibonacci numbers below 100
+start("""+++++++++++
+      >+>>>>++++++++++++++++++++++++++++++++++++++++++++
+      >++++++++++++++++++++++++++++++++<<<<<<[>[>>>>>>+>
+      +<<<<<<<-]>>>>>>>[<<<<<<<+>>>>>>>-]<[>++++++++++[-
+      <-[>>+>+<<<-]>>>[<<<+>>>-]+<[>[-]<[-]]>[<<[>>>+<<<
+      -]>>[-]]<<]>>>[>>+>+<<<-]>>>[<<<+>>>-]+<[>[-]<[-]]
+      >[<<+>>[-]]<<<<<<<]>>>>>[+++++++++++++++++++++++++
+      +++++++++++++++++++++++.[-]]++++++++++<[->-<]>++++
+      ++++++++++++++++++++++++++++++++++++++++++++.[-]<<
+      <<<<<<<<<<[>>>+>+<<<<-]>>>>[<<<<+>>>>-]<-[>>.>.<<<
+      [-]]<<[>>+>+<<<-]>>>[<<<+>>>-]<<[<+>-]>[<+>-]<<<-]""", Map())
+
+
+//outputs the square numbers up to 10000
+start("""++++[>+++++<-]>[<+++++>-]+<+[
+    >[>+>+<<-]++>>[<<+>>-]>>>[-]++>[-]+
+    >>>+[[-]++++++>>>]<<<[[<++++++++<++>>-]+<.<[>----<-]<]
+    <<[>>>>>[>>>[-]+++++++++<[>-<-]+++++++++>[-[<->-]+[<<<]]<[>+<-]>]<<-]<<-]""", Map())
+
+
+//Collatz numbers (need to be typed in)
+start(""">,[[----------[
+      >>>[>>>>]+[[-]+<[->>>>++>>>>+[>>>>]++[->+<<<<<]]<<<]
+      ++++++[>------<-]>--[>>[->>>>]+>+[<<<<]>-],<]>]>>>++>+>>[
+      <<[>>>>[-]+++++++++<[>-<-]+++++++++>[-[<->-]+[<<<<]]<[>+<-]>]
+      >[>[>>>>]+[[-]<[+[->>>>]>+<]>[<+>[<<<<]]+<<<<]>>>[->>>>]+>+[<<<<]]
+      >[[>+>>[<<<<+>>>>-]>]<<<<[-]>[-<<<<]]>>>>>>>
+      ]>>+[[-]++++++>>>>]<<<<[[<++++++++>-]<.[-]<[-]<[-]<]<,]""", Map())
+
+
+// infinite Collatz (never stops)
+start(""">>+>+<[[->>[>>]>>>[>>]+[<<]<<<[<<]>[>[>>]>>+>[>>]<+<[<<]<<<[<
+      <]>-]>[>>]>>[<<<<[<<]>+>[>>]>>-]<<<<[<<]+>>]<<[+++++[>+++++++
+      +<-]>.<++++++[>--------<-]+<<]>>[>>]+[>>>>[<<+>+>-]<-[>+<-]+<
+      [<<->>-[<<+>>[-]]]>>>[<<<+<<+>>>>>-]<<<[>>>+<<<-]<<[[-]>+>>->
+      [<+<[<<+>>-]<[>+<-]<[>+<-]>>>>-]<[>+<-]+<[->[>>]<<[->[<+++>-[
+      <+++>-[<+++>-[<[-]++>>[-]+>+<<-[<+++>-[<+++>-[<[-]+>>>+<<-[<+
+      ++>-[<+++>-]]]]]]]]]<[>+<-]+<<]>>>+<[->[<+>-[<+>-[<+>-[<+>-[<
+      +>-[<+>-[<+>-[<+>-[<+>-[<[-]>>[-]+>+<<-[<+>-]]]]]]]]]]]<[>+<-
+      ]+>>]<<[<<]>]<[->>[->+>]<[-[<+>-[<->>+<-[<+>-[<->>+<-[<+>-[<-
+      >>+<-[<+>-[<->>+<-[<+>-[<->>+<-[<+>-[<->>+<-[<+>-[<->>+<-[<+>
+      -[<->>+<-[<+>-[<->>+<-[<+>-]]]]]]]]]]]]]]]]]]]>[<+>-]<+<[<+++
+      +++++++>-]<]>>[<+>->>]<<[>+>+<<-]>[<+>-]+>[<->[-]]<[-<<-]<<[<
+      <]]++++++[>+++++++<-]>++.------------.[-]>[>>]<<[+++++[>+++++
+      +++<-]>.<++++++[>--------<-]+<<]+<]>[<+>-]<]>>>[>>]<<[>[-]<-<
+      <]++++++++++.[-]<<<[<<]>>>+<[->[<+>-[<+>-[<+>-[<+>-[<+>-[<+>-
+      [<+>-[<+>-[<+>-[<[-]>>[-]+>+<<-]]]]]]]]]]<[>+<-]+>>]<<[<<]>>]""", Map())
+
+
+*/ 
+
+}
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/template3/re.scala	Thu Dec 07 12:04:31 2017 +0000
@@ -0,0 +1,127 @@
+// Part 1 about Regular Expression Matching
+//==========================================
+
+
+object CW8a {
+
+abstract class Rexp
+case object ZERO extends Rexp
+case object ONE extends Rexp
+case class CHAR(c: Char) extends Rexp
+case class ALT(r1: Rexp, r2: Rexp) extends Rexp   // alternative 
+case class SEQ(r1: Rexp, r2: Rexp) extends Rexp   // sequence
+case class STAR(r: Rexp) extends Rexp             // star
+
+
+// some convenience for typing in regular expressions
+
+import scala.language.implicitConversions    
+import scala.language.reflectiveCalls 
+
+def charlist2rexp(s: List[Char]): Rexp = s match {
+  case Nil => ONE
+  case c::Nil => CHAR(c)
+  case c::s => SEQ(CHAR(c), charlist2rexp(s))
+}
+implicit def string2rexp(s: String): Rexp = charlist2rexp(s.toList)
+
+implicit def RexpOps (r: Rexp) = new {
+  def | (s: Rexp) = ALT(r, s)
+  def % = STAR(r)
+  def ~ (s: Rexp) = SEQ(r, s)
+}
+
+implicit def stringOps (s: String) = new {
+  def | (r: Rexp) = ALT(s, r)
+  def | (r: String) = ALT(s, r)
+  def % = STAR(s)
+  def ~ (r: Rexp) = SEQ(s, r)
+  def ~ (r: String) = SEQ(s, r)
+}
+
+// (1a) Complete the function nullable according to
+// the definition given in the coursework; this 
+// function checks whether a regular expression
+// can match the empty string and Returns a boolean
+// accordingly.
+
+//def nullable (r: Rexp) : Boolean = ...
+
+
+// (1b) Complete the function der according to
+// the definition given in the coursework; this
+// function calculates the derivative of a 
+// regular expression w.r.t. a character.
+
+//def der (c: Char, r: Rexp) : Rexp = ...
+
+
+// (1c) Complete the simp function according to
+// the specification given in the coursework; this
+// function simplifies a regular expression from
+// the inside out, like you would simplify arithmetic 
+// expressions; however it does not simplify inside 
+// STAR-regular expressions.
+
+//def simp(r: Rexp) : Rexp = ... 
+
+
+// (1d) Complete the two functions below; the first 
+// calculates the derivative w.r.t. a string; the second
+// is the regular expression matcher taking a regular
+// expression and a string and checks whether the
+// string matches the regular expression
+
+//def ders (s: List[Char], r: Rexp) : Rexp = ... 
+
+//def matcher(r: Rexp, s: String): Boolean = ...
+
+
+// (1e) Complete the size function for regular
+// expressions according to the specification 
+// given in the coursework.
+
+//def size(r: Rexp): Int = ...
+
+
+// some testing data
+
+/*
+matcher(("a" ~ "b") ~ "c", "abc")  // => true
+matcher(("a" ~ "b") ~ "c", "ab")   // => false
+
+// the supposedly 'evil' regular expression (a*)* b
+val EVIL = SEQ(STAR(STAR(CHAR('a'))), CHAR('b'))
+
+matcher(EVIL, "a" * 1000 ++ "b")   // => true
+matcher(EVIL, "a" * 1000)          // => false
+
+// size without simplifications
+size(der('a', der('a', EVIL)))             // => 28
+size(der('a', der('a', der('a', EVIL))))   // => 58
+
+// size with simplification
+size(simp(der('a', der('a', EVIL))))           // => 8
+size(simp(der('a', der('a', der('a', EVIL))))) // => 8
+
+// Java needs around 30 seconds for matching 28 a's with EVIL. 
+//
+// Lets see how long it takes to match strings with 
+// 0.5 Million a's...it should be in the range of some
+// seconds.
+
+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 <- 0 to 5000000 by 500000) {
+  println(i + " " + "%.5f".format(time_needed(2, matcher(EVIL, "a" * i))))
+}
+
+*/
+
+
+}
--- a/testing2/knight1.scala	Tue Dec 05 00:34:14 2017 +0000
+++ b/testing2/knight1.scala	Thu Dec 07 12:04:31 2017 +0000
@@ -1,91 +1,133 @@
+
 // Part 1 about finding and counting Knight's tours
 //==================================================
 
-object CW7a {
+object CW7a extends App{
 
 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
-  for (i <- 0 until dim) {
-    for (j <- 0 until dim) {
-      print(f"${path.reverse.indexOf((j, dim - i - 1))}%3.0f ")
-    }
-    println
-  } 
-}
+//(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 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)
-
-assert(is_legal(8, Nil)((3,4)) == true)
-assert(is_legal(8, List((4,1), (1,0)))((4,1)) == false)
-assert(is_legal(2, Nil)((0,0)) == true)
+def is_legal(dim: Int, path: Path)(x: Pos) : Boolean = {
+  
+// if ((x._1<dim && x._2<dim) && (x._1>0 || x._2>0)) false else !path.contains(x)
+ 
+  if (x._1 < 0 || x._2 < 0) false 
+  else if (x._1 < dim && x._2 < dim && !path.contains(x)) true 
+  else false
+ 
+  
+}
 
 
 
-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 count_tours(dim: Int, path: Path): Int = {
-  if (path.length == dim * dim) 1
-  else 
-    (for (x <- legal_moves(dim, path, path.head)) yield count_tours(dim, x::path)).sum
-}
-
-def count_tours(dim: Int, path : Path) : Int = {
+def legal_moves(dim: Int, path: Path, x: Pos) : List[Pos] = {
+  
+  val allPossibleMoves = List((x._1+1, x._2+2), (x._1+2, x._2+1), (x._1+2, x._2-1), (x._1+1, x._2-2), (x._1-1, x._2-2), (x._1-2, x._2-1), (x._1-2, x._2+1), (x._1-1, x._2+2));
+ 
+  //val finalList = allPossibleMoves.filter((a=>a._1<dim && a._2<dim && x._1 >= 0 && a._2 >= 0));
   
-  if (path.length == dim * dim) {1}
-  else 
-  val x = for (m <- legal_moves(dim,path,path.head)) yield {
+  val finalList = for(pos<-allPossibleMoves if(is_legal(dim,path)(pos))) yield pos;
+  
+  // println("Space in board: " + dim*dim + " for dim: " + dim)
+   
+  
+  finalList.toList;
     
-    count_tours(dim,m::path)
-  }
-  x.sum
   
 }
 
-def enum_tours(dim: Int, path: Path): List[Path] = {
-  if (path.length == dim * dim) List(path)
-  else 
-    (for (x <- legal_moves(dim, path, path.head)) yield enum_tours(dim, x::path)).flatten
-}
+println(legal_moves(8, Nil, (2,2)))
+println(legal_moves(8, Nil, (7,7)))
+println(legal_moves(8, List((4,1), (1,0)), (2,2)))
+println(legal_moves(8, List((6,6)), (7,7)))
+println(legal_moves(1, Nil, (0,0)))
+println(legal_moves(2, Nil, (0,0)))
+println(legal_moves(3, Nil, (0,0)))
+
+println("=================================================================================")
+println("================================Comparision output===============================")
+println("=================================================================================")
+
+println(legal_moves(8, Nil, (2,2)) == List((3,4), (4,3), (4,1), (3,0), (1,0), (0,1), (0,3), (1,4)))
+println(legal_moves(8, Nil, (7,7)) == List((6,5), (5,6)))
+println(legal_moves(8, List((4,1), (1,0)), (2,2)) == List((3,4), (4,3), (3,0), (0,1), (0,3), (1,4)))
+println(legal_moves(8, List((6,6)), (7,7)) == List((6,5), (5,6)))
+println(legal_moves(1, Nil, (0,0)) == Nil)
+println(legal_moves(2, Nil, (0,0)) == Nil)
+println(legal_moves(3, Nil, (0,0)) == List((1,2), (2,1)))
+
 
-def count_all_tours(dim: Int) = {
-  for (i <- (0 until dim).toList; 
-       j <- (0 until dim).toList) yield count_tours(dim, List((i, j)))
+def count_tours(dim: Int, path: Path) : Int = {
+     
+  val allMovesFromCurrentPosition = legal_moves(dim, path, path.head);
+  
+  if (path.length == dim*dim) 1 else  {
+    
+    if (allMovesFromCurrentPosition.size == 0 ) 0  else {
+      
+      allMovesFromCurrentPosition.map( element => count_tours(dim, element::path)).sum
+      
+      
+    }
+    
+  }
+  
 }
+    
+  
 
-def enum_all_tours(dim: Int): List[Path] = {
-  (for (i <- (0 until dim).toList; 
-        j <- (0 until dim).toList) yield enum_tours(dim, List((i, j)))).flatten
+println ( count_tours(5, List((0,0))) )
+
+def enum_tours(dim: Int, path: Path) : List[Path] = {
+  
+     val allMovesFromCurrentPosition = legal_moves(dim, path, path.head);
+  
+  if (path.length == dim*dim) List(path) else  {
+    
+  allMovesFromCurrentPosition.map( element => enum_tours(dim, element::path)).flatten ;
+      
+      
+      }
+    }
+  println ( enum_tours(6, List((0,2))).size)
 }
 
-/*
-for (dim <- 1 to 5) {
-  println(s"${dim} x ${dim} " + count_tours(dim, List((0, 0))))
-}
+
+
+
 
-for (dim <- 1 to 5) {
-  println(s"${dim} x ${dim} " + count_all_tours(dim))
-}
+ 
+ 
+//(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" manner.
+ 
+//def legal_moves(dim: Int, path: Path, x: Pos) : List[Pos] = ...
+
+
+
 
-for (dim <- 1 to 5) {
-  val ts = enum_tours(dim, List((0, 0)))
-  println(s"${dim} x ${dim} ")   
-  if (ts != Nil) {
-    print_board(dim, ts.head)
-    println(ts.head)
-  }
-}
-*/ 
+//some test cases
+//
+//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) Complete the two recursive functions below. 
+//     They exhaustively search for knight's tours starting from the 
+//     given path. The first function counts all possible tours, 
+//     and the second collects all tours in a list of paths.
+
+//def count_tours(dim: Int, path: Path) : Int = ...
+
+
+//def enum_tours(dim: Int, path: Path) : List[Path] = ...