diff -r e9d14d58be3c -r daf561a83ba6 main_testing4/knight1.scala
--- a/main_testing4/knight1.scala	Thu Jan 13 12:55:03 2022 +0000
+++ b/main_testing4/knight1.scala	Mon Apr 11 23:55:27 2022 +0100
@@ -1,13 +1,119 @@
-// Part 1 about finding and counting Knight's tours
-//==================================================
+// Main Part 4 about finding Knight's tours
+//==========================================
+import scala.annotation.tailrec
+
+
+object M4a {
 
-object M4a {   // for preparing the jar
+// If you need any auxiliary functions, feel free to 
+// implement them, but do not make any changes to the
+// templates below. Also have a look whether the functions
+// at the end of the file are of any help.
+
+
 
 type Pos = (Int, Int)    // a position on a chessboard 
 type Path = List[Pos]    // a path...a list of positions
 
+//(1) Complete the function that tests whether the position x
+//    is inside the board and not yet element in the path.
 
-// for measuring time in the JAR
+def is_legal(dim: Int, path: Path, x: Pos) : Boolean = {
+  (x._1 < dim) && (x._2 < dim) && (!path.contains(x))
+}
+
+
+
+//(2) Complete the function that calculates for a position x
+//    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] = {
+    val movesets = 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)
+  )
+  movesets.filter(is_legal(dim, path, _))
+}
+
+
+//some testcases
+//
+//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)))
+
+
+//(3) 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 = {
+  if (dim <= 4) 0 else {
+    if (path.length >= (dim * dim)) 1 else {
+      val movesets = legal_moves(dim, path, path.head)
+      (for (move <- movesets) yield count_tours(dim, move :: path)).sum
+    }
+  }
+}
+
+def enum_tours(dim: Int, path: Path) : List[Path] = {
+  if (dim <= 4) Nil else {
+    if (path.length >= (dim * dim)) List(path) else {
+      val movesets = legal_moves(dim, path, path.head)
+      (for (move <- movesets) yield enum_tours(dim, move :: path)).flatten
+    }
+  }
+}
+
+
+//(4) 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. If possible,
+//    calculate f(x) only once.
+
+@tailrec
+def first(xs: List[Pos], f: Pos => Option[Path]) : Option[Path] = {
+  xs match {
+    case Nil => None
+    case head :: rest => {
+      val result = f(head)
+      if (result.isEmpty) first(rest, f) else result
+    }
+  }
+}
+
+
+// testcases
+//
+//def foo(x: (Int, Int)) = if (x._1 > 3) Some(List(x)) else None
+//
+//first(List((1, 0),(2, 0),(3, 0),(4, 0)), foo)   // Some(List((4,0)))
+//first(List((1, 0),(2, 0),(3, 0)), foo)          // None
+
+
+//(5) Implement a function that uses the first-function from (4) for
+//    trying out onward moves, and searches recursively for a
+//    knight tour on a dim * dim-board.
+
+def first_tour(dim: Int, path: Path) : Option[Path] = ???
+ 
+
+
+/* Helper functions
+
+
+// for measuring time
 def time_needed[T](code: => T) : T = {
   val start = System.nanoTime()
   val result = code
@@ -16,6 +122,14 @@
   result
 }
 
+// can be called for example with
+//
+//     time_needed(count_tours(dim, List((0, 0))))
+//
+// in order to print out the time that is needed for 
+// running count_tours
+
+
 // for printing a board
 def print_board(dim: Int, path: Path): Unit = {
   println()
@@ -27,145 +141,7 @@
   } 
 }
 
-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)
 
-// testcases
-//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 add_pair(x: Pos, y: Pos): Pos = 
-  (x._1 + y._1, x._2 + y._2)
-
-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, _))
-
-
-
-// testcases
-//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)))
-//assert(legal_moves(8, Nil, (0,1)) == List((1,3), (2,2), (2,0)))
-//assert(legal_moves(1, Nil, (0,0)) == List())
-//assert(legal_moves(2, Nil, (0,0)) == List())
-//assert(legal_moves(3, Nil, (0,0)) == List((1,2), (2,1)))
-
-
-def tcount_tours(dim: Int, path: Path): Int = {
-  if (path.length == dim * dim) 1
-  else 
-    (for (x <- legal_moves(dim, path, path.head)) yield tcount_tours(dim, x::path)).sum
-}
-
-def count_tours(dim: Int, path: Path) =
-  time_needed(tcount_tours(dim: Int, path: Path))
-
-
-def tenum_tours(dim: Int, path: Path): List[Path] = {
-  if (path.length == dim * dim) List(path)
-  else 
-    (for (x <- legal_moves(dim, path, path.head)) yield tenum_tours(dim, x::path)).flatten
-}
-
-def enum_tours(dim: Int, path: Path) =
-  time_needed(tenum_tours(dim: Int, path: Path))
-
-// test cases
-
-/*
-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 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("Number of tours starting from (0, 0)")
-
-for (dim <- 1 to 5) {
-  println(s"${dim} x ${dim} " + time_needed(0, count_tours(dim, List((0, 0)))))
-}
-
-println("Number of tours starting from all fields")
-
-for (dim <- 1 to 5) {
-  println(s"${dim} x ${dim} " + time_needed(0, count_all_tours(dim)))
-}
-
-for (dim <- 1 to 5) {
-  val ts = enum_tours(dim, List((0, 0)))
-  println(s"${dim} x ${dim} ")   
-  if (ts != Nil) {
-    print_board(dim, ts.head)
-    println(ts.head)
-  }
-}
 */
 
-
-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)
-  }
 }
-
-// test cases
-//def foo(x: (Int, Int)) = if (x._1 > 3) Some(List(x)) else None
-//
-//first(List((1, 0),(2, 0),(3, 0),(4, 0)), foo)
-//first(List((1, 0),(2, 0),(3, 0)), foo)
-
-
-def tfirst_tour(dim: Int, path: Path): Option[Path] = {
-  if (path.length == dim * dim) Some(path)
-  else
-    first(legal_moves(dim, path, path.head), (x:Pos) => tfirst_tour(dim, x::path))
-}
-
-def first_tour(dim: Int, path: Path) = 
-  time_needed(tfirst_tour(dim: Int, path: Path))
-
-
-/*
-for (dim <- 1 to 8) {
-  val t = first_tour(dim, List((0, 0)))
-  println(s"${dim} x ${dim} " + (if (t == None) "" else { print_board(dim, t.get) ; "" }))
-}
-*/
-
-// 15 secs for 8 x 8
-//val ts1 = time_needed(0,first_tour(8, List((0, 0))).get)
-//??val ts1 = time_needed(0,first_tour(8, List((1, 1))).get)
-
-// no result for 4 x 4
-//val ts2 = time_needed(0, first_tour(4, List((0, 0))))
-
-// 0.3 secs for 6 x 6
-//val ts3 = time_needed(0, first_tour(6, List((0, 0))))
-
-// 15 secs for 8 x 8
-//time_needed(0, print_board(8, first_tour(8, List((0, 0))).get))
-
-
-
-
-
-}
-
-