--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/pre_solution4/knight1.scala	Wed Nov 25 17:29:11 2020 +0000
@@ -0,0 +1,177 @@
+// Part 1 about finding and counting Knight's tours
+//==================================================
+
+object CW8a {   // for preparing the jar
+
+type Pos = (Int, Int)    // a position on a chessboard 
+type Path = List[Pos]    // a path...a list of positions
+
+
+// for measuring time in the JAR
+def time_needed[T](code: => T) : T = {
+  val start = System.nanoTime()
+  val result = code
+  val end = System.nanoTime()
+  println(f"Time needed: ${(end - start) / 1.0e9}%3.3f secs.")
+  result
+}
+
+// for printing a board
+def print_board(dim: Int, path: Path): Unit = {
+  println()
+  for (i <- 0 until dim) {
+    for (j <- 0 until dim) {
+      print(f"${path.reverse.indexOf((j, dim - i - 1))}%3.0f ")
+    }
+    println()
+  } 
+}
+
+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, _))
+
+// 1 mark
+
+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)))
+
+// 2 marks
+
+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 ${dim} ")   
+  if (ts != Nil) {
+    print_board(dim, ts.head)
+    println(ts.head)
+  }
+}
+*/
+
+// 1 mark
+
+def first(xs: List[Pos], f: Pos => Option[Path]): Option[Path] = xs match {
+  case Nil => None
+  case x::xs => {
+    val result = f(x)
+    if (result.isDefined) result else first(xs, f)
+  }
+}
+
+// test cases
+//def foo(x: (Int, Int)) = if (x._1 > 3) Some(List(x)) else None
+//
+//first(List((1, 0),(2, 0),(3, 0),(4, 0)), foo)
+//first(List((1, 0),(2, 0),(3, 0)), foo)
+
+
+// 1 mark
+
+def tfirst_tour(dim: Int, path: Path): Option[Path] = {
+  if (path.length == dim * dim) Some(path)
+  else
+    first(legal_moves(dim, path, path.head), (x:Pos) => tfirst_tour(dim, x::path))
+}
+
+def first_tour(dim: Int, path: Path) = 
+  time_needed(tfirst_tour(dim: Int, path: Path))
+
+
+/*
+for (dim <- 1 to 8) {
+  val t = first_tour(dim, List((0, 0)))
+  println(s"${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))
+
+
+
+
+
+}
+
+