// Part 1 about finding and counting Knight's tours

//==================================================

object CW9a {   // 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))

}