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

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

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

  } 

}

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 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))

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(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 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 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

}

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

}

def add_pair_urban(x: Pos)(y: Pos): Pos = 

  (x._1 + y._1, x._2 + y._2)

def is_legal_urban(dim: Int, path: Path)(x: Pos): Boolean = 

  0 <= x._1 && 0 <= x._2 && x._1 < dim && x._2 < dim && !path.contains(x)

def moves_urban(x: Pos): List[Pos] = 

  List(( 1,  2),( 2,  1),( 2, -1),( 1, -2),

       (-1, -2),(-2, -1),(-2,  1),(-1,  2)).map(add_pair_urban(x))

def legal_moves_urban(dim: Int, path: Path, x: Pos): List[Pos] = 

  moves_urban(x).filter(is_legal_urban(dim, path))

def correct_urban(dim: Int)(p: Path): Boolean = p match {

  case Nil => true

  case x::Nil => true

  case x::y::p => if (legal_moves_urban(dim, p, y).contains(x)) correct_urban(dim)(y::p) else false

}

enum_tours(5, List((0, 2))).map(correct_urban(5)).forall(_ == true)

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))

}

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)

  }

}