// Part 2 about finding a single tour for a board

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

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((i, j))}%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)

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 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)

  }

}

first(List((1, 0),(2, 0),(3, 0),(4, 0)), (x => if (x._1 > 3) Some(List(x)) else None))

first(List((1, 0),(2, 0),(3, 0)), (x => if (x._1 > 3) Some(List(x)) else None))

def first_tour(dim: Int, path: Path): Option[Path] = {

  if (path.length == dim * dim) Some(path)

  else

    first(legal_moves(dim, path, path.head), (x: Pos) => first_tour(dim, x::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) ; "" }))

}

*/

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

}

val ts1 = first_tour(8, List((0, 0))).get

  assert(correct_urban(8)(ts1) == true)

val ts2 = first_tour(4, List((0, 0)))

assert(ts2 == None)  

print_board(8, first_tour(8, List((0, 0))).get)