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

     2 //================================================

     3 

     4 type Pos = (Int, Int)    // a position on a chessboard 

     5 type Path = List[Pos]    // a path...a list of positions

     6 

     7 def print_board(dim: Int, path: Path): Unit = {

     8   println

     9   for (i <- 0 until dim) {

    10     for (j <- 0 until dim) {

    11       print(f"${path.reverse.indexOf((i, j))}%3.0f ")

    12     }

    13     println

    14   } 

    15 }

    16 

    17 def add_pair(x: Pos)(y: Pos): Pos = 

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

    19 

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

    21   0 <= x._1 && 0 <= x._2 && x._1 < dim && x._2 < dim && !path.contains(x)

    22 

    23 def moves(x: Pos): List[Pos] = 

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

    25        (-1, -2),(-2, -1),(-2,  1),(-1,  2)).map(add_pair(x))

    26 

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

    28   moves(x).filter(is_legal(dim, path))

    29 

    30 

    31 def first(xs: List[Pos], f: Pos => Option[Path]): Option[Path] = xs match {

    32   case Nil => None

    33   case x::xs => {

    34     val result = f(x)

    35     if (result.isDefined) result else first(xs, f)

    36   }

    37 }

    38 

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

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

    41 

    42 

    43 

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

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

    46   else

    47     first(legal_moves(dim, path, path.head), (x: Pos) => first_tour(dim, x::path))

    48 }

    49 

    50 /*

    51 for (dim <- 1 to 8) {

    52   val t = first_tour(dim, List((0, 0)))

    53   println(s"${dim} x ${dim} " + (if (t == None) "" else { print_board(dim, t.get) ; "" }))

    54 }

    55 */

    56 

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

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

    59 

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

    61   0 <= x._1 && 0 <= x._2 && x._1 < dim && x._2 < dim && !path.contains(x)

    62 

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

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

    65        (-1, -2),(-2, -1),(-2,  1),(-1,  2)).map(add_pair_urban(x))

    66 

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

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

    69 

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

    71   case Nil => true

    72   case x::Nil => true

    73   case x::y::p => 

    74     if (legal_moves_urban(dim, p, y).contains(x)) correct_urban(dim)(y::p) else false

    75 }

    76 

    77 

    78 

    79  

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

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

    82 

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

    84 assert(ts2 == None)  

    85 

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

    87 

    88