// Part 1 about finding Knight's tours

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

// If you need any auxiliary function, feel free to 

// implement it, but do not make any changes to the

// templates below. Also have a look whether the functions

// at the end are of any help.

//(1) Complete the function that tests whether the position x

//    is inside the board and not yet element in the path.

def is_legal(dim: Int, path: Path, x: Pos) : Boolean = ((!(path.contains(x))) && (x._1 < dim) && (x._2 < dim))

//(2) Complete the function that calculates for a position x

//    all legal onward moves that are not already in the path. 

//    The moves should be ordered in a "clockwise" manner.

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

  val y = List((x._1 + 1, x._2 + 2),

               (x._1 + 2, x._2 + 1),

               (x._1 + 2, x._2 - 1),

               (x._1 + 1, x._2 - 2),

               (x._1 - 1, x._2 - 2),

               (x._1 - 2, x._2 - 1),

               (x._1 - 2, x._2 + 1),

               (x._1 - 1, x._2 + 2)

   )

  y.filter(next => is_legal(dim, path, next))

}

//some test cases

//

//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)))

//(3) Complete the two recursive functions below. 

//    They exhaustively search for knight's tours starting from the 

//    given path. The first function counts all possible tours, 

//    and the second collects all tours in a list of paths.

def count_tours(dim: Int, path: Path) : Int = {

  if(path.length == dim*dim) 1 else

    (for(i <- legal_moves(dim, path, path.head)) yield

      count_tours(dim, i :: path)

    ).sum

}

def enum_tours(dim: Int, path: Path) : List[Path] ={

  if(path.length == dim*dim) List(path) else

    (for(i <- legal_moves(dim, path, path.head)) yield

      enum_tours(dim, i :: path)

    ).flatten

}

//(5) Implement a first-function that finds the first 

//    element, say x, in the list xs where f is not None. 

//    In that case Return f(x), otherwise None. If possible,

//    calculate f(x) only once.

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

  if(xs == Nil) None

  else(

    for(x <- xs) yield{

      val a = f(x)

      if(a != None) a

      else first(xs.drop(1), f)

    }

  ).head

}

// 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)   // Some(List((4,0)))

//first(List((1, 0),(2, 0),(3, 0)), foo)          // None

//(6) Implement a function that uses the first-function from (5) for

//    trying out onward moves, and searches recursively for a

//    knight tour on a dim * dim-board.

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

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

// }

/* Helper functions

// for measuring time

// can be called for example with

//     time_needed(count_tours(dim, List((0, 0))))

// in order to print out the time that is needed for 

// running count_tours