// 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.

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

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

//(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 = ...

//(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] = ...

//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 = ...

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

//(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] = ...

// 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] = ...

/* Helper functions

// for measuring time

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

}

// 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

// 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

  } 

}

*/