// Preliminary Part about finding Knight's tours//===============================================object CW8a {// 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 = { if ((!(path.contains(x))) && (x._1 >= 0) && (x._2 >= 0) && (x._1 < dim) && (x._2 < dim)) true else false}//(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] = {//List[Pos] val changes = List((1,2),(2,1),(2,-1),(1,-2),(-1,-2),(-2,-1),(-2,1),(-1,2)) val returnList = (for ((y,z) <- changes) yield( //println(y,z)-2,-1 if ((is_legal(dim,path,((x._1 + y) , (x._2 + z)))) == true) Some(x._1 + y , x._2 + z) else None )) returnList.flatten}//some testcases////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 = (dim,path) match {//Int case (_, Nil) => 0 case (0, path) => 0 case (dim, path) => { if (legal_moves(dim,path, path.head).size == 0) if(path.size < dim*dim) 0 else 1 else (for (j <- legal_moves(dim,path, path.head)) yield count_tours(dim,j::path)).sum }}def enum_tours(dim: Int, path: Path) : List[Path] = (dim,path) match { case (_, Nil) => Nil case (0, path) => Nil case (dim, path) => { if (legal_moves(dim,path, path.head).size == 0) if(path.size < dim*dim) Nil else List(path) else (for (j <- legal_moves(dim,path, path.head)) yield enum_tours(dim,j::path)).flatten }}//(4) 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] = ...// testcases////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//(5) 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 timedef time_needed[T](code: => T) : T = { val start = System.nanoTime() val result = code val end = System.nanoTime() println(f"Time needed: ${(end - start) / 1.0e9}%3.3f secs.") result}// can be called for example with// time_needed(count_tours(dim, List((0, 0))))// in order to print out the time that is needed for // running count_tours// for printing a boarddef print_board(dim: Int, path: Path): Unit = { println for (i <- 0 until dim) { for (j <- 0 until dim) { print(f"${path.reverse.indexOf((j, dim - i - 1))}%3.0f ") } println } }*/}