// Main Part 4 about finding Knight's tours//==========================================object M4a {// If you need any auxiliary functions, feel free to // implement them, but do not make any changes to the// templates below. Also have a look whether the functions// at the end of the file 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 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 = ???def enum_tours(dim: Int, path: Path) : List[Path] = ???//(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 (4) 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() } }*/}