2 //================================================== |
2 //================================================== |
3 |
3 |
4 type Pos = (Int, Int) // a position on a chessboard |
4 type Pos = (Int, Int) // a position on a chessboard |
5 type Path = List[Pos] // a path...a list of positions |
5 type Path = List[Pos] // a path...a list of positions |
6 |
6 |
7 //(1a) Complete the function that tests whether the position |
7 def print_board(dim: Int, path: Path): Unit = { |
8 // is inside the board and not yet element in the path. |
8 println |
9 |
9 for (i <- 0 until dim) { |
10 def is_legal(dim: Int, path: Path)(x: Pos): Boolean = ... |
10 for (j <- 0 until dim) { |
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11 print(f"${path.reverse.indexOf((j, dim - i - 1))}%3.0f ") |
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12 } |
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13 println |
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14 } |
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15 } |
11 |
16 |
12 |
17 |
13 //(1b) Complete the function that calculates for a position |
18 // 1 mark |
14 // all legal onward moves that are not already in the path. |
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15 // The moves should be ordered in a "clockwise" order. |
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16 |
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17 def legal_moves(dim: Int, path: Path, x: Pos): List[Pos] = ... |
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18 |
19 |
19 //assert(legal_moves(8, Nil, (2,2)) == |
20 def is_legal(dim: Int, path: Path, x: Pos): Boolean = |
20 // List((3,4), (4,3), (4,1), (3,0), (1,0), (0,1), (0,3), (1,4))) |
21 0 <= x._1 && 0 <= x._2 && x._1 < dim && x._2 < dim && !path.contains(x) |
21 //assert(legal_moves(8, Nil, (7,7)) == List((6,5), (5,6))) |
22 |
22 //assert(legal_moves(8, List((4,1), (1,0)), (2,2)) == |
23 assert(is_legal(8, Nil)((3,4)) == true) |
23 // List((3,4), (4,3), (3,0), (0,1), (0,3), (1,4))) |
24 assert(is_legal(8, List((4,1), (1,0)))((4,1)) == false) |
24 //assert(legal_moves(8, List((6,6)), (7,7)) == List((6,5), (5,6))) |
25 assert(is_legal(2, Nil)((0,0)) == true) |
25 |
26 |
26 |
27 |
27 //(1c) Complete the two recursive functions below. |
28 def add_pair(x: Pos)(y: Pos): Pos = |
28 // They exhaustively search for open tours starting from the |
29 (x._1 + y._1, x._2 + y._2) |
29 // given path. The first function counts all possible open tours, |
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30 // and the second collects all open tours in a list of paths. |
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31 |
30 |
32 def count_tours(dim: Int, path: Path): Int = ... |
31 def moves(x: Pos): List[Pos] = |
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32 List(( 1, 2),( 2, 1),( 2, -1),( 1, -2), |
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33 (-1, -2),(-2, -1),(-2, 1),(-1, 2)).map(add_pair(x)) |
33 |
34 |
34 def enum_tours(dim: Int, path: Path): List[Path] = ... |
35 // 1 mark |
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36 |
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37 def legal_moves(dim: Int, path: Path, x: Pos): List[Pos] = |
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38 moves(x).filter(is_legal(dim, path)) |
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39 |
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40 assert(legal_moves(8, Nil, (2,2)) == |
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41 List((3,4), (4,3), (4,1), (3,0), (1,0), (0,1), (0,3), (1,4))) |
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42 assert(legal_moves(8, Nil, (7,7)) == List((6,5), (5,6))) |
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43 assert(legal_moves(8, List((4,1), (1,0)), (2,2)) == |
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44 List((3,4), (4,3), (3,0), (0,1), (0,3), (1,4))) |
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45 assert(legal_moves(8, List((6,6)), (7,7)) == List((6,5), (5,6))) |
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46 assert(legal_moves(1, Nil, (0,0)) == List()) |
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47 assert(legal_moves(2, Nil, (0,0)) == List()) |
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48 assert(legal_moves(3, Nil, (0,0)) == List((1,2), (2,1))) |
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49 |
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50 // 2 marks |
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51 |
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52 def count_tours(dim: Int, path: Path): Int = { |
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53 if (path.length == dim * dim) 1 |
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54 else |
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55 (for (x <- legal_moves(dim, path, path.head)) yield count_tours(dim, x::path)).sum |
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56 } |
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57 |
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58 def enum_tours(dim: Int, path: Path): List[Path] = { |
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59 if (path.length == dim * dim) List(path) |
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60 else |
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61 (for (x <- legal_moves(dim, path, path.head)) yield enum_tours(dim, x::path)).flatten |
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62 } |
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63 |
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64 // as far as tasks go |
35 |
65 |
36 |
66 |
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67 |
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68 def count_all_tours(dim: Int) = { |
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69 for (i <- (0 until dim).toList; |
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70 j <- (0 until dim).toList) yield count_tours(dim, List((i, j))) |
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71 } |
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72 |
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73 def enum_all_tours(dim: Int): List[Path] = { |
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74 (for (i <- (0 until dim).toList; |
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75 j <- (0 until dim).toList) yield enum_tours(dim, List((i, j)))).flatten |
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76 } |
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77 |
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78 |
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79 println("Number of tours starting from (0, 0)") |
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80 |
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81 for (dim <- 1 to 5) { |
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82 println(s"${dim} x ${dim} " + count_tours(dim, List((0, 0)))) |
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83 } |
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84 |
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85 for (dim <- 1 to 5) { |
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86 println(s"${dim} x ${dim} " + count_all_tours(dim)) |
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87 } |
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88 |
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89 for (dim <- 1 to 5) { |
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90 val ts = enum_tours(dim, List((0, 0))) |
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91 println(s"${dim} x ${dim} ") |
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92 if (ts != Nil) { |
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93 print_board(dim, ts.head) |
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94 println(ts.head) |
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95 } |
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96 } |
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97 |
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98 |