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1 // Part 1 about finding and counting Knight's tours |
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2 //================================================== |
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3 |
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4 type Pos = (Int, Int) // a position on a chessboard |
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5 type Path = List[Pos] // a path...a list of positions |
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6 |
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7 def print_board(dim: Int, path: Path): Unit = { |
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8 println |
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9 for (i <- 0 until dim) { |
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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 } |
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16 |
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17 def add_pair(x: Pos)(y: Pos): Pos = |
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18 (x._1 + y._1, x._2 + y._2) |
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19 |
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20 def is_legal(dim: Int, path: Path)(x: Pos): Boolean = |
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21 0 <= x._1 && 0 <= x._2 && x._1 < dim && x._2 < dim && !path.contains(x) |
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22 |
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23 assert(is_legal(8, Nil)((3,4)) == true) |
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24 assert(is_legal(8, List((4,1), (1,0)))((4,1)) == false) |
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25 assert(is_legal(2, Nil)((0,0)) == true) |
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26 |
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27 def moves(x: Pos): List[Pos] = |
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28 List(( 1, 2),( 2, 1),( 2, -1),( 1, -2), |
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29 (-1, -2),(-2, -1),(-2, 1),(-1, 2)).map(add_pair(x)) |
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30 |
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31 def legal_moves(dim: Int, path: Path, x: Pos): List[Pos] = |
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32 moves(x).filter(is_legal(dim, path)) |
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33 |
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34 assert(legal_moves(8, Nil, (2,2)) == |
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35 List((3,4), (4,3), (4,1), (3,0), (1,0), (0,1), (0,3), (1,4))) |
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36 assert(legal_moves(8, Nil, (7,7)) == List((6,5), (5,6))) |
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37 assert(legal_moves(8, List((4,1), (1,0)), (2,2)) == |
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38 List((3,4), (4,3), (3,0), (0,1), (0,3), (1,4))) |
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39 assert(legal_moves(8, List((6,6)), (7,7)) == List((6,5), (5,6))) |
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40 assert(legal_moves(1, Nil, (0,0)) == List()) |
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41 assert(legal_moves(2, Nil, (0,0)) == List()) |
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42 assert(legal_moves(3, Nil, (0,0)) == List((1,2), (2,1))) |
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43 |
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44 |
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45 def count_tours(dim: Int, path: Path): Int = { |
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46 if (path.length == dim * dim) 1 |
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47 else |
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48 (for (x <- legal_moves(dim, path, path.head)) yield count_tours(dim, x::path)).sum |
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49 } |
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50 |
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51 def enum_tours(dim: Int, path: Path): List[Path] = { |
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52 if (path.length == dim * dim) List(path) |
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53 else |
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54 (for (x <- legal_moves(dim, path, path.head)) yield enum_tours(dim, x::path)).flatten |
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55 } |
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56 |
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57 def count_all_tours(dim: Int) = { |
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58 for (i <- (0 until dim).toList; |
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59 j <- (0 until dim).toList) yield count_tours(dim, List((i, j))) |
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60 } |
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61 |
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62 def enum_all_tours(dim: Int): List[Path] = { |
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63 (for (i <- (0 until dim).toList; |
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64 j <- (0 until dim).toList) yield enum_tours(dim, List((i, j)))).flatten |
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65 } |
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66 |
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67 |
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68 def add_pair_urban(x: Pos)(y: Pos): Pos = |
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69 (x._1 + y._1, x._2 + y._2) |
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70 |
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71 def is_legal_urban(dim: Int, path: Path)(x: Pos): Boolean = |
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72 0 <= x._1 && 0 <= x._2 && x._1 < dim && x._2 < dim && !path.contains(x) |
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73 |
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74 def moves_urban(x: Pos): List[Pos] = |
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75 List(( 1, 2),( 2, 1),( 2, -1),( 1, -2), |
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76 (-1, -2),(-2, -1),(-2, 1),(-1, 2)).map(add_pair_urban(x)) |
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77 |
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78 def legal_moves_urban(dim: Int, path: Path, x: Pos): List[Pos] = |
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79 moves_urban(x).filter(is_legal_urban(dim, path)) |
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80 |
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81 def correct_urban(dim: Int)(p: Path): Boolean = p match { |
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82 case Nil => true |
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83 case x::Nil => true |
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84 case x::y::p => if (legal_moves_urban(dim, p, y).contains(x)) correct_urban(dim)(y::p) else false |
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85 } |
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86 |
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87 enum_tours(5, List((0, 2))).map(correct_urban(5)).forall(_ == true) |
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88 |
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89 |
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90 for (dim <- 1 to 5) { |
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91 println(s"${dim} x ${dim} " + count_tours(dim, List((0, 0)))) |
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92 } |
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93 |
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94 for (dim <- 1 to 5) { |
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95 println(s"${dim} x ${dim} " + count_all_tours(dim)) |
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96 } |
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97 |
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98 for (dim <- 1 to 5) { |
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99 val ts = enum_tours(dim, List((0, 0))) |
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100 println(s"${dim} x ${dim} ") |
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101 if (ts != Nil) { |
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102 print_board(dim, ts.head) |
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103 println(ts.head) |
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104 } |
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105 } |
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106 |
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107 |