NQueens.ipynb

{
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  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
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   "source": [
    "from ortools.sat.python import cp_model"
   ]
  },
  {
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   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Very lightweight wrapper to allow for a simple callback function to be used to collect results\n",
    "class Collector(cp_model.CpSolverSolutionCallback):\n",
    "    def __init__(self, func):\n",
    "        cp_model.CpSolverSolutionCallback.__init__(self)\n",
    "        self.num_solutions = 0\n",
    "        self.func = func\n",
    "        \n",
    "    def OnSolutionCallback(self):\n",
    "        self.num_solutions += 1\n",
    "        self.func(self, self.num_solutions)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def solve_nqueens(board_size):\n",
    "    model = cp_model.CpModel()\n",
    "\n",
    "    # Variable for the queen in column i representing it's row j in [0,boardsize)\n",
    "    queens = [model.NewIntVar(0, board_size-1, 'q%d' % (i)) for i in range(board_size)]\n",
    "\n",
    "    # Implicit - Queens are in different columns because we are solving for the row of each column\n",
    "    \n",
    "    # Explicit - Queens are in different rows \n",
    "    model.AddAllDifferent(queens)\n",
    "    \n",
    "    # Explicit - Queens are in different diagonals\n",
    "    # q[i] = j therefore q[i] + i = j + i for the queen in column i\n",
    "    # If any two queens j_1 + i_1 == j_2 + i_2 then they lay on an up-right diagonal\n",
    "    # If any two queens j_1 - i_1 == j_2 - i_2 then they lay on a down-right diagonal\n",
    "    \n",
    "    asc = []\n",
    "    dsc = []\n",
    "    for i in range(board_size):\n",
    "        # Create variables to hold the results of our equations j + i and j - i for the current i\n",
    "        j_add_i = model.NewIntVar(          0, board_size*2, 'j_add_%d' % (i))\n",
    "        j_sub_i = model.NewIntVar(-board_size, board_size  , 'j_sub_%d' % (i))\n",
    "        \n",
    "        # Constrain that these variables must be equal to the results of the equations\n",
    "        model.Add(j_add_i == queens[i] + i)\n",
    "        model.Add(j_sub_i == queens[i] - i)\n",
    "        \n",
    "        # Append the variables which equal the results of j + i and j - i respectively to seperate lists\n",
    "        asc += [j_add_i]\n",
    "        dsc += [j_sub_i]\n",
    "    \n",
    "    # Constraining that each list contains all different values ensures that no two share the same value\n",
    "    # This asserts that queens can never be on the same ascending or descending diagonals\n",
    "    model.AddAllDifferent(asc)\n",
    "    model.AddAllDifferent(dsc)\n",
    "    \n",
    "    # Callback function to pretty print the solution as a grid\n",
    "    def on_solution(solution, count):\n",
    "        print('Solution:', count)\n",
    "        for i in range(board_size):\n",
    "            j = int(solution.Value(queens[i]))\n",
    "            line = '|'.join(['Q' if (k == j) else ' ' for k in range(board_size)])\n",
    "            print(line + (('\\n' + '-'*len(line)) if (i != board_size-1) else ''))\n",
    "        print()\n",
    "    \n",
    "    # Run the solver\n",
    "    solver = cp_model.CpSolver()\n",
    "    status = solver.SearchForAllSolutions(model, Collector(on_solution))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Solution: 1\n",
      " | |Q| | | \n",
      "-----------\n",
      " | | | | |Q\n",
      "-----------\n",
      " |Q| | | | \n",
      "-----------\n",
      " | | | |Q| \n",
      "-----------\n",
      "Q| | | | | \n",
      "-----------\n",
      " | | |Q| | \n",
      "\n",
      "Solution: 2\n",
      " | | |Q| | \n",
      "-----------\n",
      "Q| | | | | \n",
      "-----------\n",
      " | | | |Q| \n",
      "-----------\n",
      " |Q| | | | \n",
      "-----------\n",
      " | | | | |Q\n",
      "-----------\n",
      " | |Q| | | \n",
      "\n",
      "Solution: 3\n",
      " |Q| | | | \n",
      "-----------\n",
      " | | |Q| | \n",
      "-----------\n",
      " | | | | |Q\n",
      "-----------\n",
      "Q| | | | | \n",
      "-----------\n",
      " | |Q| | | \n",
      "-----------\n",
      " | | | |Q| \n",
      "\n",
      "Solution: 4\n",
      " | | | |Q| \n",
      "-----------\n",
      " | |Q| | | \n",
      "-----------\n",
      "Q| | | | | \n",
      "-----------\n",
      " | | | | |Q\n",
      "-----------\n",
      " | | |Q| | \n",
      "-----------\n",
      " |Q| | | | \n",
      "\n"
     ]
    }
   ],
   "source": [
    "solve_nqueens(6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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