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Solve the Defective Chessboard Problem using Dynamic Programming, with the board size 2n x 2n.

The Code must be written in Python.

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SOURCE CODE:

from __future__ import print_function import sys from ortools.sat.python import cp_model def main(board_size): model = cp_model.CpModel() # Creates the variables. # The array index is the column, and the value is the row. queens = [model.NewIntVar(0, board_size - 1, 'x%i' % i) for i in range(board_size)] # Creates the constraints. # The following sets the constraint that all queens are in different rows. model.AddAllDifferent(queens) # Note: all queens must be in different columns because the indices of queens are all different. # The following sets the constraint that no two queens can be on the same diagonal. for i in range(board_size): # Note: is not used in the inner loop. diag1 = [] diag2 = [] for j in range(board_size): # Create variable array for queens(j) + j. q1 = model.NewIntVar(0, 2 * board_size, 'diag1_%i' % i) diag1.append(q1) model.Add(q1 == queens[j] + j) # Create variable array for queens(j) - j. q2 = model.NewIntVar(-board_size, board_size, 'diag2_%i' % i) diag2.append(q2) model.Add(q2 == queens[j] - j) model.AddAllDifferent(diag1) model.AddAllDifferent(diag2) ### Solve model. solver = cp_model.CpSolver() solution_printer = SolutionPrinter(queens) status = solver.SearchForAllSolutions(model, solution_printer) print() print('Solutions found : %i' % solution_printer.SolutionCount()) class SolutionPrinter(cp_model.CpSolverSolutionCallback): """Print intermediate solutions.""" def __init__(self, variables): cp_model.CpSolverSolutionCallback.__init__(self) self.__variables = variables self.__solution_count = 0 def OnSolutionCallback(self): self.__solution_count += 1 for v in self.__variables: print('%s = %i' % (v, self.Value(v)), end = ' ') print() def SolutionCount(self): return self.__solution_count

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class DiagramPrinter(cp_model.CpSolverSolutionCallback): def __init__(self, variables): cp_model.CpSolverSolutionCallback.__init__(self) self.__variables = variables self.__solution_count = 0 def OnSolutionCallback(self): self.__solution_count += 1 for v in self.__variables: queen_col = int(self.Value(v)) board_size = len(self.__variables) # Print row with queen. for j in range(board_size): if j == queen_col: # There is a queen in column j, row i. print("Q", end=" ") else: print("_", end=" ") print() print() def SolutionCount(self): return self.__solution_count if __name__ == '__main__': # By default, solve the 8x8 problem. board_size = 8 if len(sys.argv) > 1: board_size = int(sys.argv[1]) main(board_size)

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