



from math import isclose, sqrt







def 	lowerCamelCase_   (  _a    :		float					,						_a    :		float					,						_a    :		float  ):



    '''simple docstring'''




    UpperCAmelCase_			:    List[str]   						= point_y / 4 / point_x
    UpperCAmelCase_			:    int   						= 2 * normal_gradient / (1 + normal_gradient * normal_gradient)
    UpperCAmelCase_			:    List[Any]   						= (1 - normal_gradient * normal_gradient) / (
        1 + normal_gradient * normal_gradient
    )
    UpperCAmelCase_			:    List[str]   						= (sa - ca * incoming_gradient) / (ca + sa * incoming_gradient)

    # to find the next point, solve the simultaeneous equations:
    # y^2 + 4x^2 = 100
    # y - b = m * (x - a)
    # ==> A x^2 + B x + C = 0
    UpperCAmelCase_			:    List[str]   						= outgoing_gradient**2 + 4
    UpperCAmelCase_			:    Optional[int]   						= 2 * outgoing_gradient * (point_y - outgoing_gradient * point_x)
    UpperCAmelCase_			:    Any   						= (point_y - outgoing_gradient * point_x) ** 2 - 100

    UpperCAmelCase_			:    Any   						= (
        -linear_term - sqrt(linear_term**2 - 4 * quadratic_term * constant_term  )
    ) / (2 * quadratic_term)
    UpperCAmelCase_			:    Optional[int]   						= (
        -linear_term + sqrt(linear_term**2 - 4 * quadratic_term * constant_term  )
    ) / (2 * quadratic_term)

    # two solutions, one of which is our input point
    UpperCAmelCase_			:    Tuple   						= x_minus if isclose(__SCREAMING_SNAKE_CASE					,						__SCREAMING_SNAKE_CASE  ) else x_plus
    UpperCAmelCase_			:    List[str]   						= point_y + outgoing_gradient * (next_x - point_x)

    return next_x, next_y, outgoing_gradient







def 	lowerCamelCase_   (  _a    :		float = 1.4					,						_a    :		float = -9.6  ):



    '''simple docstring'''




    UpperCAmelCase_			:    int   						= 0
    UpperCAmelCase_			:    float   						= first_x_coord
    UpperCAmelCase_			:    float   						= first_y_coord
    UpperCAmelCase_			:    float   						= (1_0.1 - point_y) / (0.0 - point_x)

    while not (-0.0_1 <= point_x <= 0.0_1 and point_y > 0):
        UpperCAmelCase_			:    Dict   						= next_point(__SCREAMING_SNAKE_CASE					,						__SCREAMING_SNAKE_CASE					,						__SCREAMING_SNAKE_CASE  )
        num_reflections += 1

    return num_reflections


if __name__ == "__main__":
      print(F"{solution() = }")

"""simple docstring"""







from __future__ import annotations

__SCREAMING_SNAKE_CASE    	=[]



def 			lowercase__(       __SCREAMING_SNAKE_CASE							:     list[list[int]]     ,			__SCREAMING_SNAKE_CASE							:     int     ,			__SCREAMING_SNAKE_CASE							:     int      ):
     for i in range(len(__SCREAMING_SNAKE_CASE      )      ):
          if board[row][i] == 1:
               return False
     for i in range(len(__SCREAMING_SNAKE_CASE      )      ):
          if board[i][column] == 1:
               return False
     for i, j in zip(range(__SCREAMING_SNAKE_CASE     ,			-1     ,			-1      )     ,			range(__SCREAMING_SNAKE_CASE     ,			-1     ,			-1      )      ):
          if board[i][j] == 1:
               return False
     for i, j in zip(range(__SCREAMING_SNAKE_CASE     ,			-1     ,			-1      )     ,			range(__SCREAMING_SNAKE_CASE     ,			len(__SCREAMING_SNAKE_CASE      )      )      ):
          if board[i][j] == 1:
               return False
     return True



def 			lowercase__(       __SCREAMING_SNAKE_CASE							:     list[list[int]]     ,			__SCREAMING_SNAKE_CASE							:     int      ):
     if row >= len(__SCREAMING_SNAKE_CASE      ):
          solution.append(__SCREAMING_SNAKE_CASE      )
          printboard(__SCREAMING_SNAKE_CASE      )
          print()
          return True
     for i in range(len(__SCREAMING_SNAKE_CASE      )      ):
          if is_safe(__SCREAMING_SNAKE_CASE     ,			__SCREAMING_SNAKE_CASE     ,			__SCREAMING_SNAKE_CASE      ):
               lowercase_	:				int													=	1
               solve(__SCREAMING_SNAKE_CASE     ,			row + 1      )
               lowercase_	:				Dict													=	0
     return False



def 			lowercase__(       __SCREAMING_SNAKE_CASE							:     list[list[int]]      ):
     for i in range(len(__SCREAMING_SNAKE_CASE      )      ):
          for j in range(len(__SCREAMING_SNAKE_CASE      )      ):
               if board[i][j] == 1:
                    print('Q'     ,			end=' '      )
               else:
                    print('.'     ,			end=' '      )
          print()


# n=int(input("The no. of queens"))
__SCREAMING_SNAKE_CASE    	=8
__SCREAMING_SNAKE_CASE    	=[[0 for i in range(n)] for j in range(n)]
solve(board, 0)







print("The total no. of solutions are :", len(solution))

