

from __future__ import annotations

from scipy.special import comb  # type: ignore


class       SCREAMING_SNAKE_CASE__					:





    def __init__(      self						,   a):
         lowercase__    :   Tuple      =       list_of_points
         # Degree determines the flexibility of the curve.
         # Degree = 1 will produce a straight line.
         lowercase__    :   Any      =       len(lowerCamelCase_) - 1





    def       snake_case_				(      self						,   a):
         assert 0 <= t <= 1, "Time t must be between 0 and 1."
         lowercase__    :   List[str]      =       []
         for i in range(len(self.list_of_points)):
              # basis function for each i
              output_values.append(
                  comb(self.degree						,   lowerCamelCase_) * ((1 - t) ** (self.degree - i)) * (t**i))
         # the basis must sum up to 1 for it to produce a valid Bezier curve.
         assert round(sum(lowerCamelCase_)						,   5) == 1
         return output_values





    def       snake_case_				(      self						,   a):
         assert 0 <= t <= 1, "Time t must be between 0 and 1."

         lowercase__    :   Union[str, Any]      =       self.basis_function(lowerCamelCase_)
         lowercase__    :   Union[str, Any]      =       0.0
         lowercase__    :   Any      =       0.0
         for i in range(len(self.list_of_points)):
              # For all points, sum up the product of i-th basis function and i-th point.
              x += basis_function[i] * self.list_of_points[i][0]
              y += basis_function[i] * self.list_of_points[i][1]
         return (x, y)





    def       snake_case_				(      self						,   a = 0.01):
         from matplotlib import pyplot as plt  # type: ignore

         lowercase__    :   List[str]      =       []  # x coordinates of points to plot
         lowercase__    :   List[Any]      =       []  # y coordinates of points to plot

         lowercase__    :   Optional[int]      =       0.0
         while t <= 1:
              lowercase__    :   int      =       self.bezier_curve_function(lowerCamelCase_)
              to_plot_x.append(value[0])
              to_plot_y.append(value[1])
              t += step_size

         lowercase__    :   List[str]      =       [i[0] for i in self.list_of_points]
         lowercase__    :   Union[str, Any]      =       [i[1] for i in self.list_of_points]

         plt.plot(
             lowerCamelCase_						,   lowerCamelCase_						,   color='blue'						,   label='Curve of Degree ' + str(self.degree)						,   )
         plt.scatter(lowerCamelCase_						,   lowerCamelCase_						,   color='red'						,   label='Control Points')
         plt.legend()
         plt.show()


if __name__ == "__main__":
       import doctest

       doctest.testmod()

       BezierCurve([(1, 2), (3, 5)]).plot_curve()  # degree 1
       BezierCurve([(0, 0), (5, 5), (5, 0)]).plot_curve()  # degree 2
       BezierCurve([(0, 0), (5, 5), (5, 0), (2.5, -2.5)]).plot_curve()  # degree 3






from __future__ import annotations

_lowercase: Tuple      			=		list[list[int]]

# assigning initial values to the grid
_lowercase: Matrix      			=		[
    [3, 0, 6, 5, 0, 8, 4, 0, 0],
    [5, 2, 0, 0, 0, 0, 0, 0, 0],
    [0, 8, 7, 0, 0, 0, 0, 3, 1],
    [0, 0, 3, 0, 1, 0, 0, 8, 0],
    [9, 0, 0, 8, 6, 3, 0, 0, 5],
    [0, 5, 0, 0, 9, 0, 6, 0, 0],
    [1, 3, 0, 0, 0, 0, 2, 5, 0],
    [0, 0, 0, 0, 0, 0, 0, 7, 4],
    [0, 0, 5, 2, 0, 6, 3, 0, 0],
]

# a grid with no solution
_lowercase: Matrix      			=		[
    [5, 0, 6, 5, 0, 8, 4, 0, 3],
    [5, 2, 0, 0, 0, 0, 0, 0, 2],
    [1, 8, 7, 0, 0, 0, 0, 3, 1],
    [0, 0, 3, 0, 1, 0, 0, 8, 0],
    [9, 0, 0, 8, 6, 3, 0, 0, 5],
    [0, 5, 0, 0, 9, 0, 6, 0, 0],
    [1, 3, 0, 0, 0, 0, 2, 5, 0],
    [0, 0, 0, 0, 0, 0, 0, 7, 4],
    [0, 0, 5, 2, 0, 6, 3, 0, 0],
]






def     a(      A       :    Matrix ,     A       :    int ,     A       :    int ,     A       :    int       )		->       bool:
					"""simple docstring"""





					for i in range(9       ):
										if grid[row][i] == n or grid[i][column] == n:
															return False

					for i in range(3       ):
										for j in range(3       ):
															if grid[(row - row % 3) + i][(column - column % 3) + j] == n:
																				return False

					return True






def     a(      A       :    Matrix       )		->       tuple[int, int] | None:
					"""simple docstring"""





					for i in range(9       ):
										for j in range(9       ):
															if grid[i][j] == 0:
																				return i, j
					return None






def     a(      A       :    Matrix       )		->       Matrix | None:
					"""simple docstring"""





					if location := find_empty_location(A       ):
										a     ,				a					     =   location
					else:
										# If the location is ``None``, then the grid is solved.
										return grid

					for digit in range(1 ,     10       ):
										if is_safe(A ,     A ,     A ,     A       ):
															a					     =   digit

															if sudoku(A       ) is not None:
																				return grid

															a					     =   0

					return None






def     a(      A       :    Matrix       )		->       None:
					"""simple docstring"""





					for row in grid:
										for cell in row:
															print(A ,     end=" "       )
										print()


if __name__ == "__main__":
				# make a copy of grid so that you can compare with the unmodified grid
				for example_grid in (initial_grid, no_solution):
								print("\nExample grid:\n" + "=" * 20)
								print_solution(example_grid)
								print("\nExample grid solution:")
								_lowercase: List[str]      			=		sudoku(example_grid)
								if solution is not None:
												print_solution(solution)
								else:
												print("Cannot find a solution.")
