


'''simple docstring'''
from __future__ import annotations

__SCREAMING_SNAKE_CASE					:				Any									=		list[list[int]]

# assigning initial values to the grid
__SCREAMING_SNAKE_CASE					:				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
__SCREAMING_SNAKE_CASE					:				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_				(	UpperCamelCase_						,  UpperCamelCase_						,  UpperCamelCase_						,  UpperCamelCase_ ):
							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_				(	UpperCamelCase_ ):
							for i in range(9 ):
														for j in range(9 ):
																					if grid[i][j] == 0:
																												return i, j
							return None







def       a_				(	UpperCamelCase_ ):
							if location := find_empty_location(UpperCamelCase_ ):
														A_    ,					A_			    =		location
							else:
														# If the location is ``None``, then the grid is solved.
														return grid

							for digit in range(1						,  1_0 ):
														if is_safe(UpperCamelCase_						,  UpperCamelCase_						,  UpperCamelCase_						,  UpperCamelCase_ ):
																					A_			    =		digit

																					if sudoku(UpperCamelCase_ ) is not None:
																												return grid

																					A_			    =		0

							return None







def       a_				(	UpperCamelCase_ ):
							for row in grid:
														for cell in row:
																					print(UpperCamelCase_						,  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:''')
														__SCREAMING_SNAKE_CASE					:				str									=		sudoku(example_grid)
														if solution is not None:
																					print_solution(solution)
														else:
																					print('''Cannot find a solution.''')









'''simple docstring'''
def       a_				(	UpperCamelCase_ ):

							if length <= 0 or not isinstance(UpperCamelCase_						,  UpperCamelCase_ ):
														raise ValueError("Length must be a positive integer." )
							return [n * (2 * n - 1) for n in range(UpperCamelCase_ )]


if __name__ == "__main__":
							print(hexagonal_numbers(length=5))
							print(hexagonal_numbers(length=10))





