


'''simple docstring'''




from math import isclose, sqrt





def 	a_					( lowerCamelCase	:     float	, lowerCamelCase	:     float	, lowerCamelCase	:     float				):
   lowerCAmelCase     =			point_y / 4 / point_x
   lowerCAmelCase     =			2 * normal_gradient / (1 + normal_gradient * normal_gradient)
   lowerCAmelCase     =			(1 - normal_gradient * normal_gradient) / (
       1 + normal_gradient * normal_gradient
   )
   lowerCAmelCase     =			(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
   lowerCAmelCase     =			outgoing_gradient**2 + 4
   lowerCAmelCase     =			2 * outgoing_gradient * (point_y - outgoing_gradient * point_x)
   lowerCAmelCase     =			(point_y - outgoing_gradient * point_x) ** 2 - 100

   lowerCAmelCase     =			(
       -linear_term - sqrt(linear_term**2 - 4 * quadratic_term * constant_term				)
   ) / (2 * quadratic_term)
   lowerCAmelCase     =			(
       -linear_term + sqrt(linear_term**2 - 4 * quadratic_term * constant_term				)
   ) / (2 * quadratic_term)

   # two solutions, one of which is our input point
   lowerCAmelCase     =			x_minus if isclose(lowerCamelCase	, lowerCamelCase				) else x_plus
   lowerCAmelCase     =			point_y + outgoing_gradient * (next_x - point_x)

   return next_x, next_y, outgoing_gradient





def 	a_					( lowerCamelCase	:     float = 1.4	, lowerCamelCase	:     float = -9.6				):
   lowerCAmelCase     =			0
   lowerCAmelCase     =			first_x_coord
   lowerCAmelCase     =			first_y_coord
   lowerCAmelCase     =			(10.1 - point_y) / (0.0 - point_x)

   while not (-0.01 <= point_x <= 0.01 and point_y > 0):
      lowerCAmelCase     ,  lowerCAmelCase     ,  lowerCAmelCase     =			next_point(lowerCamelCase	, lowerCamelCase	, lowerCamelCase				)
      num_reflections += 1

   return num_reflections


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













from __future__ import annotations

UpperCAmelCase__				   =		list[list[int]]

# assigning initial values to the grid
UpperCAmelCase__				   =		[
    [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
UpperCAmelCase__				   =		[
    [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:
 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:
 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:
 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:
 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:")
          UpperCAmelCase__				   =		sudoku(example_grid)
          if solution is not None:
               print_solution(solution)
          else:
               print("Cannot find a solution.")


