

from itertools import permutations
def   _lowerCAmelCase(						__A = 5      ):
  if gon_side < 3 or gon_side > 5:
    raise ValueError("gon_side must be in the range [3, 5]"      )

  # Since it's 16, we know 10 is on the outer ring
  # Put the big numbers at the end so that they are never the first number
  UpperCAmelCase          =	list(range(gon_side + 1 ,							0 ,							-1      )      )
  UpperCAmelCase          =	list(range(gon_side + 2 ,							gon_side * 2 + 1      )      )

  for perm in permutations(small_numbers + big_numbers      ):
    UpperCAmelCase          =	generate_gon_ring(__A ,							list(__A      )      )
    if is_magic_gon(__A      ):
      return int("".join(str(__A      ) for n in numbers      )      )

  UpperCAmelCase          =	F"Magic {gon_side}-gon ring is impossible"
  raise ValueError(__A      )
def   _lowerCAmelCase(						__A ,							__A      ):
  UpperCAmelCase          =	[0] * (gon_side * 3)
  UpperCAmelCase          =	perm[0:3]
  perm.append(perm[1]      )

  UpperCAmelCase          =	1 if gon_side < 5 else 2

  for i in range(1 ,							len(__A      ) // 3 + magic_number      ):
    UpperCAmelCase          =	perm[2 * i + 1]
    UpperCAmelCase          =	result[3 * i - 1]
    UpperCAmelCase          =	perm[2 * i + 2]

  return result
def   _lowerCAmelCase(						__A      ):
  if len(__A      ) % 3 != 0:
    raise ValueError("a gon ring should have a length that is a multiple of 3"      )

  if min(numbers[::3]      ) != numbers[0]:
    return False

  UpperCAmelCase          =	sum(numbers[:3]      )

  return all(sum(numbers[i : i + 3]      ) == total for i in range(3 ,							len(__A      ) ,							3      )      )


if __name__ == "__main__":
      print(solution())






from typing import SupportsIndex

import numpy as np
from scipy.ndimage import convolve





def 				__a							(       __A			,							__A			,							__A						):
    SCREAMING_SNAKE_CASE__         =    np.stack((horizontal_flow, vertical_flow)			,							2						)

    # Create a grid of all pixel coordinates and subtract the flow to get the
    # target pixels coordinates
    SCREAMING_SNAKE_CASE__         =    np.stack(
        np.meshgrid(np.arange(0			,							image.shape[1]						)			,							np.arange(0			,							image.shape[0]						)						)			,							2						)
    SCREAMING_SNAKE_CASE__         =    np.round(grid - flow						).astype(np.intaa						)

    # Find the locations outside of the original image
    SCREAMING_SNAKE_CASE__         =    (grid < 0) | (grid >= np.array([image.shape[1], image.shape[0]]						))
    SCREAMING_SNAKE_CASE__         =    0

    SCREAMING_SNAKE_CASE__         =    image[grid[:, :, 1], grid[:, :, 0]]

    # Set pixels at invalid locations to 0
    SCREAMING_SNAKE_CASE__         =    0

    return warped





def 				__a							(       __A			,							__A			,							__A			,							__A = None			,							):
    if alpha is None:
        SCREAMING_SNAKE_CASE__         =    0.1

    # Initialize flow
    SCREAMING_SNAKE_CASE__         =    np.zeros_like(__A						)
    SCREAMING_SNAKE_CASE__         =    np.zeros_like(__A						)

    # Prepare kernels for the calculation of the derivatives and the average velocity
    SCREAMING_SNAKE_CASE__         =    np.array([[-1, 1], [-1, 1]]						) * 0.25
    SCREAMING_SNAKE_CASE__         =    np.array([[-1, -1], [1, 1]]						) * 0.25
    SCREAMING_SNAKE_CASE__         =    np.array([[1, 1], [1, 1]]						) * 0.25
    SCREAMING_SNAKE_CASE__         =    np.array(
        [[1 / 12, 1 / 6, 1 / 12], [1 / 6, 0, 1 / 6], [1 / 12, 1 / 6, 1 / 12]]						)

    # Iteratively refine the flow
    for _ in range(__A						):
        SCREAMING_SNAKE_CASE__         =    warp(__A			,							__A			,							__A						)
        SCREAMING_SNAKE_CASE__         =    convolve(__A			,							__A						) + convolve(__A			,							__A						)
        SCREAMING_SNAKE_CASE__         =    convolve(__A			,							__A						) + convolve(__A			,							__A						)
        SCREAMING_SNAKE_CASE__         =    convolve(__A			,							__A						) + convolve(__A			,							-kernel_t						)

        SCREAMING_SNAKE_CASE__         =    convolve(__A			,							__A						)
        SCREAMING_SNAKE_CASE__         =    convolve(__A			,							__A						)

        # This updates the flow as proposed in the paper (Step 12)
        SCREAMING_SNAKE_CASE__         =    (
            derivative_x * avg_horizontal_velocity
            + derivative_y * avg_vertical_velocity
            + derivative_t
        )
        SCREAMING_SNAKE_CASE__         =    update / (alpha**2 + derivative_x**2 + derivative_y**2)

        SCREAMING_SNAKE_CASE__         =    avg_horizontal_velocity - derivative_x * update
        SCREAMING_SNAKE_CASE__         =    avg_vertical_velocity - derivative_y * update

    return horizontal_flow, vertical_flow


if __name__ == "__main__":
  import doctest

  doctest.testmod()


