#### Knight Tour

K
D
T
E
and 1 more contributors
```# Knight Tour Intro: https://www.youtube.com/watch?v=ab_dY3dZFHM

from __future__ import annotations

def get_valid_pos(position: tuple[int, int], n: int) -> list[tuple[int, int]]:
"""
Find all the valid positions a knight can move to from the current position.

>>> get_valid_pos((1, 3), 4)
[(2, 1), (0, 1), (3, 2)]
"""

y, x = position
positions = [
(y + 1, x + 2),
(y - 1, x + 2),
(y + 1, x - 2),
(y - 1, x - 2),
(y + 2, x + 1),
(y + 2, x - 1),
(y - 2, x + 1),
(y - 2, x - 1),
]
permissible_positions = []

for position in positions:
y_test, x_test = position
if 0 <= y_test < n and 0 <= x_test < n:
permissible_positions.append(position)

return permissible_positions

def is_complete(board: list[list[int]]) -> bool:
"""
Check if the board (matrix) has been completely filled with non-zero values.

>>> is_complete([[1]])
True

>>> is_complete([[1, 2], [3, 0]])
False
"""

return not any(elem == 0 for row in board for elem in row)

def open_knight_tour_helper(
board: list[list[int]], pos: tuple[int, int], curr: int
) -> bool:
"""
Helper function to solve knight tour problem.
"""

if is_complete(board):
return True

for position in get_valid_pos(pos, len(board)):
y, x = position

if board[y][x] == 0:
board[y][x] = curr + 1
if open_knight_tour_helper(board, position, curr + 1):
return True
board[y][x] = 0

return False

def open_knight_tour(n: int) -> list[list[int]]:
"""
Find the solution for the knight tour problem for a board of size n. Raises
ValueError if the tour cannot be performed for the given size.

>>> open_knight_tour(1)
[[1]]

>>> open_knight_tour(2)
Traceback (most recent call last):
...
ValueError: Open Kight Tour cannot be performed on a board of size 2
"""

board = [[0 for i in range(n)] for j in range(n)]

for i in range(n):
for j in range(n):
board[i][j] = 1
if open_knight_tour_helper(board, (i, j), 1):
return board
board[i][j] = 0

msg = f"Open Kight Tour cannot be performed on a board of size {n}"
raise ValueError(msg)

if __name__ == "__main__":
import doctest

doctest.testmod()
```