



'''simple docstring'''

from __future__ import annotations







def a_     (    __UpperCAmelCase					)				->					str:





			"""simple docstring"""







			snake_case:    Dict      							=len(__UpperCAmelCase					)
			# We need to create solution object to save path.
			snake_case:    Tuple      							=[[0 for _ in range(__UpperCAmelCase					)] for _ in range(__UpperCAmelCase					)]
			snake_case:    int      							=run_maze(__UpperCAmelCase				,							0				,							0				,							__UpperCAmelCase					)
			if solved:
						print('\n'.join(str(__UpperCAmelCase					) for row in solutions					)					)
			else:
						print('No solution exists!'					)
			return solved







def a_     (    __UpperCAmelCase				,							__UpperCAmelCase				,							__UpperCAmelCase				,							__UpperCAmelCase					)				->					Optional[int]:





			"""simple docstring"""







			snake_case:    Any      							=len(__UpperCAmelCase					)
			# Final check point.
			if i == j == (size - 1):
						snake_case:    Union[str, Any]      							=1
						return True

			snake_case:    int      							=(not i < 0) and (not j < 0)  # Check lower bounds
			snake_case:    int      							=(i < size) and (j < size)  # Check upper bounds

			if lower_flag and upper_flag:
						# check for already visited and block points.
						snake_case:    Any      							=(not solutions[i][j]) and (not maze[i][j])
						if block_flag:
									# check visited
									snake_case:    Tuple      							=1

									# check for directions
									if (
									    run_maze(__UpperCAmelCase				,							i + 1				,							__UpperCAmelCase				,							__UpperCAmelCase					)
									    or run_maze(__UpperCAmelCase				,							__UpperCAmelCase				,							j + 1				,							__UpperCAmelCase					)
									    or run_maze(__UpperCAmelCase				,							i - 1				,							__UpperCAmelCase				,							__UpperCAmelCase					)
									    or run_maze(__UpperCAmelCase				,							__UpperCAmelCase				,							j - 1				,							__UpperCAmelCase					)
									):
												return True

									snake_case:    Tuple      							=0
									return False
			return False


if __name__ == "__main__":
	import doctest

	doctest.testmod()











'''simple docstring'''






from __future__ import annotations

import numpy as np
def   UpperCamelCase__ (						lowerCAmelCase    ):





      """simple docstring"""


      _lowerCAmelCase	,				_lowerCAmelCase					=				np.shape(lowerCAmelCase    )
      if rows != columns:
            _lowerCAmelCase					=				(
                """'table' has to be of square shaped array but got a """
                f"{rows}x{columns} array:\n{table}"
            )
            raise ValueError(lowerCAmelCase    )

      _lowerCAmelCase					=				np.zeros((rows, columns)    )
      _lowerCAmelCase					=				np.zeros((rows, columns)    )
      for i in range(lowerCAmelCase    ):
            for j in range(lowerCAmelCase    ):
                  _lowerCAmelCase					=				sum(lower[i][k] * upper[k][j] for k in range(lowerCAmelCase    )    )
                  if upper[j][j] == 0:
                        raise ArithmeticError("""No LU decomposition exists"""    )
                  _lowerCAmelCase					=				(table[i][j] - total) / upper[j][j]
            _lowerCAmelCase					=				1
            for j in range(lowerCAmelCase				,      lowerCAmelCase    ):
                  _lowerCAmelCase					=				sum(lower[i][k] * upper[k][j] for k in range(lowerCAmelCase    )    )
                  _lowerCAmelCase					=				table[i][j] - total
      return lower, upper


if __name__ == "__main__":
      import doctest

      doctest.testmod()






