



"""simple docstring"""





from __future__ import annotations
def      a__			(   lowerCAmelCase__   ):
						UpperCAmelCase_    				=							len(lowerCAmelCase__   )
						# We need to create solution object to save path.
						UpperCAmelCase_    				=							[[0 for _ in range(lowerCAmelCase__   )] for _ in range(lowerCAmelCase__   )]
						UpperCAmelCase_    				=							run_maze(lowerCAmelCase__			,    0			,    0			,    lowerCAmelCase__   )
						if solved:
												print("\n".join(str(lowerCAmelCase__   ) for row in solutions   )   )
						else:
												print("No solution exists!"   )
						return solved
def      a__			(   lowerCAmelCase__			,    lowerCAmelCase__			,    lowerCAmelCase__			,    lowerCAmelCase__   ):
						UpperCAmelCase_    				=							len(lowerCAmelCase__   )
						# Final check point.
						if i == j == (size - 1):
												UpperCAmelCase_    				=							1
												return True

						UpperCAmelCase_    				=							(not i < 0) and (not j < 0)  # Check lower bounds
						UpperCAmelCase_    				=							(i < size) and (j < size)  # Check upper bounds

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

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

																		UpperCAmelCase_    				=							0
																		return False
						return False


if __name__ == "__main__":
		import doctest

		doctest.testmod()









"""simple docstring"""





from __future__ import annotations

import math
def      a__			(   lowerCAmelCase__   ):

      if 1 < number < 4:
            # 2 and 3 are primes
            return True
      elif number < 2 or number % 2 == 0 or number % 3 == 0:
            # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
            return False

      # All primes number are in format of 6k +/- 1
      for i in range(5			,    int(math.sqrt(lowerCAmelCase__   ) + 1   )			,    6   ):
            if number % i == 0 or number % (i + 2) == 0:
                  return False
      return True


lowerCamelCase      =		[num for num in range(3, 100_001, 2) if not is_prime(num)]
def      a__			(   lowerCAmelCase__   ):
      if not isinstance(lowerCAmelCase__			,    lowerCAmelCase__   ):
            raise ValueError("n must be an integer"   )
      if n <= 0:
            raise ValueError("n must be >= 0"   )

      UpperCAmelCase_    				=							[]
      for num in range(len(lowerCAmelCase__   )   ):
            UpperCAmelCase_    				=							0
            while 2 * i * i <= odd_composites[num]:
                  UpperCAmelCase_    				=							odd_composites[num] - 2 * i * i
                  if is_prime(lowerCAmelCase__   ):
                        break
                  i += 1
            else:
                  list_nums.append(odd_composites[num]   )
                  if len(lowerCAmelCase__   ) == n:
                        return list_nums

      return []
def      a__			(   ):
      return compute_nums(1   )[0]


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




