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"""

Given a partially filled 9×9 2D array, the objective is to fill a 9×9

square grid with digits numbered 1 to 9, so that every row, column, and

and each of the nine 3×3 sub-grids contains all of the digits.

This can be solved using Backtracking and is similar to n-queens.

We check to see if a cell is safe or not and recursively call the

function on the next column to see if it returns True. if yes, we

have solved the puzzle. else, we backtrack and place another number

in that cell and repeat this process.

"""

from __future__ import annotations

Matrix = list[list[int]]

# assigning initial values to the grid

initial_grid: Matrix = [

[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

no_solution: Matrix = [

[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 is_safe(grid: Matrix, row: int, column: int, n: int) -> bool:

"""

This function checks the grid to see if each row,

column, and the 3x3 subgrids contain the digit 'n'.

It returns False if it is not 'safe' (a duplicate digit

is found) else returns True if it is 'safe'

"""

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 find_empty_location(grid: Matrix) -> tuple[int, int] | None:

"""

This function finds an empty location so that we can assign a number

for that particular row and column.

"""

for i in range(9):

for j in range(9):

if grid[i][j] == 0:

return i, j

return None

def sudoku(grid: Matrix) -> Matrix | None:

"""

Takes a partially filled-in grid and attempts to assign values to

all unassigned locations in such a way to meet the requirements

for Sudoku solution (non-duplication across rows, columns, and boxes)

>>> sudoku(initial_grid) # doctest: +NORMALIZE_WHITESPACE

[[3, 1, 6, 5, 7, 8, 4, 9, 2],

[5, 2, 9, 1, 3, 4, 7, 6, 8],

[4, 8, 7, 6, 2, 9, 5, 3, 1],

[2, 6, 3, 4, 1, 5, 9, 8, 7],

[9, 7, 4, 8, 6, 3, 1, 2, 5],

[8, 5, 1, 7, 9, 2, 6, 4, 3],

[1, 3, 8, 9, 4, 7, 2, 5, 6],

[6, 9, 2, 3, 5, 1, 8, 7, 4],

[7, 4, 5, 2, 8, 6, 3, 1, 9]]

>>> sudoku(no_solution) is None

True

"""

if location := find_empty_location(grid):

row, column = location

else:

# If the location is ``None``, then the grid is solved.

return grid

for digit in range(1, 10):

if is_safe(grid, row, column, digit):

grid[row][column] = digit

if sudoku(grid) is not None:

return grid

grid[row][column] = 0

return None

def print_solution(grid: Matrix) -> None:

"""

A function to print the solution in the form

of a 9x9 grid

"""

for row in grid:

for cell in row:

print(cell, 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:")

solution = sudoku(example_grid)

if solution is not None:

print_solution(solution)

else:

print("Cannot find a solution.")