Given a partially filled 99 2D array, the objective is to fill a 99 square grid with digits numbered 1 to 9, so that every row, column, and and each of the nine 33 subgrids contains all of the digits. This can be solved using Backtracking and is similar to nqueens. 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. assigning initial values to the grid a grid with no solution 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' This function finds an empty location so that we can assign a number for that particular row and column. Takes a partially filledin grid and attempts to assign values to all unassigned locations in such a way to meet the requirements for Sudoku solution nonduplication across rows, columns, and boxes  sudokuinitialgrid   doctest: NORMALIZEWHITESPACE 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  sudokunosolution is None True If the location is None, then the grid is solved. A function to print the solution in the form of a 9x9 grid make a copy of grid so that you can compare with the unmodified grid
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 n in {grid[row][i], grid[i][column]}:
            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.")
