Sudoku is a logic-based, combinatorial number-placement puzzle. The objective is to fill a 9×9 grid with digits so that each column, each row, and each of the nine 3×3 sub-grids that compose the grid contain all of the digits from 1 to 9. The rule of the game is that you can not use the same number more then once in a given row or column.

To help us solve the puzzle in the above image we will put the numbers or the lack of it into a list of list or a 2D array, like so. As a side note, we pre-fill the empty fields we have in the puzzle with zeros.

grid [[5,3,0,0,7,0,0,0,0],

And to helps us to a bit with the readability of the matrix we are going to utilize a package call numpy, which prints the matrix as we have above:

import numpy as np

To solve the puzzle we are going to use recursion. But first we need to write a helper function to check if it is actually possible to add a given number into a cell.

def possible(y, x, n):
    global grid

    # Check if a given number could be added in the row
    for i in range(0, 9):
        if grid[y][i] == n:
            return False

    # Check if a given number could be added in the column
    for i in range(0, 9):
        if grid[i][x] == n:
            return False

    # Then we check if we can add a given number into the sub-grid (square)
    x0 = (x // 3) * 3
    y0 = (y // 3) * 3

    for i in range(0, 3):
        for j in range(0, 3):
            if grid[y0 + i][x0 + j] == n:
                return False

    return True

Now to actually solve the puzzle we iterate over the grid using recursion and backtracking.

def solve():
    global grid

    for y in range(9):
        for x in range(9):
            if grid[y][x] == 0:
                for n in range(1, 10):
                    if possible(y, x, n):
                        grid[y][x] = n
                        solve()  # Here we enter into our recursion
                        grid[y][x] = 0  # And if we got a bad solution we backtrack.
                return  # We end up in a dead end we return

    # Print solution

    # We can use the follow "trick" to re-run the code to find more solutions
    input("Solve again?")