Added sudoku solving program in backtracking algorithms (#1128)

* Added sudoku solver in backtracking

* Added sudoku solver program

* Added sudoku solver

* Added sudoku solver

* Format with black, add doctests, cleanup main

backtracking/sudoku.py

@@ -0,0 +1,151 @@
+"""
+
+    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.
+
+"""
+
+# assigning initial values to the grid
+initial_grid = [
+    [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 = [
+    [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, row, column, n):
+    """
+    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 is_completed(grid):
+    """
+    This function checks if the puzzle is completed or not.
+    it is completed when all the cells are assigned with a number(not zero)
+    and There is no repeating number in any column, row or 3x3 subgrid.
+
+    """
+
+    for row in grid:
+        for cell in row:
+            if cell == 0:
+                return False
+
+    return True
+
+
+def find_empty_location(grid):
+    """
+    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
+
+
+def sudoku(grid):
+    """
+    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)
+     False
+     """
+
+    if is_completed(grid):
+        return grid
+
+    row, column = find_empty_location(grid)
+
+    for digit in range(1, 10):
+        if is_safe(grid, row, column, digit):
+            grid[row][column] = digit
+
+            if sudoku(grid):
+                return grid
+
+            grid[row][column] = 0
+
+    return False
+
+
+def print_solution(grid):
+    """
+    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 grid in (initial_grid, no_solution):
+        grid = list(map(list, grid))
+        solution = sudoku(grid)
+        if solution:
+            print("grid after solving:")
+            print_solution(solution)
+        else:
+            print("Cannot find a solution.")