sudoku Algorithm
The Sudoku Solver Algorithm is an advanced technique designed to solve Sudoku puzzles by implementing various logical strategies and computational methods. It aims to find the correct numerical sequence for each row, column, and 3x3 grid in the puzzle without violating the basic rules of Sudoku. The algorithm is capable of solving puzzles of varying difficulty levels, from simple to highly complex ones. The core idea behind this algorithm is to use a combination of logic-based techniques, such as naked singles, hidden singles, locked candidates, and more, along with backtracking, which is a form of trial-and-error, to deduce the correct values for each cell in the puzzle.
One of the most popular techniques used in the Sudoku Solver Algorithm is backtracking, which involves placing a number in a cell and then proceeding to the next cell to place another number. If the algorithm reaches a point where the current number cannot be placed without violating the puzzle's rules, it will backtrack to the previous cell and try a different number. This process continues until the solution is found or all possibilities are exhausted. Another essential aspect of the algorithm is the elimination method, which systematically narrows down the possible values for each cell by examining the numbers present in the corresponding row, column, and 3x3 grid. By combining these techniques, the Sudoku Solver Algorithm can efficiently solve even the most challenging puzzles, making it a valuable tool for both casual players and competitive Sudoku enthusiasts.
"""
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 non-zero number.
>>> is_completed([[0]])
False
>>> is_completed([[1]])
True
>>> is_completed([[1, 2], [0, 4]])
False
>>> is_completed([[1, 2], [3, 4]])
True
>>> is_completed(initial_grid)
False
>>> is_completed(no_solution)
False
"""
return all(all(cell != 0 for cell in row) for row in grid)
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.")
