# Coming Back to Old Problems: How I Finally Wrote a Sudoku Solving Algorithm

###
Ali Spittel
Jun 26 '18
*Updated on Mar 25, 2019 *
・10 min read

Quick warning, this article will be part technical, part personal story, part cultural critique. If you are just here for the code and explanation, jump to the The Initial Approach header!

This story starts a few years ago in a college computer science classroom. I had an untraditional path to writing code -- I randomly enrolled in a computer science class during my sophomore year of college because I had an extra credit hour and I was curious what it was about. I thought we would learn how to use Microsoft Word and Excel -- I genuinely had no idea what code was. My high school definitely did not have any coding classes, they barely had functioning computers! I didn't play video games or engage in activities that traditionally lead to kids learning how to code either. So coding was brand new to me when I took that Python class in college.

As soon as I walked into the classroom, they had us type Python code into Idle, a text editor that comes with the Python language. They had printed the code and just had us type it in and run it -- I was immediately hooked. Over the course of that class, I built a Tic Tac Toe script with a GUI to input pieces and a Flappy Bird clone. It honestly came pretty easy to me, and I had a ton of fun. I quickly decided to minor in computer science, and I just wanted to write more code.

The next semester, I enrolled in a Data Structures and Algorithms course which was next in the computer science sequence. The class was taught in C++, which, unbeknown to me was supposed to be learned over the summer before the class. It quickly became obvious that the professors were trying to use the class to filter out students -- so many of the enrollees on day one made it through the semester. We even changed classrooms from a lecture hall to a break out room. My pride was the only thing keeping me in the class. I felt completely lost in pretty much every lesson. I spent many all-nighters working on projects and studying for the exams.

One problem in particular really got me -- we were supposed to build a program in C++ that would solve any Sudoku problem. Again, I spent countless hours on the assignment trying to get the code working. By the time the project was due, my solution worked for some of the test cases but not all of them. I ended up getting a C+ on my assignment -- one of my worst grades in all of college.

After that semester, I abandoned my idea of minoring in computer science, completely quit coding, and stuck to what I thought I was good at -- writing and politics.

Of course, funny things happen in life and I obviously started coding again, but it took me a long time to feel like I was a competent programmer.

All that being said, a few years later into my programming journey, I decided to retry implementing the Sudoku solving algorithm to prove it to myself that I could implement it now. The code isn't perfect, but it will solve pretty much any Sudoku puzzle. Let's walk through the algorithm and then the implementation.

## Sudoku Puzzles

In case you haven't played Sudoku puzzles before, they are number puzzles in which each row, column, and 3x3 square in the puzzle must have the numbers 1-9 represented exactly once. There are lots of approaches to solving these puzzles, many of which can be duplicated by a computer instead of a person. Usually, when we solve them using a computer, we will use nested arrays to represent the Sudoku board like so:

```
puzzle = [[5, 3, 0, 0, 7, 0, 0, 0, 0],
[6, 0, 0, 1, 9, 5, 0, 0, 0],
[0, 9, 8, 0, 0, 0, 0, 6, 0],
[8, 0, 0, 0, 6, 0, 0, 0, 3],
[4, 0, 0, 8, 0, 3, 0, 0, 1],
[7, 0, 0, 0, 2, 0, 0, 0, 6],
[0, 6, 0, 0, 0, 0, 2, 8, 0],
[0, 0, 0, 4, 1, 9, 0, 0, 5],
[0, 0, 0, 0, 8, 0, 0, 7, 9]]
```

When solved, the zeros will be filled in with actual numbers:

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

## The Initial Approach

Because I didn't feel like writing a full test suite with different puzzles, I used the challenges on CodeWars to test myself. The first problem I tried was this -- where all of the puzzles were "easy" Sudokus that could be solved without a more complex algorithm.

I decided to try and solve the Sudokus in the way I personally do -- where I would find the possible numbers for a space, keep track of them, and if there is only one possible number plug it into that spot. Since these were easier Sudokus, this approach worked fine for this Kata, and I passed.

Here's my (uncleaned) code!

```
class SudokuSolver:
def __init__(self, puzzle):
self.puzzle = puzzle
self.box_size = 3
def find_possibilities(self, row_number, column_number):
possibilities = set(range(1, 10))
row = self.get_row(row_number)
column = self.get_column(column_number)
box = self.get_box(row_number, column_number)
for item in row + column + box:
if not isinstance(item, list)and item in possibilities:
possibilities.remove(item)
return possibilities
def get_row(self, row_number):
return self.puzzle[row_number]
def get_column(self, column_number):
return [row[column_number] for row in self.puzzle]
def get_box(self, row_number, column_number):
start_y = column_number // 3 * 3
start_x = row_number // 3 * 3
if start_x < 0:
start_x = 0
if start_y < 0:
start_y = 0
box = []
for i in range(start_x, self.box_size + start_x):
box.extend(self.puzzle[i][start_y:start_y+self.box_size])
return box
def find_spot(self):
unsolved = True
while unsolved:
unsolved = False
for row_number, row in enumerate(self.puzzle):
for column_number, item in enumerate(row):
if item == 0:
unsolved = True
possibilities = self.find_possibilities(
row_number, column_number)
if len(possibilities) == 1:
self.puzzle[row_number][column_number] = list(possibilities)[
0]
return self.puzzle
def sudoku(puzzle):
sudoku = SudokuSolver(puzzle)
return sudoku.find_spot()
```

Of course, I also wanted to solve more difficult Sudoku puzzles, so I decided to also implement a more complex algorithm in order to solve those puzzles.

## The Algorithm

One algorithm to solve Sudoku puzzles is the backtracking algorithm. Essentially, you keep trying numbers in empty spots until there aren't any that are possible, then you backtrack and try different numbers in the previous slots.

####
*Shoutout to Wikipedia for the awesome visualization!*

The first thing that I did was continue my "easy" Sudoku solver's approach of finding the possible values for each square based on which values were already in that square's row, column, and box. I stored all of these values in a list so that I could quickly refer to them while backtracking or finding which value to use in that square.

Next, I needed to implement the forward moving and backtracking of putting items in each space. I put markers on each non-given space (so the ones that were zeros when the game started) so that those spaces would be included in the backtracking and given spots wouldn't be. I then iterated through those un-solved spots. I would put the first item of the possible value list in that spot and then move to the next unsolved spot. I would then put the first possible value of that spot in its place. If it conflicted with the value of the previous slot, I would then move to the second item in the list of possible values and then move to the next slot. That process would continue until there was no possible move for a given spot -- i.e. the end of the possible value list was reached and none of the values worked in that row, column, or box. Then, the backtracking algorithm kicked in.

Within the backtracking implementation, the code would move back to the last spot that was filled in and move to the next possible value and then start moving forward again. If the last of the possible values was reached at that spot as well, the backtracking algorithm would keep moving backwards until there was a spot that could be incremented.

Once the end of the puzzle was reached with correct values in each square, the puzzle was solved!

## My Approach

I like object oriented approaches, so I have two different classes in my solution: one for the cell and one for the Sudoku board. My very imperfect code looks like this:

```
class Cell:
"""One individual cell on the Sudoku board"""
def __init__(self, column_number, row_number, number, game):
# Whether or not to include the cell in the backtracking
self.solved = True if number > 0 else False
self.number = number # the current value of the cell
# Which numbers the cell could potentially be
self.possibilities = set(range(1, 10)) if not self.solved else []
self.row = row_number # the index of the row the cell is in
self.column = column_number # the index of the column the cell is in
self.current_index = 0 # the index of the current possibility
self.game = game # the sudoku game the cell belongs to
if not self.solved: # runs the possibility checker
self.find_possibilities()
def check_area(self, area):
"""Checks to see if the cell's current value is a valid sudoku move"""
values = [item for item in area if item != 0]
return len(values) == len(set(values))
def set_number(self):
"""changes the number attribute and also changes the cell's value in the larger puzzle"""
if not self.solved:
self.number = self.possibilities[self.current_index]
self.game.puzzle[self.row][self.column] = self.possibilities[self.current_index]
def handle_one_possibility(self):
"""If the cell only has one possibility, set the cell to that value and mark it as solved"""
if len(self.possibilities) == 1:
self.solved = True
self.set_number()
def find_possibilities(self):
"""filter the possible values for the cell"""
for item in self.game.get_row(self.row) + self.game.get_column(self.column) + self.game.get_box(self.row, self.column):
if not isinstance(item, list) and item in self.possibilities:
self.possibilities.remove(item)
self.possibilities = list(self.possibilities)
self.handle_one_possibility()
def is_valid(self):
"""checks to see if the current number is valid in its row, column, and box"""
for unit in [self.game.get_row(self.row), self.game.get_column(self.column), self.game.get_box(self.row, self.column)]:
if not self.check_area(unit):
return False
return True
def increment_value(self):
"""move number to the next possibility while the current number is invalid and there are possibilities left"""
while not self.is_valid() and self.current_index < len(self.possibilities) - 1:
self.current_index += 1
self.set_number()
class SudokuSolver:
"""contains logic for solving a sudoku puzzle -- even very difficult ones using a backtracking algorithm"""
def __init__(self, puzzle):
self.puzzle = puzzle # the 2d list of spots on the board
self.solve_puzzle = [] # 1d list of the Cell objects
# the size of the boxes within the puzzle -- 3 for a typical puzzle
self.box_size = int(len(self.puzzle) ** .5)
self.backtrack_coord = 0 # what index the backtracking is currently at
def get_row(self, row_number):
"""Get the full row from the puzzle based on the row index"""
return self.puzzle[row_number]
def get_column(self, column_number):
"""Get the full column"""
return [row[column_number] for row in self.puzzle]
def find_box_start(self, coordinate):
"""Get the start coordinate for the small sudoku box"""
return coordinate // self.box_size * self.box_size
def get_box_coordinates(self, row_number, column_number):
"""Get the numbers of the small sudoku box"""
return self.find_box_start(column_number), self.find_box_start(row_number)
def get_box(self, row_number, column_number):
"""Get the small sudoku box for an x and y coordinate"""
start_y, start_x = self.get_box_coordinates(row_number, column_number)
box = []
for i in range(start_x, self.box_size + start_x):
box.extend(self.puzzle[i][start_y:start_y+self.box_size])
return box
def initialize_board(self):
"""create the Cells for each item in the puzzle and get its possibilities"""
for row_number, row in enumerate(self.puzzle):
for column_number, item in enumerate(row):
self.solve_puzzle.append(
Cell(column_number, row_number, item, self))
def move_forward(self):
"""Move forwards to the next cell"""
while self.backtrack_coord < len(self.solve_puzzle) - 1 and self.solve_puzzle[self.backtrack_coord].solved:
self.backtrack_coord += 1
def backtrack(self):
"""Move forwards to the next cell"""
self.backtrack_coord -= 1
while self.solve_puzzle[self.backtrack_coord].solved:
self.backtrack_coord -= 1
def set_cell(self):
"""Set the current cell to work on"""
cell = self.solve_puzzle[self.backtrack_coord]
cell.set_number()
return cell
def reset_cell(self, cell):
"""set a cell back to zero"""
cell.current_index = 0
cell.number = 0
self.puzzle[cell.row][cell.column] = 0
def decrement_cell(self, cell):
"""runs the backtracking algorithm"""
while cell.current_index == len(cell.possibilities) - 1:
self.reset_cell(cell)
self.backtrack()
cell = self.solve_puzzle[self.backtrack_coord]
cell.current_index += 1
def change_cells(self, cell):
"""move forwards or backwards based on the validity of a cell"""
if cell.is_valid():
self.backtrack_coord += 1
else:
self.decrement_cell(cell)
def solve(self):
"""run the other functions necessary for solving the sudoku puzzle"""
self.move_forward()
cell = self.set_cell()
cell.increment_value()
self.change_cells(cell)
def run_solve(self):
"""runs the solver until we are at the end of the puzzle"""
while self.backtrack_coord <= len(self.solve_puzzle) - 1:
self.solve()
def solve(puzzle):
solver = SudokuSolver(puzzle)
solver.initialize_board()
solver.run_solve()
return solver.puzzle
```

## My Takeaways

Sometimes it just takes time and practice -- the Sudoku solver I spent countless college hours on took me less than an hour a few years later. I will say that computer science programs don't tend to start in a way that allows people who didn't write code earlier in life to participate, which in a few years with computer science education policy changes may be okay, but for now eliminates people who grew up in small towns, who weren't interested in coding growing up, or who went to weaker high schools. In part, this definitely contributes to the success of coding bootcamps which start with the fundamentals and teach the less conceptual web development skills rather than heavy algorithms.

I can now write the Sudoku solving algorithm, but I don't think its a necessary skill for developers to have -- I still was a successful software engineer shortly after the time period that I couldn't implement the Sudoku solver. I do think that some computer science fundamentals can be very helpful, even for new developers. For example, the concepts behind Big-O notation can be really helpful for deciding between approaches. That being said, most data structures and algorithms aren't used on a day to day basis, so why are they the basis for interviews and computer science classes instead of the more important things used every day?

I'm happy to see my own personal growth in coding; however, I can't wait for a day where developers aren't jumping through imaginary hoops to prove themselves and learning environments are much more constructive.

From how you described the school you went to, I can say that the education system in Saudi Arabia is the same. We don't have serious computer or programming classes. In best cases they would teach you how to use MS Office and that's it. I started learning programming earlier in my last year of elementary school. It was all self-learning. I kept studying and implementing some simple code using PHP and MySQL. A few years later I lost interest and gradually moved away from programming.

When I went to college I chose a major that is related to computers (MIS), and I was forced to go back to programming, specially since my graduation project was to design a whole system using Java by my own. Now I'm back again and studying so intensively. I'm happy to be programming and learning more stuff about it, but I still regret that I had stopped doing so for long years, I could've been a much better programming now.

Thank you for sharing this Ali (by the way Ali is a male name in Arabic :p)

Sounds familiar to me. I went to school in Mumbai (Bombay), India where they had 'computers' as a subject since 5th grade. They used to take us to a computer tab filled with computers in late 90s, running software from 1980s and let us play Dragon Ball and Pacman on those monochrome monitors. The syllabus used to start with the chapter 'Introduction to Computers', every alternate year and if consisted of repetitive things like "constants and variables". They made us memorize a few QBASIC programs with no clear explanation and we were supposed to 'spit' it out during our practical exams.

The entire environment was to make students believe that computers are boring, hard and the only good use they had was to play video-games.

It took me long to realize the potential of computer programming when I picked it up during my second year of college that was supposed to be based on Electronics Engineering. I find myself lucky to even realize how fun programming can be and it all happened in class when a teacher was pretending to explain us a C program that added two numbers. Majority of my classmates flunked the subject at the end of the term, probably because of the way the subject was presented (mis-represented) to us, but I have been writing programs since that day in 2004.

Adding to that, it is till today that I mostly focus on what I can make a computer do for me rather than to learn theory that makes it all sound more complicated than it is supposed to be.

In retrospect every school I've been to, no matter how much they focused on programming, they always lacked the overall "if a computer can do it, why shouldn't it?" attitude, and were way too much focused on building imaginary systems instead of solving actual problems.

Once I ran into your GitHub account and I saw the sudoku repo I didn't open it though because I don't like spoilers hahaha so I decided to make my own sudoku solver that like your first version only solve easy ones...I tested it using the Gnome sudoku from ubuntu 14.04 and It solves easy and medium level of difficultly this is the repo github.com/lmbarr/sudoku-solver

PD: I didn't know about the backtracking algorithm, now I do know thank to you.

Nice job! Always rewarding to go back and solve problems that stumped in the past. You know... now that I think about it... I'm not sure I ever succeeded at writing a Sudoku solver 🤔 Perhaps it's time to fix that for me, too!

👏 My story to a T

Small town, the computer class was "Advanced Excel" (pivot tables and maybe vlookup), most of the school was on free lunch so computers at home weren't a thing... I'm still dealling with that feeling of not knowing a lick about anything anyone is saying.

I ended up in a couple programming classes due to my math program, but overall, it was just playing around (nothing to show of it either) while unemployed after graduation that helped show my interviewers that I knew how to Google and learn on the job enough to hire me.

Another one! I wasn't given a Sudoku solver to do as part of a class (I trained in engineering), but have done one recently simply because it's really interesting. Pretty sure this one is efficient (github.com/rantydave/sudoku), took a

lotlonger than a couple of hours :)Thanks for sharing Ali. I think it's unfortunate that those first CS courses are often used to

weed outstudents. I'm glad you persevered, even if you did end up dropping the CS minor!Amazing, the sudoku part almost exactly mirrors my experience.

I first wrote it in javascript, with sets of possible numbers for every cell and human-like non-destructive transforms being applied (only answers we are sure about). There were quite a few different transforms, because I picked a modified version of Sudoku which had these additional blob sum constraints, which were interesting because they were indeed sums and not sets of the

`[1,9]`

range.For the UI, it used Vue, but really the renderer was mostly handwritten svg. It solved the lesser difficulties, but then I got bored.

Later, I stumbled upon someone quoting the backtracking algorithm as described in wikipedia. So I implemented it in Rust, with the additional constraints. The first time I wrote it, it solved them all in negligible time. I also tried it 3x4 and 4x4 boards. 4x4 gave noticeable time, and anything above 4x4 took too long to run.

Perhaps applying human-like transforms is computationally valuable at bigger board sizes to reduce the iteration space dramatically.

Maybe I'll still port those, or make the rust code an API or a WASM embed to the UI. Probably not though.

here's the solution (and the explanation) by Peter Norvig, director of research at Google: norvig.com/sudoku.html

The are also some quite efficient, well studied solutions for this problem. For example, check this repo which uses AC3: github.com/msanpe/Sudoku-Backtrack...

Wow! Nice article!

Really loved reading it, ❤️

someday I'll implement this challenge as well!

But currently I don't even know how to play it!

'...less than an hour...' - wow.

To be clear, that was for the easy solver! Not the hard one -- that took a few hours + cleanup