# Sudoku Puzzle Solver

I have written a program to solve Sudoku puzzles as a (fun?) way to learn Python. It solves puzzles as a human would, using reasoning algorithms rather than brute force/backtracking, as I thought it would a more interesting challenge that way.

I really looking for any feedback on how to reduce indentation, lines of code, or increase efficiency. I'm sure the functions that I've repeated for columns/rows/subgrids can be merged into one, but I just can't think of an elegant way. Any general feedback would be more than welcome too, please lay into the code as much as possible. I'm a 2nd year computing (CS but lite) student if that makes any difference but probably not.

I appreciate this involves domain specific knowledge i.e. how to solve Sudoku puzzles. I created all the algorithms using only the following resource and all of the techniques contained in the program are described here for anyone interested.

import itertools

# TERMINOLOGY
# grid:         The main 9x9 grid of the puzzle.
# sub-grid:     A 3x3 sub-grid within the main grid.
# square:       A single square within the grid.
# row:          A whole row of squares within the main grid.
# column:       A whole column of squares within the main grid.
# sub-row:      A row of three squares in a sub-grid.
# sub-column:   A column of three squares in a sub-grid.
# candidate:    A possible solution for a square.
# solved:       A square is solved when the correct value is filled.
#
# For in-depth descriptions of the various sudoku solving techniques
# used in this program, visit:
# http://www.paulspages.co.uk/sudoku/howtosolve/index.htm
# This is the sole resource used to generate the techniques
# found in this program.

# Holds puzzle itself as 2d list. Blank squares represented as 0.
# Individual square access: main_grid[x][y]
main_grid = [[] for x in range(0, 9)]

# Holds all possible candidates for each square as a 2d list of sets.
# Individual square set access: candidates_grid[x][y]
candidates_grid = [[set() for y in range(0, 9)] for x in range(0, 9)]

# Holds all solved values in an individual row/col/sub-grid
col_set = [set() for x in range(0, 9)]  # Access: col_set[x]
row_set = [set() for x in range(0, 9)]  # Access: row_set[y]
sub_grid_set = [[set() for y in range(0, 3)] for x in range(0, 3)]

# Misc sets used for solving techniques/optimisation
full_set = set({1, 2, 3, 4, 5, 6, 7, 8, 9})
coordinates_set = set({0, 1, 2, 3, 4, 5, 6, 7, 8})

# Populates main_grid, candidates_grid, row/col/block sets from data file
def init():
puzzle = open('../puzzles/extreme2.txt', 'r')
for y in range(0, 9):
for x in range(0, 9):
main_grid[x].append(int(next_line[x]))
if next_line[x] != '0':

for y in range(0, 9):
for x in range(0, 9):
if main_grid[x][y] == 0:
candidatesSet = set.union(row_set[y], col_set[x],
sub_grid_set[int(x / 3)][int(y / 3)])
candidates_grid[x][y] = full_set.difference(candidatesSet)

def iter_over_subgrids(func, *args):
for sub_grid_y in range(0, 3):
for sub_grid_x in range(0, 3):
func(sub_grid_x, sub_grid_y, *args)

def iter_over_line(func, *args):
for square in range(0, 9):
func(square, *args)

def print_main_grid():
for y in range(0, 9):
for x in range(0, 9):
print(main_grid[x][y], end="")
if x % 3 == 2:
print(" ", end="")
print("")

def print_candidates_grid():
for y in range(0, 9):
for x in range(0, 9):
print(candidates_grid[x][y], " ", end="")
print("")

def is_solved():
for y in range(0, 9):
if len(row_set[y]) != 9:
return 0
return 1

# Writes solution to main_grid, updates sets and tables.
def pencil_in(solution, x, y, func):
sub_grid_x = int(x / 3)
sub_grid_y = int(y / 3)
main_grid[x][y] = solution
candidates_grid[x][y].clear()

for sg_y in range(sub_grid_y * 3, sub_grid_y * 3 + 3):
for sg_x in range(sub_grid_x * 3, sub_grid_x * 3 + 3):
for i in range(0, 9):

# Solves squares that have only one candidate.
def single_candidate_square(y):
for x in range(0, 9):
if len(candidates_grid[x][y]) == 1:
pencil_in(candidates_grid[x][y].pop(), x, y,
single_candidate_square)

# Solves squares where candidate appears only once in a row.
def single_sq_candidate_row(y):
for candidate in full_set.difference(row_set[y]):  # Skip solved values
count = 0
prev_x = 0
for x in range(0, 9):
if candidate in candidates_grid[x][y]:
count += 1
prev_x = x
if count == 1:
pencil_in(candidate, prev_x, y, single_sq_candidate_row)

# As single_sq_candidate_row, for columns.
def single_sq_candidate_col(x):
for candidate in full_set.difference(col_set[x]):  # Skip solved values
count = 0
prev_y = 0
for y in range(0, 9):
if candidate in candidates_grid[x][y]:
count += 1
prev_y = y
if count == 1:
pencil_in(candidate, x, prev_y, single_sq_candidate_col)

# As single_sq_candidate_row, for subgrids.
def single_sq_candidate_subgrid(sub_grid_x, sub_grid_y):
for candidate in full_set.difference(sub_grid_set[sub_grid_x][sub_grid_y]):
count = 0
prev_coords = [0, 0]
for y in range(sub_grid_y * 3, sub_grid_y * 3 + 3):
for x in range(sub_grid_x * 3, sub_grid_x * 3 + 3):
if candidate in candidates_grid[x][y]:
count += 1
prev_coords[0] = x
prev_coords[1] = y
if count == 1:
pencil_in(candidate, prev_coords[0], prev_coords[1],
single_sq_candidate_subgrid)

# Finds candidates in block that lie only on one subrow,
# removes candidates from rest of row.
def number_claiming_row(sub_grid_x, sub_grid_y):
# Get set of all candidates each subrow
subrow_sets = [set(), set(), set()]
for y in range(sub_grid_y * 3, sub_grid_y * 3 + 3):
for x in range(sub_grid_x * 3, sub_grid_x * 3 + 3):
subrow_sets[y % 3] = subrow_sets[y % 3].union(candidates_grid[x][y])

# Get candidates which only appear in one subrow
claimed = [subrow_sets[0].difference(subrow_sets[1], subrow_sets[2])]
claimed.append(subrow_sets[1].difference(subrow_sets[0], subrow_sets[2]))
claimed.append(subrow_sets[2].difference(subrow_sets[0], subrow_sets[1]))

# Remove candidates from other subrows in parent row
for sub_row in range(0, 3):
for claimant in set(claimed[sub_row]):
for x in range(0, 9):
if int(x / 3) != sub_grid_x:

# As number_claiming_row, but for columns
def number_claiming_col(sub_grid_x, sub_grid_y):
# Get set of all candidates each subcolumn
subcol_sets = [set(), set(), set()]
for x in range(sub_grid_x * 3, sub_grid_x * 3 + 3):
for y in range(sub_grid_y * 3, sub_grid_y * 3 + 3):
subcol_sets[x % 3] = subcol_sets[x % 3].union(candidates_grid[x][y])

# Get candidates which only appear in one subcolumn
claimed = [subcol_sets[0].difference(subcol_sets[1], subcol_sets[2])]
claimed.append(subcol_sets[1].difference(subcol_sets[0], subcol_sets[2]))
claimed.append(subcol_sets[2].difference(subcol_sets[0], subcol_sets[1]))

# Remove candidates from other subcolumns in parent column
for sub_col in range(0, 3):
for claimant in set(claimed[sub_col]):
for y in range(0, 9):
if int(y / 3) != sub_grid_y:

# Finds sets of n squares in a row where:
# - No squares contain more than n candidates each.
# - The cardinality of the set of all the candidates in squares is n.
# All candidates in that set can be assumed to lie in those squares,
# so the set of candidates can be removed from all other squares in
# that row. Sudoku solvers may already know disjoint subsets as "pairs"
# or "triples".
#
# Basic example: three squares in a row contain the candidate sets
# {2,4}, {2,7} and {4,7} respectively. All three squares contain no
# more than three candidates, and the set of all candidates is {2,4,7},
# which has a cardinality of three. It can then be assumed that those
# squares MUST contain 2, 4 and 7 and nothing else. Any squares outside
# those three in the row can then have the candidates 2, 4 and 7 removed.
def disjoint_subsets_row(y, n):
sets = []
# Get all candidate sets in row with cardinality no greater than n
for x in range(0, 9):
if 1 < len(candidates_grid[x][y]) <= n:
sets.append(candidates_grid[x][y])

# For all disjoint subsets found, remove candidates from other squares
for d in get_disjoint_subsets(sets, n):
for x in range(0, 9):
if not candidates_grid[x][y].issubset(d):
candidates_grid[x][y] = candidates_grid[x][y].difference(d)

# As disjoint_subsets_row, for columns
def disjoint_subsets_col(x, n):
sets = []
# Get all candidate sets in row with cardinality no greater than n
for y in range(0, 9):
if 1 < len(candidates_grid[x][y]) <= n:
sets.append(candidates_grid[x][y])

# For all disjoint subsets found, remove candidates from other squares
for d in get_disjoint_subsets(sets, n):
for y in range(0, 9):
if not candidates_grid[x][y].issubset(d):
candidates_grid[x][y] = candidates_grid[x][y].difference(d)

# As disjoint_subsets_row, for sub-grids.
def disjoint_subsets_subgrid(sub_grid_x, sub_grid_y, n):
sets = []
# Get all candidate sets in row with cardinality no greater than n
for y in range(sub_grid_y * 3, sub_grid_y * 3 + 3):
for x in range(sub_grid_x * 3, sub_grid_x * 3 + 3):
if 1 < len(candidates_grid[x][y]) <= n:
sets.append(candidates_grid[x][y])

# For all disjoint subsets found, remove candidates from other squares
for d in get_disjoint_subsets(sets, n):
for y in range(sub_grid_y * 3, sub_grid_y * 3 + 3):
for x in range(sub_grid_x * 3, sub_grid_x * 3 + 3):
if not candidates_grid[x][y].issubset(d):
candidates_grid[x][y] = candidates_grid[x][y].difference(d)

def get_disjoint_subsets(sets, n):
disjoint_subsets = set()
# For each combination of n sets in sets
for combination in itertools.combinations(sets, n):
superset = set()
# For each individual set in combination
for c in combination:
superset = superset.union(c)
if len(superset) == n:
# Cardinality of candidate superset in combination is n, is djss.
return disjoint_subsets

# Runs through solving techniques until puzzle solved or no possible solution.
def solve():
for x in range(0, 100):
iter_over_line(single_candidate_square)
iter_over_line(single_sq_candidate_row)
iter_over_line(single_sq_candidate_col)
iter_over_subgrids(single_sq_candidate_subgrid)
iter_over_subgrids(number_claiming_row)
iter_over_subgrids(number_claiming_col)
for n in range(2, 5):
iter_over_line(disjoint_subsets_row, n)
iter_over_line(disjoint_subsets_col, n)
iter_over_subgrids(disjoint_subsets_subgrid, n)
if is_solved() == 1:
print_main_grid()
break

init()
solve()


Puzzles are solved in the text file in the format:

400700900
000004000
070280006
500000010
301060408
020000009
900036020
000800000
004001003


Example of a Sudoku game:

4 - - | 7 - - | 9 - -
- - - | - - 4 | - - -
- 7 - | 2 8 - | - - 6
------|-------|------
5 - - | - - - | - 1 -
3 - 1 | - 6 - | 4 - 8
- 2 - | - - - | - - 9
------|-------|------
9 - - | - 3 6 | - 2 -
- - - | 8 - - | - - -
- - 4 | - - 1 | - - 3


The above is an "extreme" level puzzle.

### Working with file handles

Don't do this:

puzzle = open('../puzzles/extreme2.txt', 'r')


Use with instead:

with open('../puzzles/extreme2.txt') as puzzle:
# ...


In your version you forget to close the file handle. That's a bad practice. When using with, the file handle is automatically closed.

### Use booleans

Instead of returning 0 for failure and 1 for success:

def is_solved():
for y in range(0, 9):
if len(row_set[y]) != 9:
return 0
return 1


Use booleans:

def is_solved():
for y in range(0, 9):
if len(row_set[y]) != 9:
return False
return True


### Usability

It would be nice if the puzzle file was a command line argument, not the hardcoded path ../puzzles/extreme2.txt.

### Documentation

It's nice that you included explanation about the terminology. But it would be better to use proper doc strings.

### Simplifications

You can replace this:

full_set = set({1, 2, 3, 4, 5, 6, 7, 8, 9})


With a set literal:

full_set = {1, 2, 3, 4, 5, 6, 7, 8, 9}


You can write range(9) instead of range(0, 9)

Instead of int(x / 3), it's better to use x // 3

• Great suggestions - I feel like I have learned a lot of useful things in a short time, thank you. Jul 13 '15 at 22:24

You can make it much tighter by using list comprehensions. The function below will perform all of the puzzle checks and return all possible moves for a particular row and column given a particular puzzle, although I used characters instead of ints (you can transform by switching that part to m+1 for m in range(9)).

def puzzle_moves(row,col,puzzle):
same_row = {puzzle[row][c] for c in range(9)}
same_col = {puzzle[r][col] for r in range(9)}
# box row/col refer to the row/col of the box that row/col is in
box_row = 3*(row/3)
box_col = 3*(col/3)
same_box = {puzzle[box_row+r][box_col+c] for r in range(3) for c in range(3)}
return {m for m in '123456789'
if m not in same_row
and m not in same_col
and m not in same_box}


My solution can solve large numbers of 'extreme' puzzles in seconds, and I think it's pretty easy to read (YMMV!).

[EDIT: I guess it's not as clear as I thought!]:

To understand this, look at the return statement: We want to return any possible 'moves' that are not in the same row, same column, or in the same box. All the lines before the return statement are defining what it means to be in the same row/col/box.

same_row = {puzzle[row][c] for c in range(9)}


We pass in the puzzle, row and column, we want to know which moves are in this current row of the puzzle. So for example, say puzzle is a Sudoku puzzle that looks like this:

If we pass in row 0 and col 0, we expect for same row to get back everything in the 0th (first) row: {2, 7, 8, 9}

To try it for yourself with the puzzle you posted, open a python prompt and type in puzzle_str=""", then paste your puzzle, followed by """ again:

>>>puzzle_str = """400700900...etc..."""


You can then construct the puzzle as a list-of-lists:

puzzle = [[c for c in line.strip()] for line in puzzle_str.strip().split('\n')]


That line is splitting the puzzle string by newline characters (after removing any extras at the end or beginning), and then returning each character.

Then you can call the above function by passing in the puzzle with any row, col and the above puzzle. You can also try each line of code, so let's look at same_row again:

row = 0
col = 0
{puzzle[row][c] for c in range(9)}  # same_row, from above function


So this returns {'0', '4', '7', '9'}, the distinct characters that are in the first row of your puzzle. same_col follows the same logic.

Getting the same_box is trickier. To understand this part, think of the puzzle as having 3 'box rows' and 3 'box columns', which define the 9 boxes. To get the box rows/cols, you take the row/col that was passed in, and divide it by 3, which (since it's an int) will truncate it. To see this in action, look at this:

[i/3 for i in range(9)]
[0, 0, 0, 1, 1, 1, 2, 2, 2]


Once you've determined box_row and box_col, the same_box expression is just the same logic as same_row and same_col. The for r in range(3) for c in range(3) is giving you all the combinations for a 3x3 box.

I've rambled enough...please let me know if there is another specific part I can explain?

[EDIT #2] Ok, here's another version that's still pretty concise, but runs a little faster (about 10% faster). This time it's an OO design, which isn't my typical go-to, but hey, at least it's pretty!

import copy

possible_moves = set([m for m in '123456789'])

class Sudoku(object):
def __init__(self, puzzle_str):
self.rows = [set() for r in range(9)]
self.cols = [set() for c in range(9)]
self.boxes = [[set() for bc in range(3)] for br in range(3)]
self.grid = [[c for c in line.strip()] for line in puzzle_str.strip().split('\n')]
for r, row in enumerate(self.grid):
for c, m in enumerate(row):
self.move(r, c, m)

def move(self, r, c, m):
self.grid[r][c] = m

def moves(self, r, c):
return possible_moves - self.rows[r] - self.cols[c] - self.boxes[int(r/3)][int(c/3)]

def __str__(self):
return '\n'.join(['.'.join([c for c in line]) for line in self.grid])

worst_best_move = ((10),(),())

def solve_sudoku(puzzle):
best_move = worst_best_move
found_move = True
finished = True

while found_move:
finished = True
found_move = False
best_move = worst_best_move
for r in range(0,9):
for c in range(0,9):
if puzzle.grid[r][c]=='0':
possible_moves = puzzle.moves(r,c)
num_moves = len(possible_moves)
if num_moves==0:
return None
if num_moves==1:
found_move=True
puzzle.move(r, c, possible_moves.pop())
else:
finished = False
if num_moves < best_move[0]:
best_move =(num_moves,(r,c), possible_moves)
if finished:
return puzzle

target = best_move[1]
for move in best_move[2]:
guess_puzzle = copy.deepcopy(puzzle)
guess_puzzle.move(target[0], target[1], move)
result = solve_sudoku(guess_puzzle)
if result is not None:
return result
return None

def run_sudoku_test():
puzzle_str = """
400700900
000004000
070280006
500000010
301060408
020000009
900036020
000800000
004001003"""
puzzle = Sudoku(puzzle_str)
#print(str(puzzle))
result = solve_sudoku(puzzle)
if result is None:
print 'Failed to solve %s'%puzzle_num
else:
pass#print_sudoku_puzzle(result)

if __name__=='__main__':
import timeit
print(timeit.timeit('run_sudoku_test()',setup='from __main__ import run_sudoku_test',number=50))

• I'm not sure I understand this code, could you explain what it's doing? Jul 13 '15 at 19:14
• I think the efficiency trade off is too great to do this. OK, so this returns all unsolved values for a squares parent row, column, and block, how does this help? I already have sets (row_set[], col_set[], block_set[]) that contain all solved values so a set.difference() will give the unsolved values anyway. Those sets are initialised once at the start based on the initial puzzle structure, then single elements are added to the sets as squares are solved. Surely your way is much more inefficient as you are essentially building the sets again and again for every square? Jul 14 '15 at 11:23
• My solution solves your puzzle 500 times in 3.06 seconds. I'm sure it could be faster, but it does the trick. With 'easy' puzzles it goes much faster. Jul 14 '15 at 17:37
• Scratch that - under 2.4 seconds. Definitely it's slower, by about an order of magnitude, I thought you were looking for easy to read. I'm sure it could be better, but with all I/O, timing, whitespace, pretty-print, etc., the size is < 1/4 your code. =I wrote it years ago for Project Euler, it was 'good enough' to move on to the next puzzle. I guess it depends on what you're trying to optimize? Jul 14 '15 at 18:08
• If only I could make it the length of your program with the efficiency of mine :P In seriousness though, your way is more than "good enough". I'm sure there is some kind of way to merge your method and mine to avoid tradeoffs, I'll give it a think over. Jul 14 '15 at 19:21