# Algorithm for solving Sudoku

Here is my sudoku solver in python. It uses simple backtracking algorithm to solve the puzzle.
For simplicity no input validations or fancy output is done. It’s the bare minimum code which solves the problem.

## Algorithm

1. Find all legal values of a given cell
2. For each legal value, Go recursively and try to solve the grid

## Solution

It takes 9X9 grid partially filled with numbers. A cell with value 0 indicates that it is not filled.

## Code

def findNextCellToFill(grid, i, j):
for x in range(i,9):
for y in range(j,9):
if grid[x][y] == 0:
return x,y
for x in range(0,9):
for y in range(0,9):
if grid[x][y] == 0:
return x,y
return -1,-1

def isValid(grid, i, j, e):
rowOk = all([e != grid[i][x] for x in range(9)])
if rowOk:
columnOk = all([e != grid[x][j] for x in range(9)])
if columnOk:
# finding the top left x,y co-ordinates of the section containing the i,j cell
secTopX, secTopY = 3 *(i//3), 3 *(j//3) #floored quotient should be used here.
for x in range(secTopX, secTopX+3):
for y in range(secTopY, secTopY+3):
if grid[x][y] == e:
return False
return True
return False

def solveSudoku(grid, i=0, j=0):
i,j = findNextCellToFill(grid, i, j)
if i == -1:
return True
for e in range(1,10):
if isValid(grid,i,j,e):
grid[i][j] = e
if solveSudoku(grid, i, j):
return True
# Undo the current cell for backtracking
grid[i][j] = 0
return False


## Testing the code


>>> input = [[5,1,7,6,0,0,0,3,4],[2,8,9,0,0,4,0,0,0],[3,4,6,2,0,5,0,9,0],[6,0,2,0,0,0,0,1,0],[0,3,8,0,0,6,0,4,7],[0,0,0,0,0,0,0,0,0],[0,9,0,0,0,0,0,7,8],[7,0,3,4,0,0,5,6,0],[0,0,0,0,0,0,0,0,0]]
>>> solveSudoku(input)
True
>>> input
[[5, 1, 7, 6, 9, 8, 2, 3, 4], [2, 8, 9, 1, 3, 4, 7, 5, 6], [3, 4, 6, 2, 7, 5, 8, 9, 1], [6, 7, 2, 8, 4, 9, 3, 1, 5], [1, 3, 8, 5, 2, 6, 9, 4, 7], [9, 5, 4, 7, 1, 3, 6, 8, 2], [4, 9, 5, 3, 6, 2, 1, 7, 8], [7, 2, 3, 4, 8, 1, 5, 6, 9], [8, 6, 1, 9, 5, 7, 4, 2, 3]]



The above one is very basic backtracking algorithm which is explained at many places. But the most interesting and natural of the sudoku solving strategies I came across is this one from here