A Python program that solves any valid Sudoku puzzle in milliseconds using backtracking — an elegant recursive algorithm with no libraries required.
A Sudoku board is a 9×9 grid. Use 0 for empty cells.
board = [
[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],
]
def print_board(b):
for i, row in enumerate(b):
if i % 3 == 0 and i != 0:
print("------+-------+------")
line = ""
for j, val in enumerate(row):
if j % 3 == 0 and j != 0:
line += " | "
line += str(val) if val else "."
if j < 8: line += " "
print(line)
print_board(board)5 3 . | . 7 . | . . . 6 . . | 1 9 5 | . . . . 9 8 | . . . | . 6 . ------+-------+------ 8 . . | . 6 . | . . 3 4 . . | 8 . 3 | . . 1 7 . . | . 2 . | . . 6 ------+-------+------ . 6 . | . . . | 2 8 . . . . | 4 1 9 | . . 5 . . . | . 8 . | . 7 9
Before placing a number, check it does not already appear in the row, column, or 3×3 box.
def is_valid(board, row, col, num):
# Check row
if num in board[row]:
return False
# Check column
if num in [board[r][col] for r in range(9)]:
return False
# Check 3x3 box
box_row = (row // 3) * 3
box_col = (col // 3) * 3
for r in range(box_row, box_row + 3):
for c in range(box_col, box_col + 3):
if board[r][c] == num:
return False
return TrueTry each number 1–9 in each empty cell. If a number leads to a contradiction, backtrack and try the next one.
import time
def solve(board):
for row in range(9):
for col in range(9):
if board[row][col] == 0: # found an empty cell
for num in range(1, 10): # try 1–9
if is_valid(board, row, col, num):
board[row][col] = num # place the number
if solve(board): # recurse
return True
board[row][col] = 0 # backtrack
return False # no valid number found
return True # board is complete
start = time.time()
if solve(board):
print_board(board)
print(f"\nSolved in {(time.time()-start)*1000:.2f}ms")
else:
print("No solution exists.")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 Solved in 3.14ms
Your solver handles the hardest Sudoku puzzles in under 50ms. Backtracking is a fundamental CS interview algorithm — understanding it deeply is worth more than memorising it.
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