class SolutionFound(BaseException):
	def __init__(self):
		self.s = []

def assign(clauses, val):
	n = []
	for c in clauses:
		if val in c:
			continue
		elif -val in c:
			n.append([x for x in c if x != -val])
		else:
			n.append(c)
	return n

def try_sat(clauses, val):
	try:
		solve_sat(assign(clauses, val))
	except SolutionFound, s:
		s.s.append(val)
		raise s

def solve_sat(clauses):
	if [] in clauses:
		return
	if len(clauses) == 0:
		raise SolutionFound()
	for x in clauses:
		if len(x) == 1:
			try_sat(clauses, x[0])
			return
	smallest = clauses[0]
	for c in clauses:
		if c[0] > 0 and (smallest[0] < 0 or len(smallest) > len(c)):
			smallest = c
	try_sat(clauses, smallest[0])
	try_sat(clauses, -smallest[0])

svals = [list(range(x, x+9)) for x in range(0, 81, 9)] + \
		[list(range(x, x+81, 9)) for x in range(9)] + \
		[[x,x+1,x+2,x+9,x+10,x+11,x+18,x+19,x+20] for x in [0,3,6,27,30,33,54,57,60]]

def group(c, p):
	p.append(c)
	for i in range(len(c)):
		for j in range(i):
			p.append([-c[i], -c[j]])

def groups(c, p):
	for i in range(9):
		group([x + 100 * i + 1 for x in c], p)

def decode(r):
	r2 = [[0 for x in range(9)] for y in range(9)]
	for v in r:
		if v > 0:
			pos = (v-1) % 100
			r2[pos // 9][pos % 9] = (v // 100) + 1
	return r2

def solve_sudoku(problem):
	p = []
	for i in range(81):
		group(list(range(i+1, 901, 100)), p)
	for s in svals:
		groups(s, p)
	pre = []
	for x in range(9):
		for y in range(9):
			if problem[x][y]:
				m = x * 9 + y + 100 * (problem[x][y] - 1) + 1
				pre.append(m)
				p = assign(p, m)
	try:
		solve_sat(p)
	except SolutionFound, s:
		return decode(s.s + pre)
	return None

import sys
sys.setrecursionlimit(10000)

if __name__ == '__main__':
	tt = [[0] * 9 for i in range(9)]
	print('\n'.join(' '.join([str(p) for p in q]) for q in solve_sudoku(tt)))
	print(' ')
	tt[0][0] = 6
	tt[3][4] = 1
	print('\n'.join(' '.join([str(p) for p in q]) for q in solve_sudoku(tt)))

Timeout