# sudoku.py

```## Solve Every Sudoku Puzzle

## See http://norvig.com/sudoku.html

## Throughout this program we have:
##   r is a row,    e.g. 'A'
##   c is a column, e.g. '3'
##   s is a square, e.g. 'A3'
##   d is a digit,  e.g. '9'
##   u is a unit,   e.g. ['A1','B1','C1','D1','E1','F1','G1','H1','I1']
##   grid is a grid,e.g. 81 non-blank chars, e.g. starting with '.18...7...
##   values is a dict of possible values, e.g. {'A1':'12349', 'A2':'8', ...}

def cross(A, B):
"Cross product of elements in A and elements in B."
return [a+b for a in A for b in B]

digits   = '123456789'
rows     = 'ABCDEFGHI'
cols     = digits
squares  = cross(rows, cols)
unitlist = ([cross(rows, c) for c in cols] +
[cross(r, cols) for r in rows] +
[cross(rs, cs) for rs in ('ABC','DEF','GHI') for cs in ('123','456','789')])
units = dict((s, [u for u in unitlist if s in u])
for s in squares)
peers = dict((s, set(sum(units[s],[]))-set([s]))
for s in squares)

################ Unit Tests ################

def test():
"A set of tests that must pass."
assert len(squares) == 81
assert len(unitlist) == 27
assert all(len(units[s]) == 3 for s in squares)
assert all(len(peers[s]) == 20 for s in squares)
assert units['C2'] == [['A2', 'B2', 'C2', 'D2', 'E2', 'F2', 'G2', 'H2', 'I2'],
['C1', 'C2', 'C3', 'C4', 'C5', 'C6', 'C7', 'C8', 'C9'],
['A1', 'A2', 'A3', 'B1', 'B2', 'B3', 'C1', 'C2', 'C3']]
assert peers['C2'] == set(['A2', 'B2', 'D2', 'E2', 'F2', 'G2', 'H2', 'I2',
'C1', 'C3', 'C4', 'C5', 'C6', 'C7', 'C8', 'C9',
'A1', 'A3', 'B1', 'B3'])
print 'All tests pass.'

################ Parse a Grid ################

def parse_grid(grid):
"""Convert grid to a dict of possible values, {square: digits}, or
return False if a contradiction is detected."""
## To start, every square can be any digit; then assign values from the grid.
values = dict((s, digits) for s in squares)
for s,d in grid_values(grid).items():
if d in digits and not assign(values, s, d):
return False ## (Fail if we can't assign d to square s.)
return values

def grid_values(grid):
"Convert grid into a dict of {square: char} with '0' or '.' for empties."
chars = [c for c in grid if c in digits or c in '0.']
assert len(chars) == 81
return dict(zip(squares, chars))

################ Constraint Propagation ################

def assign(values, s, d):
"""Eliminate all the other values (except d) from values[s] and propagate.
Return values, except return False if a contradiction is detected."""
other_values = values[s].replace(d, '')
if all(eliminate(values, s, d2) for d2 in other_values):
return values
else:
return False

def eliminate(values, s, d):
"""Eliminate d from values[s]; propagate when values or places <= 2.
Return values, except return False if a contradiction is detected."""
if d not in values[s]:
values[s] = values[s].replace(d,'')
## (1) If a square s is reduced to one value d2, then eliminate d2 from the peers.
if len(values[s]) == 0:
return False ## Contradiction: removed last value
elif len(values[s]) == 1:
d2 = values[s]
if not all(eliminate(values, s2, d2) for s2 in peers[s]):
return False
## (2) If a unit u is reduced to only one place for a value d, then put it there.
for u in units[s]:
dplaces = [s for s in u if d in values[s]]
if len(dplaces) == 0:
return False ## Contradiction: no place for this value
elif len(dplaces) == 1:
# d can only be in one place in unit; assign it there
if not assign(values, dplaces, d):
return False
return values

################ Display as 2-D grid ################

def display(values):
"Display these values as a 2-D grid."
width = 1+max(len(values[s]) for s in squares)
line = '+'.join(['-'*(width*3)]*3)
for r in rows:
print ''.join(values[r+c].center(width)+('|' if c in '36' else '')
for c in cols)
if r in 'CF': print line
print

################ Search ################

def solve(grid): return search(parse_grid(grid))

def search(values):
"Using depth-first search and propagation, try all possible values."
if values is False:
return False ## Failed earlier
if all(len(values[s]) == 1 for s in squares):
return values ## Solved!
## Chose the unfilled square s with the fewest possibilities
n,s = min((len(values[s]), s) for s in squares if len(values[s]) > 1)
return some(search(assign(values.copy(), s, d))
for d in values[s])

################ Utilities ################

def some(seq):
"Return some element of seq that is true."
for e in seq:
if e: return e
return False

def from_file(filename, sep='\n'):
"Parse a file into a list of strings, separated by sep."

def shuffled(seq):
"Return a randomly shuffled copy of the input sequence."
seq = list(seq)
random.shuffle(seq)
return seq

################ System test ################

import time, random

def solve_all(grids, name='', showif=0.0):
"""Attempt to solve a sequence of grids. Report results.
When showif is a number of seconds, display puzzles that take longer.
When showif is None, don't display any puzzles."""
def time_solve(grid):
start = time.clock()
values = solve(grid)
t = time.clock()-start
## Display puzzles that take long enough
if showif is not None and t > showif:
display(grid_values(grid))
if values: display(values)
print '(%.2f seconds)\n' % t
return (t, solved(values))
times, results = zip(*[time_solve(grid) for grid in grids])
N = len(grids)
if N > 1:
print "Solved %d of %d %s puzzles (avg %.2f secs (%d Hz), max %.2f secs)." % (
sum(results), N, name, sum(times)/N, N/sum(times), max(times))

def solved(values):
"A puzzle is solved if each unit is a permutation of the digits 1 to 9."
def unitsolved(unit): return set(values[s] for s in unit) == set(digits)
return values is not False and all(unitsolved(unit) for unit in unitlist)

def random_puzzle(N=17):
"""Make a random puzzle with N or more assignments. Restart on contradictions.
Note the resulting puzzle is not guaranteed to be solvable, but empirically
about 99.8% of them are solvable. Some have multiple solutions."""
values = dict((s, digits) for s in squares)
for s in shuffled(squares):
if not assign(values, s, random.choice(values[s])):
break
ds = [values[s] for s in squares if len(values[s]) == 1]
if len(ds) >= N and len(set(ds)) >= 8:
return ''.join(values[s] if len(values[s])==1 else '.' for s in squares)
return random_puzzle(N) ## Give up and make a new puzzle

grid1  = '003020600900305001001806400008102900700000008006708200002609500800203009005010300'
grid2  = '4.....8.5.3..........7......2.....6.....8.4......1.......6.3.7.5..2.....1.4......'
hard1  = '.....6....59.....82....8....45........3........6..3.54...325..6..................'

if __name__ == '__main__':
test()
solve_all(from_file("easy50.txt", '========'), "easy", None)
solve_all(from_file("top95.txt"), "hard", None)
solve_all(from_file("hardest.txt"), "hardest", None)
solve_all([random_puzzle() for _ in range(99)], "random", 100.0)

## References used:
## http://www.scanraid.com/BasicStrategies.htm
## http://www.sudokudragon.com/sudokustrategy.htm