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adjust sat solver to new files

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This commit is contained in:
Maximilian Keßler 2023-03-18 13:18:04 +01:00
parent 8730551055
commit ddeb19265d
Signed by: max
GPG key ID: BCC5A619923C0BA5
2 changed files with 105 additions and 117 deletions
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4
hanabi.py
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@ -233,6 +233,10 @@ class GameState():
def hand_size(self): def hand_size(self):
return self.instance.hand_size return self.instance.hand_size
@property
def deck_size(self):
return self.instance.deck_size
# Properties of GameState # Properties of GameState

216
sat.py
View file

@ -4,12 +4,9 @@ import json
from typing import List from typing import List
from concurrent.futures import ProcessPoolExecutor from concurrent.futures import ProcessPoolExecutor
from compress import DeckCard, Action, ActionType, link, decompress_deck from hanabi import DeckCard, Action, ActionType, GameState, HanabiInstance
from greedy_solver import GameState, GreedyStrategy from compress import link, decompress_deck
from greedy_solver import GreedyStrategy
COLORS = 'rygbp'
STANDARD_HAND_SIZE = {2: 5, 3: 5, 4: 4, 5: 4, 6: 3}
NUM_STRIKES_TO_LOSE = 3
# literals to model game as sat instance to check for feasibility # literals to model game as sat instance to check for feasibility
@ -17,33 +14,7 @@ NUM_STRIKES_TO_LOSE = 3
class Literals(): class Literals():
# num_suits is total number of suits, i.e. also counts the dark suits # num_suits is total number of suits, i.e. also counts the dark suits
# default distribution among all suits is assumed # default distribution among all suits is assumed
def __init__(self, num_players, num_suits, num_dark_suits=0): def __init__(self, instance: HanabiInstance):
assert ( 2 <= num_players <= 6 )
## some game parameters
self.num_players = num_players
self.num_suits = num_suits
self.num_dark_suits = num_dark_suits
self.hand_size = STANDARD_HAND_SIZE[num_players]
self.num_strikes = NUM_STRIKES_TO_LOSE
self.deck_size = 10 * num_suits - 5 * num_dark_suits
self.distributed_cards = self.num_players * self.hand_size
self.draw_pile_size = self.deck_size - self.distributed_cards
## maximum number of moves in any game that can achieve max score
# each suit gives 15 moves, as we can play and discard 5 cards each and give 5 clues. dark suits only give 5 moves, since no discards are added
# number of cards that remain in players hands after end of game. they cost 2 turns each, since we cannot discard them and also have one clue less
# 8 clues at beginning, one further clue for each suit but one (the clue of the last 5 is never useful since it is gained in the extra-round)
# subtract a further move for a second 5-clue that can't be used in 5 or 6-player games, since the extraround starts too soon
self.max_moves = 15 * num_suits - 10 * num_dark_suits \
- 2 * num_players * (self.hand_size - 1) \
+ 8 + (num_suits - 1) \
+ (-1 if num_players >= 5 else 0)
###
# note that we generate 'literals' always one out of boundary and set them to explicit truth values. This makes sat formulation easier but has no actual overhead in solving it
# move are numbered starting with 0
# clues[m][i] == "after move m we have at least i clues" # clues[m][i] == "after move m we have at least i clues"
self.clues = { self.clues = {
@ -54,20 +25,20 @@ class Literals():
**{ i: Symbol('m{}clues{}'.format(m, i)) for i in range(1, 9) }, **{ i: Symbol('m{}clues{}'.format(m, i)) for i in range(1, 9) },
9: Bool(False) # never 9 or more clues. This will implicitly forbid discarding at 8 clues later 9: Bool(False) # never 9 or more clues. This will implicitly forbid discarding at 8 clues later
} }
for m in range(self.max_moves) for m in range(instance.max_winning_moves)
} }
} }
# strikes[m][i] == "after move m we have at least i strikes" # strikes[m][i] == "after move m we have at least i strikes"
self.strikes = { self.strikes = {
-1: {i: Bool(i == 0) for i in range(0, self.num_strikes + 1)} # no strikes when we start -1: {i: Bool(i == 0) for i in range(0, instance.num_strikes + 1)} # no strikes when we start
, **{ , **{
m: { m: {
0: Bool(True), 0: Bool(True),
**{ s: Symbol('m{}strikes{}'.format(m,s)) for s in range(1, self.num_strikes) }, **{ s: Symbol('m{}strikes{}'.format(m,s)) for s in range(1, instance.num_strikes) },
self.num_strikes: Bool(False) # never so many clues that we lose. Implicitly forbids striking out instance.num_strikes: Bool(False) # never so many clues that we lose. Implicitly forbids striking out
} }
for m in range(self.max_moves) for m in range(instance.max_winning_moves)
} }
} }
@ -75,8 +46,8 @@ class Literals():
self.extraround = { self.extraround = {
-1: Bool(False) -1: Bool(False)
, **{ , **{
m: Bool(False) if m < self.draw_pile_size else Symbol('m{}extra'.format(m)) # it takes at least as many turns as cards in the draw pile to start the extra round m: Bool(False) if m < instance.draw_pile_size else Symbol('m{}extra'.format(m)) # it takes at least as many turns as cards in the draw pile to start the extra round
for m in range(0, self.max_moves) for m in range(0, instance.max_winning_moves)
} }
} }
@ -84,26 +55,26 @@ class Literals():
self.dummyturn = { self.dummyturn = {
-1: Bool(False) -1: Bool(False)
, **{ , **{
m: Bool(False) if m < self.draw_pile_size + self.num_players else Symbol('m{}dummy'.format(m)) m: Bool(False) if m < instance.draw_pile_size + instance.num_players else Symbol('m{}dummy'.format(m))
for m in range(0, self.max_moves) for m in range(0, instance.max_winning_moves)
} }
} }
# draw[m][i] == "at move m we play/discard deck[i]" # draw[m][i] == "at move m we play/discard deck[i]"
self.discard = { self.discard = {
m: {i: Symbol('m{}discard{}'.format(m, i)) for i in range(self.deck_size)} m: {i: Symbol('m{}discard{}'.format(m, i)) for i in range(instance.deck_size)}
for m in range(self.max_moves) for m in range(instance.max_winning_moves)
} }
# draw[m][i] == "at move m we draw deck card i" # draw[m][i] == "at move m we draw deck card i"
self.draw = { self.draw = {
-1: { i: Bool(i == self.distributed_cards - 1) for i in range(self.distributed_cards - 1, self.deck_size) } -1: { i: Bool(i == instance.num_dealt_cards - 1) for i in range(instance.num_dealt_cards - 1, instance.deck_size) }
, **{ , **{
m: { m: {
self.distributed_cards - 1: Bool(False), instance.num_dealt_cards - 1: Bool(False),
**{i: Symbol('m{}draw{}'.format(m, i)) for i in range(self.distributed_cards, self.deck_size)} **{i: Symbol('m{}draw{}'.format(m, i)) for i in range(instance.num_dealt_cards, instance.deck_size)}
} }
for m in range(self.max_moves) for m in range(instance.max_winning_moves)
} }
} }
@ -112,82 +83,89 @@ class Literals():
-1: Bool(False) -1: Bool(False)
, **{ , **{
m: Symbol('m{}newstrike'.format(m)) m: Symbol('m{}newstrike'.format(m))
for m in range(self.max_moves) for m in range(instance.max_winning_moves)
} }
} }
# progress[m][card = (suitIndex, rank)] == "after move m we have played in suitIndex up to rank" # progress[m][card = (suitIndex, rank)] == "after move m we have played in suitIndex up to rank"
self.progress = { self.progress = {
-1: {(s, r): Bool(r == 0) for s in range(0, self.num_suits) for r in range(0, 6)} # at start, have only played rank zero -1: {(s, r): Bool(r == 0) for s in range(0, instance.num_suits) for r in range(0, 6)} # at start, have only played rank zero
, **{ , **{
m: { m: {
**{(s, 0): Bool(True) for s in range(0, self.num_suits)}, **{(s, 0): Bool(True) for s in range(0, instance.num_suits)},
**{(s, r): Symbol('m{}progress{}{}'.format(m, s, r)) for s in range(0, self.num_suits) for r in range(1, 6)} **{(s, r): Symbol('m{}progress{}{}'.format(m, s, r)) for s in range(0, instance.num_suits) for r in range(1, 6)}
} }
for m in range(self.max_moves) for m in range(instance.max_winning_moves)
} }
} }
## Utility variables ## Utility variables
# discard_any[m] == "at move m we play/discard a card" # discard_any[m] == "at move m we play/discard a card"
self.discard_any = { m: Symbol('m{}discard_any'.format(m)) for m in range(self.max_moves) } self.discard_any = { m: Symbol('m{}discard_any'.format(m)) for m in range(instance.max_winning_moves) }
# draw_any[m] == "at move m we draw a card" # draw_any[m] == "at move m we draw a card"
self.draw_any = {m: Symbol('m{}draw_any'.format(m)) for m in range(self.max_moves)} self.draw_any = {m: Symbol('m{}draw_any'.format(m)) for m in range(instance.max_winning_moves)}
# play[m] == "at move m we play a card" # play[m] == "at move m we play a card"
self.play = {m: Symbol('m{}play'.format(m)) for m in range(self.max_moves)} self.play = {m: Symbol('m{}play'.format(m)) for m in range(instance.max_winning_moves)}
# play5[m] == "at move m we play a 5" # play5[m] == "at move m we play a 5"
self.play5 = {m: Symbol('m{}play5'.format(m)) for m in range(self.max_moves)} self.play5 = {m: Symbol('m{}play5'.format(m)) for m in range(instance.max_winning_moves)}
# incr_clues[m] == "at move m we obtain a clue" # incr_clues[m] == "at move m we obtain a clue"
self.incr_clues = {m: Symbol('m{}c+'.format(m)) for m in range(self.max_moves)} self.incr_clues = {m: Symbol('m{}c+'.format(m)) for m in range(instance.max_winning_moves)}
def solve_sat(game_state: GameState): def solve_sat(starting_state: GameState | HanabiInstance):
ls = Literals(game_state.num_players, game_state.num_suits, game_state.num_dark_suits) if isinstance(starting_state, HanabiInstance):
instance = starting_state
game_state = GameState(instance)
elif isinstance(starting_state, GameState):
instance = starting_state.instance
game_state = starting_state
else:
raise ValueError("Bad argument type")
ls = Literals(instance)
##### setup of initial game state ##### setup of initial game state
# properties used later to model valid moves # properties used later to model valid moves
num_dark_suits = game_state.num_dark_suits
num_suits = game_state.num_suits
deck = game_state.deck
next_draw = game_state.progress
starting_hands = [[card.deck_index for card in hand] for hand in game_state.hands] starting_hands = [[card.deck_index for card in hand] for hand in game_state.hands]
first_turn = len(game_state.actions) first_turn = len(game_state.actions)
# set initial clues if isinstance(starting_state, GameState):
for i in range(0,10): # have to set additional variables
ls.clues[first_turn - 1][i] = Bool(i <= game_state.clues)
# set initial strikes # set initial clues
for i in range(0, game_state.num_strikes + 1): for i in range(0,10):
ls.strikes[first_turn - 1][i] = Bool(i <= game_state.strikes) ls.clues[first_turn - 1][i] = Bool(i <= game_state.clues)
# check if extraround has started (usually not) # set initial strikes
ls.extraround[first_turn - 1] = Bool(game_state.remaining_extra_turns < game_state.num_players) for i in range(0, instance.num_strikes + 1):
ls.dummyturn[first_turn -1] = Bool(False) ls.strikes[first_turn - 1][i] = Bool(i <= game_state.strikes)
# set recent draws: important to model progress # check if extraround has started (usually not)
# we just pretend that the last card drawn was in fact drawn last turn, ls.extraround[first_turn - 1] = Bool(game_state.remaining_extra_turns < game_state.num_players)
# regardless of when it was actually drawn ls.dummyturn[first_turn -1] = Bool(False)
for neg_turn in range(1, min(9, first_turn + 2)):
for i in range(game_state.num_players * game_state.hand_size, game_state.deck_size): # set recent draws: important to model progress
ls.draw[first_turn - neg_turn][i] = Bool(neg_turn == 1 and i == game_state.progress - 1) # we just pretend that the last card drawn was in fact drawn last turn,
# forbid re-drawing of the last card drawn # regardless of when it was actually drawn
for m in range(first_turn, ls.max_moves): for neg_turn in range(1, min(9, first_turn + 2)):
ls.draw[m][game_state.progress - 1] = Bool(False) for i in range(instance.num_players * instance.hand_size, instance.deck_size):
ls.draw[first_turn - neg_turn][i] = Bool(neg_turn == 1 and i == game_state.progress - 1)
# forbid re-drawing of the last card drawn
for m in range(first_turn, instance.max_winning_moves):
ls.draw[m][game_state.progress - 1] = Bool(False)
# model initial progress # model initial progress
for s in range(0, game_state.num_suits): for s in range(0, game_state.num_suits):
for r in range(0, 6): for r in range(0, 6):
ls.progress[first_turn - 1][s, r] = Bool(r <= game_state.stacks[s]) ls.progress[first_turn - 1][s, r] = Bool(r <= game_state.stacks[s])
### Now, model all valid moves ### Now, model all valid moves
@ -196,10 +174,10 @@ def solve_sat(game_state: GameState):
Implies(ls.dummyturn[m], Not(ls.discard_any[m])), Implies(ls.dummyturn[m], Not(ls.discard_any[m])),
# definition of discard_any # definition of discard_any
Iff(ls.discard_any[m], Or(ls.discard[m][i] for i in range(ls.deck_size))), Iff(ls.discard_any[m], Or(ls.discard[m][i] for i in range(instance.deck_size))),
# definition of draw_any # definition of draw_any
Iff(ls.draw_any[m], Or(ls.draw[m][i] for i in range(next_draw, ls.deck_size))), Iff(ls.draw_any[m], Or(ls.draw[m][i] for i in range(game_state.progress, instance.deck_size))),
# ls.draw implies ls.discard (and converse true before the ls.extraround) # ls.draw implies ls.discard (and converse true before the ls.extraround)
Implies(ls.draw_any[m], ls.discard_any[m]), Implies(ls.draw_any[m], ls.discard_any[m]),
@ -209,7 +187,7 @@ def solve_sat(game_state: GameState):
Implies(ls.play[m], ls.discard_any[m]), Implies(ls.play[m], ls.discard_any[m]),
# definition of ls.play5 # definition of ls.play5
Iff(ls.play5[m], And(ls.play[m], Or(ls.discard[m][i] for i in range(ls.deck_size) if deck[i].rank == 5))), Iff(ls.play5[m], And(ls.play[m], Or(ls.discard[m][i] for i in range(instance.deck_size) if instance.deck[i].rank == 5))),
# definition of ls.incr_clues # definition of ls.incr_clues
Iff(ls.incr_clues[m], And(ls.discard_any[m], Implies(ls.play[m], And(ls.play5[m], Not(ls.clues[m-1][8]))))), Iff(ls.incr_clues[m], And(ls.discard_any[m], Implies(ls.play[m], And(ls.play5[m], Not(ls.clues[m-1][8]))))),
@ -225,38 +203,38 @@ def solve_sat(game_state: GameState):
Iff(ls.strike[m], And(ls.discard_any[m], Not(ls.play[m]), ls.clues[m-1][8])), Iff(ls.strike[m], And(ls.discard_any[m], Not(ls.play[m]), ls.clues[m-1][8])),
# change of strikes # change of strikes
*[Iff(ls.strikes[m][i], Or(ls.strikes[m-1][i], And(ls.strikes[m-1][i-1], ls.strike[m]))) for i in range(1, ls.num_strikes + 1)], *[Iff(ls.strikes[m][i], Or(ls.strikes[m-1][i], And(ls.strikes[m-1][i-1], ls.strike[m]))) for i in range(1, instance.num_strikes + 1)],
# less than 0 clues not allowed # less than 0 clues not allowed
Implies(Not(ls.discard_any[m]), Or(ls.clues[m-1][1], ls.dummyturn[m])), Implies(Not(ls.discard_any[m]), Or(ls.clues[m-1][1], ls.dummyturn[m])),
# we can only draw card i if the last ls.drawn card was i-1 # we can only draw card i if the last ls.drawn card was i-1
*[Implies(ls.draw[m][i], Or(And(ls.draw[m0][i-1], *[Not(ls.draw_any[m1]) for m1 in range(m0+1, m)]) for m0 in range(max(first_turn - 1, m-9), m))) for i in range(next_draw, ls.deck_size)], *[Implies(ls.draw[m][i], Or(And(ls.draw[m0][i-1], *[Not(ls.draw_any[m1]) for m1 in range(m0+1, m)]) for m0 in range(max(first_turn - 1, m-9), m))) for i in range(game_state.progress, instance.deck_size)],
# we can only draw at most one card (NOTE: redundant, FIXME: avoid quadratic formula) # we can only draw at most one card (NOTE: redundant, FIXME: avoid quadratic formula)
AtMostOne(ls.draw[m][i] for i in range(next_draw, ls.deck_size)), AtMostOne(ls.draw[m][i] for i in range(game_state.progress, instance.deck_size)),
# we can only discard a card if we drew it earlier... # we can only discard a card if we drew it earlier...
*[Implies(ls.discard[m][i], Or(ls.draw[m0][i] for m0 in range(m-ls.num_players, first_turn - 1, -ls.num_players))) for i in range(next_draw, ls.deck_size)], *[Implies(ls.discard[m][i], Or(ls.draw[m0][i] for m0 in range(m-instance.num_players, first_turn - 1, -instance.num_players))) for i in range(game_state.progress, instance.deck_size)],
# ...or if it was part of the initial hand # ...or if it was part of the initial hand
*[Not(ls.discard[m][i]) for i in range(0, next_draw) if i not in starting_hands[m % ls.num_players] ], *[Not(ls.discard[m][i]) for i in range(0, game_state.progress) if i not in starting_hands[m % instance.num_players] ],
# we can only discard a card if we did not discard it yet # we can only discard a card if we did not discard it yet
*[Implies(ls.discard[m][i], And(Not(ls.discard[m0][i]) for m0 in range(m-ls.num_players, first_turn - 1, -ls.num_players))) for i in range(ls.deck_size)], *[Implies(ls.discard[m][i], And(Not(ls.discard[m0][i]) for m0 in range(m-instance.num_players, first_turn - 1, -instance.num_players))) for i in range(instance.deck_size)],
# we can only discard at most one card (FIXME: avoid quadratic formula) # we can only discard at most one card (FIXME: avoid quadratic formula)
AtMostOne(ls.discard[m][i] for i in range(ls.deck_size)), AtMostOne(ls.discard[m][i] for i in range(instance.deck_size)),
# we can only play a card if it matches the progress # we can only play a card if it matches the progress
*[Implies( *[Implies(
And(ls.discard[m][i], ls.play[m]), And(ls.discard[m][i], ls.play[m]),
And( And(
Not(ls.progress[m-1][deck[i].suitIndex, deck[i].rank]), Not(ls.progress[m-1][instance.deck[i].suitIndex, instance.deck[i].rank]),
ls.progress[m-1][deck[i].suitIndex, deck[i].rank-1 ] ls.progress[m-1][instance.deck[i].suitIndex, instance.deck[i].rank-1 ]
) )
) )
for i in range(ls.deck_size) for i in range(instance.deck_size)
], ],
# change of progress # change of progress
@ -266,46 +244,46 @@ def solve_sat(game_state: GameState):
Or( Or(
ls.progress[m-1][s, r], ls.progress[m-1][s, r],
And(ls.play[m], Or(ls.discard[m][i] And(ls.play[m], Or(ls.discard[m][i]
for i in range(0, ls.deck_size) for i in range(0, instance.deck_size)
if deck[i] == DeckCard(s, r) )) if instance.deck[i] == DeckCard(s, r) ))
) )
) )
for s in range(0, ls.num_suits) for s in range(0, instance.num_suits)
for r in range(1, 6) for r in range(1, 6)
], ],
# extra round bool # extra round bool
Iff(ls.extraround[m], Or(ls.extraround[m-1], ls.draw[m-1][ls.deck_size-1])), Iff(ls.extraround[m], Or(ls.extraround[m-1], ls.draw[m-1][instance.deck_size-1])),
# dummy turn bool # dummy turn bool
*[Iff(ls.dummyturn[m], Or(ls.dummyturn[m-1], ls.draw[m-1 - ls.num_players][ls.deck_size-1])) for i in range(0,1) if m >= ls.num_players] *[Iff(ls.dummyturn[m], Or(ls.dummyturn[m-1], ls.draw[m-1 - instance.num_players][instance.deck_size-1])) for i in range(0,1) if m >= instance.num_players]
) )
win = And( win = And(
# maximum progress at each color # maximum progress at each color
*[ls.progress[ls.max_moves-1][s, 5] for s in range(0, ls.num_suits)], *[ls.progress[instance.max_winning_moves-1][s, 5] for s in range(0, instance.num_suits)],
# played every color/value combination (NOTE: redundant, but makes solving faster) # played every color/value combination (NOTE: redundant, but makes solving faster)
*[ *[
Or( Or(
And(ls.discard[m][i], ls.play[m]) And(ls.discard[m][i], ls.play[m])
for m in range(first_turn, ls.max_moves) for m in range(first_turn, instance.max_winning_moves)
for i in range(ls.deck_size) for i in range(instance.deck_size)
if game_state.deck[i] == DeckCard(s, r) if game_state.deck[i] == DeckCard(s, r)
) )
for s in range(0, ls.num_suits) for s in range(0, instance.num_suits)
for r in range(1, 6) for r in range(1, 6)
if r > game_state.stacks[s] if r > game_state.stacks[s]
] ]
) )
constraints = And(*[valid_move(m) for m in range(first_turn, ls.max_moves)], win) constraints = And(*[valid_move(m) for m in range(first_turn, instance.max_winning_moves)], win)
# print('Solving instance with {} variables, {} nodes'.format(len(get_atoms(constraints)), get_formula_size(constraints))) # print('Solving instance with {} variables, {} nodes'.format(len(get_atoms(constraints)), get_formula_size(constraints)))
model = get_model(constraints) model = get_model(constraints)
if model: if model:
# print_model(model, game_state, ls) # print_model(model, game_state, ls)
solution = toJSON(model, game_state, ls) solution = evaluate_model(model, game_state, ls)
return True, solution return True, solution
else: else:
#conj = list(conjunctive_partition(constraints)) #conj = list(conjunctive_partition(constraints))
@ -316,6 +294,7 @@ def solve_sat(game_state: GameState):
# print(f.serialize()) # print(f.serialize())
return False, None return False, None
def print_model(model, cur_game_state, ls: Literals): def print_model(model, cur_game_state, ls: Literals):
deck = cur_game_state.deck deck = cur_game_state.deck
for m in range(ls.max_moves): for m in range(ls.max_moves):
@ -330,12 +309,15 @@ def print_model(model, cur_game_state, ls: Literals):
print(', '.join(f for f in flags if model.get_py_value(getattr(ls, f)[m]))) print(', '.join(f for f in flags if model.get_py_value(getattr(ls, f)[m])))
def toJSON(model, cur_game_state: GameState, ls: Literals) -> GameState:
for m in range(len(cur_game_state.actions), ls.max_moves): # given the initial game state and the model found by the SAT solver,
# evaluates the model to produce a full game history
def evaluate_model(model, cur_game_state: GameState, ls: Literals) -> GameState:
for m in range(len(cur_game_state.actions), cur_game_state.instance.max_winning_moves):
if model.get_py_value(ls.dummyturn[m]): if model.get_py_value(ls.dummyturn[m]):
break break
if model.get_py_value(ls.discard_any[m]): if model.get_py_value(ls.discard_any[m]):
card_idx = next(i for i in range(0, ls.deck_size) if model.get_py_value(ls.discard[m][i])) card_idx = next(i for i in range(0, cur_game_state.instance.deck_size) if model.get_py_value(ls.discard[m][i]))
if model.get_py_value(ls.play[m]) or model.get_py_value(ls.strike[m]): if model.get_py_value(ls.play[m]) or model.get_py_value(ls.strike[m]):
cur_game_state.play(card_idx) cur_game_state.play(card_idx)
else: else:
@ -345,6 +327,8 @@ def toJSON(model, cur_game_state: GameState, ls: Literals) -> GameState:
return cur_game_state return cur_game_state
def run_deck(): def run_deck():
puzzle = False puzzle = False
if puzzle: if puzzle:
@ -361,7 +345,7 @@ def run_deck():
print(deck) print(deck)
gs = GameState(num_p, deck) gs = GameState(HanabiInstance(deck, num_p))
if puzzle: if puzzle:
gs.play(2) gs.play(2)
else: else:
@ -372,7 +356,7 @@ def run_deck():
solvable, sol = solve_sat(gs) solvable, sol = solve_sat(gs)
if solvable: if solvable:
print(sol) print(sol)
print(link(sol.to_json())) print(link(sol))
else: else:
print('unsolvable') print('unsolvable')