def solve_nurimaze(height, width, wall_vertical, wall_horizontal, mark, start,
goal):
solver = Solver()
is_white = solver.bool_array((height, width))
graph.active_vertices_connected(solver, is_white, acyclic=True)
solver.add_answer_key(is_white)
solver.ensure(is_white[:-1, :-1] | is_white[:-1, 1:] | is_white[1:, :-1]
| is_white[1:, 1:])
solver.ensure(~(is_white[:-1, :-1] & is_white[:-1, 1:] & is_white[1:, :-1]
& is_white[1:, 1:]))
path = solver.bool_array((height, width))
solver.ensure(path.then(is_white))
for y in range(height):
for x in range(width):
if x < width - 1 and not wall_vertical[y][x]:
solver.ensure(is_white[y, x] == is_white[y, x + 1])
if y < height - 1 and not wall_horizontal[y][x]:
solver.ensure(is_white[y, x] == is_white[y + 1, x])
if (y, x) == start or (y, x) == goal:
solver.ensure(path[y, x])
solver.ensure(count_true(path.four_neighbors(y, x)) == 1)
else:
solver.ensure(
path[y,
x].then(count_true(path.four_neighbors(y, x)) == 2))
if mark[y][x] != 0:
solver.ensure(is_white[y, x])
if mark[y][x] == 1: # pass
solver.ensure(path[y, x])
elif mark[y][x] == 2:
solver.ensure(~path[y, x])
is_sat = solver.solve()
return is_sat, is_white
def solve_magnets(height, width, to_right, to_down, cond_row, cond_col):
solver = Solver()
plus = solver.bool_array((height, width))
minus = solver.bool_array((height, width))
solver.add_answer_key(plus)
solver.add_answer_key(minus)
solver.ensure(~(plus & minus))
solver.ensure(
Array(to_right)[:, :-1].then((plus[:, :-1] == minus[:, 1:])
& (minus[:, :-1] == plus[:, 1:])))
solver.ensure(
Array(to_down)[:-1, :].then((plus[:-1, :] == minus[1:, :])
& (minus[:-1, :] == plus[1:, :])))
solver.ensure(~(plus[:-1, :] & plus[1:, :]))
solver.ensure(~(minus[:-1, :] & minus[1:, :]))
solver.ensure(~(plus[:, :-1] & plus[:, 1:]))
solver.ensure(~(minus[:, :-1] & minus[:, 1:]))
for y in range(height):
if cond_row[y][0] >= 0:
solver.ensure(count_true(plus[y, :]) == cond_row[y][0])
if cond_row[y][1] >= 0:
solver.ensure(count_true(minus[y, :]) == cond_row[y][1])
for x in range(width):
if cond_col[x][0] >= 0:
solver.ensure(count_true(plus[:, x]) == cond_col[x][0])
if cond_col[x][1] >= 0:
solver.ensure(count_true(minus[:, x]) == cond_col[x][1])
is_sat = solver.solve()
return is_sat, plus, minus
def solve_fillomino(height,
width,
problem,
checkered=False,
is_non_con=False,
is_anti_knight=False):
solver = Solver()
size = solver.int_array((height, width), 1, height * width)
solver.add_answer_key(size)
group_id = graph.division_connected_variable_groups(solver,
group_size=size)
solver.ensure(
(group_id[:, :-1] == group_id[:, 1:]) == (size[:, :-1] == size[:, 1:]))
solver.ensure(
(group_id[:-1, :] == group_id[1:, :]) == (size[:-1, :] == size[1:, :]))
for y in range(height):
for x in range(width):
if problem[y][x] >= 1:
solver.ensure(size[y, x] == problem[y][x])
if checkered:
color = solver.bool_array((height, width))
solver.ensure(
(group_id[:, :-1] == group_id[:,
1:]) == (color[:, :-1] == color[:,
1:]))
solver.ensure((group_id[:-1, :] == group_id[1:, :]) == (
color[:-1, :] == color[1:, :]))
if is_non_con:
graph.numbers_non_consecutive(solver, size)
if is_anti_knight:
graph.numbers_anti_knight(solver, size)
is_sat = solver.solve()
return is_sat, size
def solve_nurikabe(height, width, problem, unknown_low=None):
solver = Solver()
clues = []
for y in range(height):
for x in range(width):
if problem[y][x] >= 1 or problem[y][x] == -1:
clues.append((y, x, problem[y][x]))
division = solver.int_array((height, width), 0, len(clues))
roots = [None] + list(map(lambda x: (x[0], x[1]), clues))
graph.division_connected(solver, division, len(clues) + 1, roots=roots)
is_white = solver.bool_array((height, width))
solver.ensure(is_white == (division != 0))
solver.add_answer_key(is_white)
solver.ensure(
(is_white[:-1, :]
& is_white[1:, :]).then(division[:-1, :] == division[1:, :]))
solver.ensure((is_white[:, :-1]
& is_white[:, 1:]).then(division[:, :-1] == division[:,
1:]))
solver.ensure(is_white[:-1, :-1] | is_white[:-1, 1:] | is_white[1:, :-1]
| is_white[1:, 1:])
for i, (y, x, n) in enumerate(clues):
if n > 0:
solver.ensure(count_true(division == (i + 1)) == n)
elif n == -1 and unknown_low is not None:
solver.ensure(count_true(division == (i + 1)) >= unknown_low)
is_sat = solver.solve()
return is_sat, is_white
def solve_putteria(height, width, blocks):
solver = Solver()
has_number = solver.bool_array((height, width))
solver.add_answer_key(has_number)
solver.ensure((~has_number[:, :-1]) | (~has_number[:, 1:]))
solver.ensure((~has_number[:-1, :]) | (~has_number[1:, :]))
for block in blocks:
solver.ensure(count_true([has_number[y, x] for y, x in block]) == 1)
block_size = [[0 for _ in range(width)] for _ in range(height)]
for block in blocks:
for y, x in block:
block_size[y][x] = len(block)
for y in range(height):
for x1 in range(width):
for x2 in range(x1 + 1, width):
if block_size[y][x1] == block_size[y][x2]:
solver.ensure(~(has_number[y, x1] & has_number[y, x2]))
for x in range(width):
for y1 in range(height):
for y2 in range(y1 + 1, height):
if block_size[y1][x] == block_size[y2][x]:
solver.ensure(~(has_number[y1, x] & has_number[y2, x]))
is_sat = solver.solve()
return is_sat, has_number
def solve_gokigen(height, width, problem):
solver = Solver()
edge_type = solver.bool_array((height, width)) # false: /, true: \
solver.add_answer_key(edge_type)
g = graph.Graph((height + 1) * (width + 1))
edge_list = []
for y in range(height):
for x in range(width):
g.add_edge(y * (width + 1) + x, (y + 1) * (width + 1) + (x + 1))
edge_list.append(edge_type[y, x])
g.add_edge(y * (width + 1) + (x + 1), (y + 1) * (width + 1) + x)
edge_list.append(~edge_type[y, x])
graph.active_edges_acyclic(solver, Array(edge_list), g)
for y in range(height + 1):
for x in range(width + 1):
if problem[y][x] >= 0:
related = []
if 0 < y and 0 < x:
related.append(edge_type[y - 1, x - 1])
if 0 < y and x < width:
related.append(~edge_type[y - 1, x])
if y < height and 0 < x:
related.append(~edge_type[y, x - 1])
if y < height and x < width:
related.append(edge_type[y, x])
solver.ensure(count_true(related) == problem[y][x])
is_sat = solver.solve()
return is_sat, edge_type
def solve_fivecells(height, width, problem):
vertex_id = [[-1 for _ in range(width)] for _ in range(height)]
id_last = 0
for y in range(height):
for x in range(width):
if problem[y][x] >= -1:
vertex_id[y][x] = id_last
id_last += 1
g = graph.Graph(id_last)
for y in range(height):
for x in range(width):
if problem[y][x] >= -1:
if y < height - 1 and problem[y + 1][x] >= -1:
g.add_edge(vertex_id[y][x], vertex_id[y + 1][x])
if x < width - 1 and problem[y][x + 1] >= -1:
g.add_edge(vertex_id[y][x], vertex_id[y][x + 1])
solver = Solver()
group_id = graph.division_connected_variable_groups(solver,
graph=g,
group_size=5)
is_invalid = False
for y in range(height):
for x in range(width):
if problem[y][x] >= 0:
borders = []
if y > 0 and problem[y - 1][x] >= -1:
borders.append(
group_id[vertex_id[y][x]] != group_id[vertex_id[y -
1][x]])
if y < height - 1 and problem[y + 1][x] >= -1:
borders.append(
group_id[vertex_id[y][x]] != group_id[vertex_id[y +
1][x]])
if x > 0 and problem[y][x - 1] >= -1:
borders.append(
group_id[vertex_id[y][x]] != group_id[vertex_id[y][x -
1]])
if x < width - 1 and problem[y][x + 1] >= -1:
borders.append(
group_id[vertex_id[y][x]] != group_id[vertex_id[y][x +
1]])
always_border = 4 - len(borders)
solver.ensure(
count_true(borders) == problem[y][x] - always_border)
if problem[y][x] - always_border < 0:
is_invalid = True
is_border = solver.bool_array(len(g))
for i, (u, v) in enumerate(g):
solver.ensure(is_border[i] == (group_id[u] != group_id[v]))
solver.add_answer_key(is_border)
if is_invalid:
is_sat = False
else:
is_sat = solver.solve()
return is_sat, is_border
def solve_nurimisaki(height, width, problem):
solver = Solver()
is_white = solver.bool_array((height, width))
solver.add_answer_key(is_white)
graph.active_vertices_connected(solver, is_white)
solver.ensure(is_white[:-1, :-1] | is_white[1:, :-1] | is_white[:-1, 1:]
| is_white[1:, 1:])
solver.ensure(~(is_white[:-1, :-1] & is_white[1:, :-1] & is_white[:-1, 1:]
& is_white[1:, 1:]))
for y in range(height):
for x in range(width):
if problem[y][x] == -1:
solver.ensure(is_white[y, x].then(
count_true(is_white.four_neighbors(y, x)) != 1))
else:
solver.ensure(is_white[y, x])
solver.ensure(count_true(is_white.four_neighbors(y, x)) == 1)
if problem[y][x] != 0:
n = problem[y][x]
cand = []
if y == n - 1:
cand.append(fold_and(is_white[(y - n + 1):y, x]))
elif y > n - 1:
cand.append(
fold_and(is_white[(y - n + 1):y, x],
~is_white[y - n, x]))
if y == height - n:
cand.append(fold_and(is_white[(y + 1):(y + n), x]))
elif y < height - n:
cand.append(
fold_and(is_white[(y + 1):(y + n), x],
~is_white[y + n, x]))
if x == n - 1:
cand.append(fold_and(is_white[y, (x - n + 1):x]))
elif x > n - 1:
cand.append(
fold_and(is_white[y, (x - n + 1):x],
~is_white[y, x - n]))
if x == width - n:
cand.append(fold_and(is_white[y, (x + 1):(x + n)]))
elif x < width - n:
cand.append(
fold_and(is_white[y, (x + 1):(x + n)],
~is_white[y, x + n]))
solver.ensure(fold_or(cand))
is_sat = solver.solve()
return is_sat, is_white
def solve_creek(height, width, problem):
solver = Solver()
is_white = solver.bool_array((height, width))
solver.add_answer_key(is_white)
graph.active_vertices_connected(solver, is_white)
for y in range(0, height + 1):
for x in range(0, width + 1):
if problem[y][x] >= 0:
solver.ensure(
count_true(~is_white[max(y - 1, 0):min(y + 1, height),
max(x - 1, 0):min(x + 1, width)]) ==
problem[y][x])
is_sat = solver.solve()
return is_sat, is_white
def check_problem_constraints(height, width, problem, flg, circ=-1):
if flg is None and circ == -1:
return True
for clue in problem:
a = 0
for i in range(2, 6):
if clue[i] >= 0:
a += 1
solver = Solver()
roots = map(lambda x: (x[0], x[1]), problem)
division = solver.int_array((height, width), 0, len(problem) - 1)
graph.division_connected(solver, division, len(problem), roots=roots)
solver.add_answer_key(division)
for i, (y, x, u, l, d, r) in enumerate(problem):
solver.ensure(division[y, x] == i)
if flg is not None and flg[i]:
solver.ensure(count_true(division == i) >= 4)
if u >= 0:
solver.ensure(count_true(division[:y, :] == i) == u)
if d >= 0:
solver.ensure(count_true(division[(y + 1):, :] == i) == d)
if l >= 0:
solver.ensure(count_true(division[:, :x] == i) == l)
if r >= 0:
solver.ensure(count_true(division[:, (x + 1):] == i) == r)
# encircling constraint
if circ != -1:
col = solver.bool_array((height, width))
solver.ensure(col[0, :])
solver.ensure(col[-1, :])
solver.ensure(col[:, 0])
solver.ensure(col[:, -1])
solver.ensure(
((division[1:, :] != circ) &
(division[:-1, :] != circ)).then(col[1:, :] == col[:-1, :]))
solver.ensure(
((division[:, 1:] != circ) &
(division[:, :-1] != circ)).then(col[:, 1:] == col[:, :-1]))
solver.ensure(
((division[1:, 1:] != circ) &
(division[:-1, :-1] != circ)).then(col[1:, 1:] == col[:-1, :-1]))
solver.ensure(
((division[:-1, 1:] != circ) &
(division[1:, :-1] != circ)).then(col[:-1, 1:] == col[1:, :-1]))
solver.ensure(fold_or(col & (division != circ)))
solver.ensure(fold_or((~col) & (division != circ)))
sat = solver.find_answer()
return sat
def solve_norinori(height, width, blocks):
solver = Solver()
is_black = solver.bool_array((height, width))
solver.add_answer_key(is_black)
for y in range(height):
for x in range(width):
solver.ensure(is_black[y, x].then(
count_true(is_black.four_neighbors(y, x)) == 1))
for block in blocks:
solver.ensure(count_true(map(lambda p: is_black[p], block)) == 2)
is_sat = solver.solve()
return is_sat, is_black
def solve_darts(target, darts, board):
solver = Solver()
is_dart = solver.bool_array(len(board))
solver.add_answer_key(is_dart)
solver.ensure(count_true(is_dart) == darts)
def _sum(d, b):
sums = 0
for i in range(len(board)):
sums += d[i].cond(b[i], 0)
return sums
solver.ensure(_sum(is_dart, board) == target)
is_sat = solver.solve()
return is_sat, is_dart
def solve_view(height, width, problem, is_non_con=False, is_anti_knight=False):
solver = Solver()
has_number = solver.bool_array((height, width))
graph.active_vertices_connected(solver, has_number)
nums = solver.int_array((height, width), 0, height + width)
solver.add_answer_key(nums)
solver.add_answer_key(has_number)
to_up = solver.int_array((height, width), 0, height - 1)
solver.ensure(to_up[0, :] == 0)
solver.ensure(to_up[1:, :] == has_number[:-1, :].cond(0, to_up[:-1, :] + 1))
to_down = solver.int_array((height, width), 0, height - 1)
solver.ensure(to_down[-1, :] == 0)
solver.ensure(to_down[:-1, :] == has_number[1:, :].cond(0, to_down[1:, :] + 1))
to_left = solver.int_array((height, width), 0, width - 1)
solver.ensure(to_left[:, 0] == 0)
solver.ensure(to_left[:, 1:] == has_number[:, :-1].cond(0, to_left[:, :-1] + 1))
to_right = solver.int_array((height, width), 0, width - 1)
solver.ensure(to_right[:, -1] == 0)
solver.ensure(to_right[:, :-1] == has_number[:, 1:].cond(0, to_right[:, 1:] + 1))
solver.ensure(has_number.then(nums == to_up + to_left + to_down + to_right))
solver.ensure((has_number[:-1, :] & has_number[1:, :]).then(nums[:-1, :] != nums[1:, :]))
solver.ensure((has_number[:, :-1] & has_number[:, 1:]).then(nums[:, :-1] != nums[:, 1:]))
solver.ensure((~has_number).then(nums == 0))
for y in range(height):
for x in range(width):
if problem[y][x] >= 0:
solver.ensure(nums[y, x] == problem[y][x])
solver.ensure(has_number[y, x])
if is_non_con:
graph.numbers_non_consecutive(solver, nums, has_number)
if is_anti_knight:
graph.numbers_anti_knight(solver, nums, has_number)
is_sat = solver.solve()
return is_sat, nums, has_number
def solve_castle_wall(height, width, arrow, inside):
solver = Solver()
grid_frame = BoolGridFrame(solver, height - 1, width - 1)
solver.add_answer_key(grid_frame)
passed = graph.active_edges_single_cycle(solver, grid_frame)
# arrow constraints
for y in range(height):
for x in range(width):
if arrow[y][x] == '..':
continue
solver.ensure(~passed[y, x])
if arrow[y][x][0] == '^':
related_edges = grid_frame.vertical[:y, x]
elif arrow[y][x][0] == 'v':
related_edges = grid_frame.vertical[y:, x]
elif arrow[y][x][0] == '<':
related_edges = grid_frame.horizontal[y, :x]
elif arrow[y][x][0] == '>':
related_edges = grid_frame.horizontal[y, x:]
else:
continue
solver.ensure(count_true(related_edges) == int(arrow[y][x][1:]))
# inout constraints
is_inside = solver.bool_array((height - 1, width - 1))
for y in range(height - 1):
for x in range(width - 1):
if y == 0:
solver.ensure(is_inside[y, x] == grid_frame[0, x * 2 + 1])
else:
solver.ensure(is_inside[y, x] == (
is_inside[y - 1, x] != grid_frame[y * 2, x * 2 + 1]))
for y in range(height):
for x in range(width):
if inside[y][x] is True:
solver.ensure(is_inside[max(0, y - 1), max(0, x - 1)])
elif inside[y][x] is False:
solver.ensure(~is_inside[max(0, y - 1), max(0, x - 1)])
is_sat = solver.solve()
return is_sat, grid_frame
def solve_star_battle(n, blocks, k, is_anti_knight=False):
if not isinstance(blocks, Array):
blocks = Array(blocks)
solver = Solver()
has_star = solver.bool_array((n, n))
solver.add_answer_key(has_star)
for i in range(n):
solver.ensure(sum(has_star[i, :].cond(1, 0)) == k)
solver.ensure(sum(has_star[:, i].cond(1, 0)) == k)
solver.ensure(~(has_star[:-1, :] & has_star[1:, :]))
solver.ensure(~(has_star[:, :-1] & has_star[:, 1:]))
solver.ensure(~(has_star[:-1, :-1] & has_star[1:, 1:]))
solver.ensure(~(has_star[:-1, 1:] & has_star[1:, :-1]))
for i in range(n):
solver.ensure(sum((has_star & (blocks == i)).cond(1, 0)) == k)
if is_anti_knight:
graph.active_vertices_anti_knight(solver, has_star)
is_sat = solver.solve()
return is_sat, has_star
def solve_nanro(height, width, blocks, num):
block_id = [[-1 for _ in range(width)] for _ in range(height)]
for i, block in enumerate(blocks):
for y, x in block:
block_id[y][x] = i
solver = Solver()
answer = []
has_num = solver.bool_array((height, width))
for y in range(height):
row = []
for x in range(width):
v = solver.int_var(0, len(blocks[block_id[y][x]]))
solver.add_answer_key(v)
solver.ensure(has_num[y, x] == (v != 0))
row.append(v)
answer.append(row)
graph.active_vertices_connected(solver, has_num)
for i, block in enumerate(blocks):
nonempty = solver.int_var(1, len(block))
solver.ensure(nonempty == count_true(answer[y][x] != 0
for y, x in block))
for y, x in block:
solver.ensure((answer[y][x] == 0) | (answer[y][x] == nonempty))
for y in range(height):
for x in range(width):
if num[y][x] > 0:
solver.ensure(answer[y][x] == num[y][x])
if y < height - 1 and x < width - 1:
solver.ensure((answer[y][x] == 0) | (answer[y][x + 1] == 0)
| (answer[y + 1][x] == 0)
| (answer[y + 1][x + 1] == 0))
if y < height - 1 and block_id[y][x] != block_id[y + 1][x]:
solver.ensure((answer[y][x] == 0) | (answer[y + 1][x] == 0)
| (answer[y][x] != answer[y + 1][x]))
if x < width - 1 and block_id[y][x] != block_id[y][x + 1]:
solver.ensure((answer[y][x] == 0) | (answer[y][x + 1] == 0)
| (answer[y][x] != answer[y][x + 1]))
is_sat = solver.solve()
return is_sat, answer
def solve_yinyang(height, width, problem):
solver = Solver()
is_black = solver.bool_array((height, width))
solver.add_answer_key(is_black)
graph.active_vertices_connected(solver, is_black)
graph.active_vertices_connected(solver, ~is_black)
solver.ensure(is_black[:-1, :-1] | is_black[:-1, 1:] | is_black[1:, :-1]
| is_black[1:, 1:])
solver.ensure(~(is_black[:-1, :-1] & is_black[:-1, 1:] & is_black[1:, :-1]
& is_black[1:, 1:]))
# auxiliary constraint
solver.ensure(~(is_black[:-1, :-1] & is_black[1:, 1:] & ~is_black[1:, :-1]
& ~is_black[:-1, 1:]))
solver.ensure(~(~is_black[:-1, :-1] & ~is_black[1:, 1:] & is_black[1:, :-1]
& is_black[:-1, 1:]))
circ = []
for y in range(height):
circ.append(is_black[y, 0])
for x in range(1, width):
circ.append(is_black[-1, x])
for y in reversed(range(0, height - 1)):
circ.append(is_black[y, -1])
for x in reversed(range(1, width - 1)):
circ.append(is_black[0, x])
circ_switching = []
for i in range(len(circ)):
circ_switching.append(circ[i] != circ[(i + 1) % len(circ)])
solver.ensure(count_true(circ_switching) <= 2)
for y in range(height):
for x in range(width):
if problem[y][x] == 1:
solver.ensure(~is_black[y, x])
elif problem[y][x] == 2:
solver.ensure(is_black[y, x])
is_sat = solver.solve()
return is_sat, is_black
def solve_sukoro(height, width, problem, is_anti_knight=False):
solver = Solver()
has_number = solver.bool_array((height, width))
graph.active_vertices_connected(solver, has_number)
nums = solver.int_array((height, width), -1, 4)
solver.add_answer_key(nums)
solver.add_answer_key(has_number)
for y in range(height):
for x in range(width):
neighbors = []
if y > 0:
neighbors.append(has_number[y-1, x])
solver.ensure((has_number[y, x] & has_number[y-1, x]).then(nums[y, x] != nums[y-1, x]))
if y < height - 1:
neighbors.append(has_number[y+1, x])
solver.ensure((has_number[y, x] & has_number[y+1, x]).then(nums[y, x] != nums[y+1, x]))
if x > 0:
neighbors.append(has_number[y, x-1])
solver.ensure((has_number[y, x] & has_number[y, x-1]).then(nums[y, x] != nums[y, x-1]))
if x < width - 1:
neighbors.append(has_number[y, x+1])
solver.ensure((has_number[y, x] & has_number[y, x+1]).then(nums[y, x] != nums[y, x+1]))
solver.ensure(has_number[y, x].then(count_true(neighbors) == nums[y, x]))
solver.ensure((~has_number).then(nums < 0))
for y in range(height):
for x in range(width):
if problem[y][x] >= 0:
solver.ensure(nums[y, x] == problem[y][x])
solver.ensure(has_number[y, x])
if is_anti_knight:
graph.numbers_anti_knight(solver, nums, has_number)
is_sat = solver.solve()
return is_sat, nums, has_number
def solve_yajilin(height, width, problem):
solver = Solver()
grid_frame = BoolGridFrame(solver, height - 1, width - 1)
is_passed = graph.active_edges_single_cycle(solver, grid_frame)
black_cell = solver.bool_array((height, width))
graph.active_vertices_not_adjacent(solver, black_cell)
solver.add_answer_key(grid_frame)
solver.add_answer_key(black_cell)
for y in range(height):
for x in range(width):
if problem[y][x] != '..':
# clue
solver.ensure(~is_passed[y, x])
solver.ensure(~black_cell[y, x])
if problem[y][x][0] == '^':
solver.ensure(
count_true(black_cell[0:y,
x]) == int(problem[y][x][1:]))
elif problem[y][x][0] == 'v':
solver.ensure(
count_true(black_cell[(y + 1):height,
x]) == int(problem[y][x][1:]))
elif problem[y][x][0] == '<':
solver.ensure(
count_true(black_cell[y,
0:x]) == int(problem[y][x][1:]))
elif problem[y][x][0] == '>':
solver.ensure(
count_true(black_cell[y, (
x + 1):width]) == int(problem[y][x][1:]))
else:
solver.ensure(is_passed[y, x] != black_cell[y, x])
is_sat = solver.solve()
return is_sat, grid_frame, black_cell
def solve_slalom(height,
width,
origin,
is_black,
gates,
reference_sol_loop=None):
solver = Solver()
loop = BoolGridFrame(solver, height - 1, width - 1)
loop_dir = BoolGridFrame(solver, height - 1, width - 1)
solver.add_answer_key(loop.all_edges())
graph.active_edges_single_cycle(solver, loop)
gate_ord = solver.int_array((height, width), 0, len(gates))
passed = solver.bool_array((height, width))
gate_id = [[None for _ in range(width)] for _ in range(height)]
for y, x, d, l, n in gates:
if d == 0: # horizontal
gate_cells = [(y, x + i) for i in range(l)]
elif d == 1: # vertical
gate_cells = [(y + i, x) for i in range(l)]
for y2, x2 in gate_cells:
gate_id[y2][x2] = n
solver.ensure(
count_true([passed[y2, x2] for y2, x2 in gate_cells]) == 1)
solver.ensure(passed[origin])
for y in range(height):
for x in range(width):
neighbors = []
if y > 0:
neighbors.append((y - 1, x))
if y < height - 1:
neighbors.append((y + 1, x))
if x > 0:
neighbors.append((y, x - 1))
if x < width - 1:
neighbors.append((y, x + 1))
# in-degree, out-degree
solver.ensure(
count_true([
loop[y + y2, x + x2]
& (loop_dir[y + y2, x + x2] != ((y2, x2) < (y, x)))
for y2, x2 in neighbors
]) == passed[y, x].cond(1, 0))
solver.ensure(
count_true([
loop[y + y2, x + x2]
& (loop_dir[y + y2, x + x2] == ((y2, x2) < (y, x)))
for y2, x2 in neighbors
]) == passed[y, x].cond(1, 0))
if is_black[y][x]:
solver.ensure(~passed[y, x])
continue
if (y, x) == origin:
continue
if gate_id[y][x] is None:
for y2, x2 in neighbors:
solver.ensure((loop[y + y2, x + x2] &
(loop_dir[y + y2, x + x2] !=
((y2, x2) < (y, x)))).then(
(gate_ord[y2, x2] == gate_ord[y, x])))
else:
for y2, x2 in neighbors:
solver.ensure((loop[y + y2, x + x2] &
(loop_dir[y + y2, x + x2] !=
((y2, x2) < (y, x)))).then(
(gate_ord[y2,
x2] == gate_ord[y, x] - 1)))
if gate_id[y][x] >= 1:
solver.ensure(passed[y,
x].then(gate_ord[y,
x] == gate_id[y][x]))
# auxiliary constraint
for y0 in range(height):
for x0 in range(width):
for y1 in range(height):
for x1 in range(width):
if (y0, x0) < (y1, x1) and gate_id[y0][
x0] is not None and gate_id[y1][x1] is not None:
solver.ensure((passed[y0, x0] & passed[y1, x1]).then(
gate_ord[y0, x0] != gate_ord[y1, x1]))
if reference_sol_loop is not None:
avoid_reference_sol = []
for y in range(height):
for x in range(width):
if y < height - 1:
avoid_reference_sol.append(
loop.vertical[y,
x] != reference_sol_loop.vertical[y,
x].sol)
if x < width - 1:
avoid_reference_sol.append(loop.horizontal[
y, x] != reference_sol_loop.horizontal[y, x].sol)
solver.ensure(fold_or(avoid_reference_sol))
is_sat = solver.find_answer()
return is_sat, loop
else:
is_sat = solver.solve()
return is_sat, loop
def generate_slalom_initial_placement(height,
width,
n_min_gates=None,
n_max_gates=None,
n_max_isolated_black_cells=None,
no_adjacent_black_cell=False,
no_facing_length_two=False,
no_space_2x2=False,
black_cell_in_every_3x3=False,
min_go_through=0):
solver = Solver()
loop = BoolGridFrame(solver, height - 1, width - 1)
is_black = solver.bool_array((height, width))
is_horizontal = solver.bool_array((height, width))
is_vertical = solver.bool_array((height, width))
solver.ensure(~(is_black & is_horizontal))
solver.ensure(~(is_black & is_vertical))
solver.ensure(~(is_horizontal & is_vertical))
solver.ensure(~(is_horizontal[0, :]))
solver.ensure(~(is_horizontal[-1, :]))
solver.ensure(~(is_vertical[:, 0]))
solver.ensure(~(is_vertical[:, -1]))
is_passed = graph.active_edges_single_cycle(solver, loop)
# --------- board must be valid as a problem ---------
# loop constraints
for y in range(height):
for x in range(width):
if y > 0:
solver.ensure(is_black[y, x].then(~loop.vertical[y - 1, x]))
solver.ensure(is_vertical[y, x].then(~loop.vertical[y - 1, x]))
if y < height - 1:
solver.ensure(is_black[y, x].then(~loop.vertical[y, x]))
solver.ensure(is_vertical[y, x].then(~loop.vertical[y, x]))
if x > 0:
solver.ensure(is_black[y, x].then(~loop.horizontal[y, x - 1]))
solver.ensure(
is_horizontal[y, x].then(~loop.horizontal[y, x - 1]))
if x < width - 1:
solver.ensure(is_black[y, x].then(~loop.horizontal[y, x]))
solver.ensure(is_horizontal[y, x].then(~loop.horizontal[y, x]))
# gates must be closed
solver.ensure(is_vertical[1:, :].then(is_vertical[:-1, :]
| is_black[:-1, :]))
solver.ensure(is_vertical[:-1, :].then(is_vertical[1:, :]
| is_black[1:, :]))
solver.ensure(is_horizontal[:, 1:].then(is_horizontal[:, :-1]
| is_black[:, :-1]))
solver.ensure(is_horizontal[:, :-1].then(is_horizontal[:, 1:]
| is_black[:, 1:]))
# each horizontal gate must be passed exactly once
for y in range(1, height - 1):
for x in range(width):
on_loop = []
for x2 in range(width):
cond = [is_passed[y, x2]]
if x2 < x:
cond += [is_horizontal[y, i] for i in range(x2, x)]
elif x < x2:
cond += [is_horizontal[y, i] for i in range(x + 1, x2 + 1)]
on_loop.append(fold_and(cond))
solver.ensure(is_horizontal[y, x].then(count_true(on_loop) == 1))
# each vertical gate must be passed exactly once
for y in range(height):
for x in range(1, width - 1):
on_loop = []
for y2 in range(width):
cond = [is_passed[y2, x]]
if y2 < y:
cond += [is_vertical[i, x] for i in range(y2, y)]
elif y < y2:
cond += [is_vertical[i, x] for i in range(y + 1, y2 + 1)]
on_loop.append(fold_and(cond))
solver.ensure(is_vertical[y, x].then(count_true(on_loop) == 1))
# --------- loop must be canonical ---------
# for simplicity, no stacked gates (although this is not necessary for the canonicity)
solver.ensure(~(is_horizontal[:-1, :] & is_horizontal[1:, :]))
solver.ensure(~(is_vertical[:, :-1] & is_vertical[:, 1:]))
for y in range(height):
for x in range(width):
if 0 < y < height - 1:
if x == 0 or x == width - 1:
solver.ensure(is_horizontal[y,
x].then(~is_black[y - 1, x]
& ~is_black[y + 1, x]))
else:
solver.ensure((is_horizontal[y, x] &
(is_black[y - 1, x] | is_black[y + 1, x])
).then(is_horizontal[y, x - 1]
& is_horizontal[y, x + 1]
& ~is_black[y - 1, x - 1]
& ~is_black[y + 1, x - 1]
& ~is_black[y + 1, x - 1]
& ~is_black[y + 1, x + 1]))
if 0 < x < width - 1:
if y == 0 or y == height - 1:
solver.ensure(is_vertical[y,
x].then(~is_black[y, x - 1]
& ~is_black[y, x + 1]))
else:
solver.ensure(
(is_vertical[y, x] &
(is_black[y, x - 1] | is_black[y, x + 1])
).then(is_vertical[y - 1, x] & is_vertical[y + 1, x]
& ~is_black[y - 1, x - 1]
& ~is_black[y + 1, x - 1]
& ~is_black[y + 1, x - 1]
& ~is_black[y + 1, x + 1]))
# no detour
for y in range(height - 1):
for x in range(width - 1):
solver.ensure(count_true(loop.cell_neighbors(y, x)) <= 2)
solver.ensure(
fold_and(~is_black[y:y + 2, x:x + 2],
~is_horizontal[y:y + 2, x:x + 2],
~is_vertical[y:y + 2, x:x + 2]).then(
count_true(loop.cell_neighbors(y, x)) +
1 < count_true(is_passed[y:y + 2, x:x + 2])))
# no ambiguous L-shaped turning
for y in range(height - 1):
for x in range(width - 1):
for dy in [0, 1]:
for dx in [0, 1]:
solver.ensure(~fold_and([
loop.horizontal[y + dy, x], loop.vertical[
y, x + dx], ~is_vertical[y + dy, x + 1 - dx],
~is_horizontal[y + 1 - dx, x +
dx], ~is_black[y + 1 - dy, x + 1 - dx],
count_true(is_passed[y:y + 2, x:x + 2]) == 3
]))
# no ambiguous L-shaped turning involving gates
for y in range(height - 1):
for x in range(width - 2):
solver.ensure(
fold_and(
is_vertical[y:y + 2, x + 1],
~is_black[y:y + 2, x:x + 3]).then(
count_true(loop.horizontal[y, x], loop.horizontal[
y + 1,
x], loop.vertical[y, x], loop.vertical[y, x + 2]) +
1 < count_true(is_passed[y:y + 2,
x], is_passed[y:y + 2,
x + 2])))
for y in range(height - 2):
for x in range(width - 1):
solver.ensure(
fold_and(is_horizontal[y + 1, x:x + 2],
~is_black[y:y + 3, x:x + 2]).
then(
count_true(loop.vertical[y, x], loop.vertical[y, x + 1],
loop.horizontal[y, x], loop.horizontal[y + 2,
x]) +
1 < count_true(is_passed[y, x:x + 2], is_passed[y + 2,
x:x + 2])))
# no dead ends
for y in range(height):
for x in range(width):
solver.ensure((~is_black[y, x]).then(
count_true(~is_black.four_neighbors(y, x)) >= 2))
# --------- avoid "trivial" problems ---------
solver.ensure(count_true(is_vertical) > 5)
solver.ensure(count_true(is_horizontal) > 4)
if n_max_isolated_black_cells is not None:
lonely_black_cell = []
for y in range(height):
for x in range(width):
cond = [is_black[y, x]]
if y > 0:
cond.append(~is_vertical[y - 1, x])
if y < height - 1:
cond.append(~is_vertical[y + 1, x])
if x > 0:
cond.append(~is_horizontal[y, x - 1])
if x < width - 1:
cond.append(~is_horizontal[y, x + 1])
lonely_black_cell.append(fold_and(cond))
solver.ensure(
count_true(lonely_black_cell) <= n_max_isolated_black_cells)
short_gates = []
for y in range(height):
for x in range(width):
g1 = fold_and([
is_vertical[y, x], ~is_vertical[y - 1, x] if y > 0 else True,
~is_vertical[y + 1, x] if y < height - 1 else True
])
g2 = fold_and([
is_horizontal[y,
x], ~is_horizontal[y, x - 1] if x > 0 else True,
~is_horizontal[y, x + 1] if x < width - 1 else True
])
if 0 < y < height - 1 and 0 < x < width - 1:
short_gates.append(g1)
short_gates.append(g2)
solver.ensure((g1 | g2).then(~is_black[y - 1, x - 1]
& ~is_black[y - 1, x + 1]
& ~is_black[y + 1, x - 1]
& ~is_black[y + 1, x + 1]))
else:
solver.ensure(~g1)
solver.ensure(~g2)
solver.ensure(count_true(short_gates) <= 0)
for y in range(1, height - 1):
for x in range(1, width - 1):
solver.ensure(
count_true(
is_horizontal[y - 1, x] & is_black[y - 1, x - 1]
& is_black[y - 1, x + 1],
is_horizontal[y + 1, x] & is_black[y + 1, x - 1]
& is_black[y + 1, x + 1],
is_vertical[y, x - 1] & is_black[y - 1, x - 1]
& is_black[y + 1, x - 1],
is_vertical[y, x + 1] & is_black[y - 1, x + 1]
& is_black[y + 1, x + 1],
) <= 1)
# --------- ensure randomness ---------
passed_constraints = [[0 for _ in range(width)] for _ in range(height)]
for y in range(height):
for x in range(width):
if (y > 0 and passed_constraints[y - 1][x] != 0) or (
x > 0 and passed_constraints[y][x - 1] != 0):
continue
passed_constraints[y][x] = max(0, random.randint(-20, 2))
for y in range(height):
for x in range(width):
if passed_constraints[y][x] == 1:
solver.ensure(is_passed[y, x])
elif passed_constraints[y][x] == 2:
solver.ensure(~is_passed[y, x])
# --------- extra constraints ---------
if n_min_gates is not None or n_max_gates is not None:
gate_representative = []
for y in range(height):
for x in range(width):
gate_representative.append(is_horizontal[y, x] & (
~is_horizontal[y, x - 1] if x > 0 else True))
gate_representative.append(is_vertical[y, x] & (
~is_vertical[y - 1, x] if y > 0 else True))
if n_min_gates is not None:
solver.ensure(n_min_gates <= count_true(gate_representative))
if n_max_gates is not None:
solver.ensure(count_true(gate_representative) <= n_max_gates)
if min_go_through > 0:
go_through = []
for y in range(height):
for x in range(width):
if y < height - 4 and 0 < x < width - 1:
go_through.append(
fold_and(
is_horizontal[y + 1,
x], is_horizontal[y + 1, x - 1]
| is_horizontal[y + 1, x + 1],
~is_black[y + 2, x - 1], ~is_black[y + 2, x + 1],
is_horizontal[y + 3,
x], is_horizontal[y + 3, x - 1]
| is_horizontal[y + 3, x + 1],
loop.vertical[y:y + 4, x]))
if x < width - 4 and 0 < y < height - 1:
go_through.append(
fold_and(
is_vertical[y, x + 1], is_vertical[y - 1, x + 1]
| is_vertical[y + 1, x + 1],
~is_black[y - 1, x + 2], ~is_black[y + 1, x + 2],
is_vertical[y, x + 3], is_vertical[y - 1, x + 3]
| is_vertical[y + 1, x + 3],
loop.horizontal[y, x:x + 4]))
solver.ensure(count_true(go_through) >= 2)
if no_adjacent_black_cell:
solver.ensure(~(is_black[:-1, :] & is_black[1:, :]))
solver.ensure(~(is_black[:, :-1] & is_black[:, 1:]))
solver.ensure(~(is_black[:-1, :-1] & is_black[1:, 1:]))
solver.ensure(~(is_black[:-1, 1:] & is_black[1:, :-1]))
if no_facing_length_two:
for y in range(height):
for x in range(width):
if y <= height - 3 and x <= width - 4:
solver.ensure(~fold_and(
is_black[y, x], is_black[y + 2, x], is_black[y, x + 3],
is_black[y + 2, x + 3], is_horizontal[
y, x + 1], is_horizontal[y, x + 2], is_horizontal[
y + 2, x + 1], is_horizontal[y + 2, x + 2]))
if y <= height - 4 and x <= width - 3:
solver.ensure(
~fold_and(is_black[y, x], is_black[y, x + 2], is_black[
y + 3, x], is_black[y + 3, x + 2], is_vertical[
y + 1, x], is_vertical[y + 2, x], is_vertical[
y + 1, x + 2], is_vertical[y + 2, x + 2]))
if no_space_2x2:
has_some = is_black | is_vertical | is_horizontal
solver.ensure(has_some[:-1, :-1] | has_some[1:, :-1]
| has_some[:-1, 1:] | has_some[1:, 1:])
if black_cell_in_every_3x3:
for y in range(-1, height - 2):
for x in range(-1, width - 2):
solver.ensure(
fold_or(is_black[max(0, y):min(height, y + 3),
max(0, x):min(width, x + 3)]))
is_sat = solver.find_answer()
if not is_sat:
return None
return loop, is_passed, is_black, is_horizontal, is_vertical
def solve_lits(height, width, blocks):
solver = Solver()
is_black = solver.bool_array((height, width))
solver.add_answer_key(is_black)
# black cells are connected
graph.active_vertices_connected(solver, is_black)
# no 2x2 black cells
solver.ensure(~(is_black[1:, 1:] & is_black[1:, :-1] & is_black[:-1, 1:] & is_black[:-1, :-1]))
block_id = [[-1 for _ in range(width)] for _ in range(height)]
for i, block in enumerate(blocks):
for y, x in block:
block_id[y][x] = i
num_straight = solver.int_array(len(blocks), 0, 2)
has_t = solver.bool_array(len(blocks))
for i in range(len(blocks)):
solver.ensure(count_true([is_black[p] for p in blocks[i]]) == 4)
adjacent_pairs = []
is_straight = []
is_t = []
for y, x in blocks[i]:
neighbor_same_block = []
for y2, x2 in is_black.four_neighbor_indices(y, x):
if block_id[y2][x2] == i:
neighbor_same_block.append((y2, x2))
if (y, x) < (y2, x2):
adjacent_pairs.append(is_black[y, x] & is_black[y2, x2])
solver.ensure(is_black[y, x].then(fold_or([is_black[p] for p in neighbor_same_block])))
tmp = []
if 0 < y < height - 1 and block_id[y - 1][x] == i and block_id[y + 1][x] == i:
tmp.append(fold_and(is_black[y-1:y+2, x]))
if 0 < x < width - 1 and block_id[y][x - 1] == i and block_id[y][x + 1] == i:
tmp.append(fold_and(is_black[y, x-1:x+2]))
if len(tmp) >= 1:
is_straight.append(fold_or(tmp))
if len(neighbor_same_block) >= 3:
is_t.append(count_true([is_black[p] for p in neighbor_same_block]) >= 3)
solver.ensure(count_true(adjacent_pairs) == 3)
solver.ensure(num_straight[i] == count_true(is_straight))
solver.ensure(has_t[i] == fold_or(is_t))
for y in range(height):
for x in range(width):
if y < height - 1 and block_id[y][x] != block_id[y + 1][x]:
i = block_id[y][x]
j = block_id[y + 1][x]
solver.ensure((is_black[y, x] & is_black[y + 1, x]).then(
(num_straight[i] != num_straight[j]) | (has_t[i] != has_t[j])
))
if x < width - 1 and block_id[y][x] != block_id[y][x + 1]:
i = block_id[y][x]
j = block_id[y][x + 1]
solver.ensure((is_black[y, x] & is_black[y, x + 1]).then(
(num_straight[i] != num_straight[j]) | (has_t[i] != has_t[j])
))
is_sat = solver.solve()
return is_sat, is_black
def solve_akari(height, width, problem, is_anti_knight=True):
solver = Solver()
has_light = solver.bool_array((height, width))
solver.add_answer_key(has_light)
for y in range(height):
for x in range(width):
if problem[y][x] >= -1:
continue
if y == 0 or problem[y - 1][x] >= -1:
group = []
for y2 in range(y, height):
if problem[y2][x] < -1:
group.append((y2, x))
else:
break
solver.ensure(count_true([has_light[p] for p in group]) <= 1)
if x == 0 or problem[y][x - 1] >= -1:
group = []
for x2 in range(x, width):
if problem[y][x2] < -1:
group.append((y, x2))
else:
break
solver.ensure(count_true([has_light[p] for p in group]) <= 1)
for y in range(height):
for x in range(width):
if problem[y][x] < -1:
sight = [(y, x)]
for y2 in range(y - 1, -1, -1):
if problem[y2][x] < -1:
sight.append((y2, x))
else:
break
for y2 in range(y + 1, height, 1):
if problem[y2][x] < -1:
sight.append((y2, x))
else:
break
for x2 in range(x - 1, -1, -1):
if problem[y][x2] < -1:
sight.append((y, x2))
else:
break
for x2 in range(x + 1, width, 1):
if problem[y][x2] < -1:
sight.append((y, x2))
else:
break
solver.ensure(fold_or([has_light[p] for p in sight]))
else:
solver.ensure(~has_light[y, x])
if problem[y][x] >= 0:
neighbors = []
if y > 0 and problem[y - 1][x] < -1:
neighbors.append((y - 1, x))
if y < height - 1 and problem[y + 1][x] < -1:
neighbors.append((y + 1, x))
if x > 0 and problem[y][x - 1] < -1:
neighbors.append((y, x - 1))
if x < width - 1 and problem[y][x + 1] < -1:
neighbors.append((y, x + 1))
solver.ensure(count_true([has_light[p] for p in neighbors]) == problem[y][x])
if is_anti_knight:
graph.active_vertices_anti_knight(solver, has_light)
is_sat = solver.solve()
return is_sat, has_light