def main():
"""Shikaku solver example."""
sym = grilops.make_number_range_symbol_set(0, HEIGHT * WIDTH - 1)
lattice = grilops.get_rectangle_lattice(HEIGHT, WIDTH)
sg = grilops.SymbolGrid(lattice, sym)
rc = grilops.regions.RegionConstrainer(
lattice,
solver=sg.solver,
rectangular=True,
min_region_size=min(GIVENS.values()),
max_region_size=max(GIVENS.values())
)
for p in lattice.points:
sg.solver.add(sg.cell_is(p, rc.region_id_grid[p]))
region_size = GIVENS.get(p)
if region_size:
sg.solver.add(rc.parent_grid[p] == grilops.regions.R)
sg.solver.add(rc.region_size_grid[p] == region_size)
if sg.solve():
sg.print()
print()
if sg.is_unique():
print("Unique solution")
else:
print("Alternate solution")
sg.print()
else:
print("No solution")
def shikaku(givens):
"""Solver for Shikaku minipuzzles."""
sym = grilops.make_number_range_symbol_set(0, SIZE * SIZE - 1)
sg = grilops.SymbolGrid(LATTICE, sym)
rc = grilops.regions.RegionConstrainer(LATTICE,
solver=sg.solver,
rectangular=True)
shifter = Shifter(sg.solver)
for p in LATTICE.points:
sg.solver.add(sg.cell_is(p, rc.region_id_grid[p]))
given = givens[p.y][p.x]
if given > 0:
given = shifter.given(p, given)
sg.solver.add(rc.parent_grid[p] == grilops.regions.R)
sg.solver.add(rc.region_size_grid[p] == given)
else:
sg.solver.add(rc.parent_grid[p] != grilops.regions.R)
assert sg.solve()
sg.print()
print()
shifter.print_shifts()
print()
return shifter.eval_binary()
def main():
"""Spiral Galaxies solver example."""
# The grid symbols will be the region IDs from the region constrainer.
sym = grilops.make_number_range_symbol_set(0, HEIGHT * WIDTH - 1)
lattice = grilops.get_rectangle_lattice(HEIGHT, WIDTH)
sg = grilops.SymbolGrid(lattice, sym)
rc = grilops.regions.RegionConstrainer(lattice, sg.solver)
for p in sg.grid:
sg.solver.add(sg.cell_is(p, rc.region_id_grid[p]))
# Make the upper-left-most cell covered by a circle the root of its region.
roots = {(int(math.floor(y)), int(math.floor(x))) for (y, x) in GIVENS}
r = grilops.regions.R
for y in range(HEIGHT):
for x in range(WIDTH):
sg.solver.add((rc.parent_grid[Point(y, x)] == r) == ((y,
x) in roots))
# Ensure that each cell has a "partner" within the same region that is
# rotationally symmetric with respect to that region's circle.
for p in lattice.points:
or_terms = []
for (gy, gx) in GIVENS:
region_id = lattice.point_to_index(
Point(int(math.floor(gy)), int(math.floor(gx))))
partner = Point(int(2 * gy - p.y), int(2 * gx - p.x))
if lattice.point_to_index(partner) is None:
continue
or_terms.append(
And(
rc.region_id_grid[p] == region_id,
rc.region_id_grid[partner] == region_id,
))
sg.solver.add(Or(*or_terms))
def show_cell(unused_p, region_id):
rp = lattice.points[region_id]
for i, (gy, gx) in enumerate(GIVENS):
if int(math.floor(gy)) == rp.y and int(math.floor(gx)) == rp.x:
return chr(65 + i)
raise Exception("unexpected region id")
if sg.solve():
sg.print(show_cell)
print()
if sg.is_unique():
print("Unique solution")
else:
print("Alternate solution")
sg.print(show_cell)
else:
print("No solution")
def main():
"""Fillomino solver example."""
givens = [
[0, 0, 0, 3, 0, 0, 0, 0, 5],
[0, 0, 8, 3, 10, 0, 0, 5, 0],
[0, 3, 0, 0, 0, 4, 4, 0, 0],
[1, 3, 0, 3, 0, 0, 2, 0, 0],
[0, 2, 0, 0, 3, 0, 0, 2, 0],
[0, 0, 2, 0, 0, 3, 0, 1, 3],
[0, 0, 4, 4, 0, 0, 0, 3, 0],
[0, 4, 0, 0, 4, 3, 3, 0, 0],
[6, 0, 0, 0, 0, 1, 0, 0, 0],
]
sym = grilops.make_number_range_symbol_set(1, 10)
lattice = grilops.get_rectangle_lattice(len(givens), len(givens[0]))
sg = grilops.SymbolGrid(lattice, sym)
rc = grilops.regions.RegionConstrainer(lattice, solver=sg.solver)
for p in lattice.points:
# The filled number must match the size of the region.
sg.solver.add(sg.grid[p] == rc.region_size_grid[p])
# The size of the region must match the clue.
given = givens[p.y][p.x]
if given != 0:
sg.solver.add(rc.region_size_grid[p] == given)
# Different regions of the same size may not be orthogonally adjacent.
region_sizes = [
n.symbol
for n in lattice.edge_sharing_neighbors(rc.region_size_grid, p)
]
region_ids = [
n.symbol
for n in lattice.edge_sharing_neighbors(rc.region_id_grid, p)
]
for region_size, region_id in zip(region_sizes, region_ids):
sg.solver.add(
Implies(rc.region_size_grid[p] == region_size,
rc.region_id_grid[p] == region_id))
if sg.solve():
sg.print()
print()
if sg.is_unique():
print("Unique solution")
else:
print("Alternate solution")
sg.print()
else:
print("No solution")
def main():
"""Hex kakuro solver example."""
points = []
points.extend([Point(1, i) for i in range(-3, 4, 2)])
points.extend([Point(2, i) for i in range(-4, 5, 2)])
points.extend([Point(3, -5), Point(3, -3), Point(3, 3), Point(3, 5)])
points.extend([
Point(4, -6),
Point(4, -4),
Point(4, -2),
Point(4, 2),
Point(4, 4),
Point(4, 6)
])
points.extend([Point(5, -5), Point(5, -3), Point(5, 3), Point(5, 5)])
points.extend([Point(6, i) for i in range(-4, 5, 2)])
points.extend([Point(7, i) for i in range(-3, 4, 2)])
lattice = PointyToppedHexagonalLattice(points)
sym = grilops.make_number_range_symbol_set(1, 9)
sg = grilops.SymbolGrid(lattice, sym)
dirs_by_name = dict(sg.lattice.edge_sharing_directions())
for entry in SUMS:
(y, x), dirname, given = entry
d = dirs_by_name[dirname]
s = grilops.sightlines.count_cells(sg,
Point(y, x),
d,
count=lambda c: c)
sg.solver.add(given == s)
for p in lattice.points:
for d in [Vector(0, 2), Vector(1, 1), Vector(1, -1)]:
if p.translate(d.negate()) not in sg.grid:
q = p
ps = []
while q in sg.grid:
ps.append(sg.grid[q])
q = q.translate(d)
sg.solver.add(Distinct(*ps))
if sg.solve():
sg.print()
print()
if sg.is_unique():
print("Unique solution")
else:
print("Alternate solution")
sg.print()
else:
print("No solution")
def killer_sudoku(cages: List[str], cage_sum_grid: List[List[int]]) -> str:
"""Solver for Killer Sudoku minipuzzles."""
sym = grilops.make_number_range_symbol_set(1, SIZE)
sg = grilops.SymbolGrid(LATTICE, sym)
shifter = Shifter(sg.solver)
# Add normal sudoku constraints.
for y in range(SIZE):
sg.solver.add(Distinct(*[sg.grid[Point(y, x)] for x in range(SIZE)]))
for x in range(SIZE):
sg.solver.add(Distinct(*[sg.grid[Point(y, x)] for y in range(SIZE)]))
for z in range(9):
top = (z // 3) * 3
left = (z % 3) * 3
cells = [
sg.grid[Point(y, x)] for y in range(top, top + 3)
for x in range(left, left + 3)
]
sg.solver.add(Distinct(*cells))
# Build a map from each cage label to the cells within that cage.
cage_cells = defaultdict(list)
for p in LATTICE.points:
cage_cells[cages[p.y][p.x]].append(sg.grid[p])
# The digits used in each cage must be unique.
for cells_in_cage in cage_cells.values():
sg.solver.add(Distinct(*cells_in_cage))
cage_sums = {}
for p in LATTICE.points:
cage_sum = cage_sum_grid[p.y][p.x]
if cage_sum > 0:
shifted_cage_sum = shifter.given(p, cage_sum)
cage_label = cages[p.y][p.x]
assert cage_label not in cage_sums
cage_sums[cage_label] = shifted_cage_sum
# Add constraints for cages with given sums.
for cage_label, shifted_cage_sum in cage_sums.items():
sg.solver.add(Sum(*cage_cells[cage_label]) == shifted_cage_sum)
assert sg.solve()
sg.print()
print()
shifter.print_shifts()
print()
return shifter.eval_binary()
def skyscraper(givens: Dict[Direction, List[int]]) -> str:
"""Solver for Skyscraper minipuzzles."""
sym = grilops.make_number_range_symbol_set(1, SIZE)
sg = grilops.SymbolGrid(LATTICE, sym)
shifter = Shifter(sg.solver)
# Each row and each column contains each building height exactly once.
for y in range(SIZE):
sg.solver.add(Distinct(*[sg.grid[Point(y, x)] for x in range(SIZE)]))
for x in range(SIZE):
sg.solver.add(Distinct(*[sg.grid[Point(y, x)] for y in range(SIZE)]))
# We'll use the sightlines accumulator to keep track of a tuple storing:
# the tallest building we've seen so far
# the number of visible buildings we've encountered
Acc = Datatype("Acc") # pylint: disable=C0103
Acc.declare("acc", ("tallest", IntSort()), ("num_visible", IntSort()))
Acc = Acc.create() # pylint: disable=C0103
def accumulate(a, height):
return Acc.acc(
If(height > Acc.tallest(a), height, Acc.tallest(a)),
If(height > Acc.tallest(a),
Acc.num_visible(a) + 1, Acc.num_visible(a)))
for d, gs in givens.items():
for i, g in enumerate(gs):
if d.vector.dy != 0:
g = g - shifter.col_shifts.get(i, 0)
p = Point(0 if d.vector.dy < 0 else SIZE - 1, i)
elif d.vector.dx != 0:
g = g - shifter.row_shifts.get(i, 0)
p = Point(i, 0 if d.vector.dx < 0 else SIZE - 1)
sg.solver.add(g == Acc.num_visible( # type: ignore[attr-defined]
grilops.sightlines.reduce_cells(
sg,
p,
LATTICE.opposite_direction(d),
Acc.acc(0, 0), # type: ignore[attr-defined]
accumulate)))
assert sg.solve()
sg.print()
print()
shifter.print_shifts()
print()
return shifter.eval_binary()
def main():
"""Sudoku solver example."""
givens = [
[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],
]
sym = grilops.make_number_range_symbol_set(1, 9)
sg = grilops.SymbolGrid(grilops.get_square_lattice(9), sym)
for y, given_row in enumerate(givens):
for x, given in enumerate(given_row):
if given != 0:
sg.solver.add(sg.cell_is(Point(y, x), sym[given]))
for y in range(9):
sg.solver.add(Distinct(*[sg.grid[Point(y, x)] for x in range(9)]))
for x in range(9):
sg.solver.add(Distinct(*[sg.grid[Point(y, x)] for y in range(9)]))
for z in range(9):
top = (z // 3) * 3
left = (z % 3) * 3
cells = [sg.grid[Point(y, x)] for y in range(top, top + 3) for x in range(left, left + 3)]
sg.solver.add(Distinct(*cells))
if sg.solve():
sg.print()
print()
if sg.is_unique():
print("Unique solution")
else:
print("Alternate solution")
sg.print()
else:
print("No solution")
def main():
"""Araf solver example."""
min_given_value = min(GIVENS.values())
max_given_value = max(GIVENS.values())
# The grid symbols will be the region IDs from the region constrainer.
sym = grilops.make_number_range_symbol_set(0, HEIGHT * WIDTH - 1)
lattice = grilops.get_rectangle_lattice(HEIGHT, WIDTH)
sg = grilops.SymbolGrid(lattice, sym)
rc = grilops.regions.RegionConstrainer(
lattice, sg.solver,
min_region_size=min_given_value + 1,
max_region_size=max_given_value - 1
)
for point in lattice.points:
sg.solver.add(sg.cell_is(point, rc.region_id_grid[point]))
# Exactly half of the givens must be region roots. As an optimization, add
# constraints that the smallest givens must not be region roots, and that the
# largest givens must be region roots.
undetermined_given_locations = []
num_undetermined_roots = len(GIVENS) // 2
for p, v in GIVENS.items():
if v == min_given_value:
sg.solver.add(Not(rc.parent_grid[p] == grilops.regions.R))
elif v == max_given_value:
sg.solver.add(rc.parent_grid[p] == grilops.regions.R)
num_undetermined_roots -= 1
else:
undetermined_given_locations.append(p)
sg.solver.add(
PbEq(
[
(rc.parent_grid[p] == grilops.regions.R, 1)
for p in undetermined_given_locations
],
num_undetermined_roots
)
)
# Non-givens must not be region roots
for point in lattice.points:
if point not in GIVENS:
sg.solver.add(Not(rc.parent_grid[point] == grilops.regions.R))
add_given_pair_constraints(sg, rc)
region_id_to_label = {
lattice.point_to_index(point): chr(65 + i)
for i, point in enumerate(GIVENS.keys())
}
def show_cell(unused_loc, region_id):
return region_id_to_label[region_id]
if sg.solve():
sg.print(show_cell)
print()
if sg.is_unique():
print("Unique solution")
else:
print("Alternate solution")
sg.print(show_cell)
else:
print("No solution")
"""Skyscraper solver example.
Example puzzle can be found at https://www.puzzlemix.com/Skyscraper.
"""
from z3 import Datatype, Distinct, If, IntSort
import grilops
import grilops.sightlines
from grilops.geometry import Point, Vector
SIZE = 5
SYM = grilops.make_number_range_symbol_set(1, SIZE)
GIVEN_TOP = [4, 2, 1, 2, 3]
GIVEN_LEFT = [3, 2, 3, 2, 1]
GIVEN_RIGHT = [3, 4, 1, 2, 2]
GIVEN_BOTTOM = [1, 4, 3, 2, 2]
def main():
"""Skyscraper solver example."""
lattice = grilops.get_square_lattice(SIZE)
sg = grilops.SymbolGrid(lattice, SYM)
# Each row and each column contains each building height exactly once.
for y in range(SIZE):
sg.solver.add(Distinct(*[sg.grid[Point(y, x)] for x in range(SIZE)]))
for x in range(SIZE):
sg.solver.add(Distinct(*[sg.grid[Point(y, x)] for y in range(SIZE)]))
# We'll use the sightlines accumulator to keep track of a tuple storing:
def main():
"""Greater Than Killer Sudoku solver example."""
cages = [
"AABCCDDEE",
"FFBGGHIJK",
"FLMNHHIJK",
"LLMNNOIPP",
"QQRNSOTTU",
"VWRSSXTUU",
"VWYYSXZaa",
"bccddXZee",
"bbbdffZgg",
]
cage_sums = {
"B": 6,
"D": 16,
"F": 14,
"H": 17,
"I": 9,
"J": 12,
"K": 9,
"L": 20,
"M": 13,
"N": 29,
"O": 4,
"R": 8,
"S": 12,
"V": 8,
"W": 14,
"Y": 17,
"b": 11,
"d": 11,
"e": 8,
}
sym = grilops.make_number_range_symbol_set(1, 9)
lattice = grilops.get_square_lattice(9)
sg = grilops.SymbolGrid(lattice, sym)
add_sudoku_constraints(sg)
# Build a map from each cage label to the cells within that cage.
cage_cells = defaultdict(list)
for p in lattice.points:
cage_cells[cages[p.y][p.x]].append(sg.grid[p])
# The digits used in each cage must be unique.
for cells_in_cage in cage_cells.values():
sg.solver.add(Distinct(*cells_in_cage))
# Add constraints for cages with given sums.
for cage_label, cage_sum in cage_sums.items():
sg.solver.add(Sum(*cage_cells[cage_label]) == cage_sum)
# Add constraints between cage sums.
def cage_sum_greater(a, b):
sg.solver.add(Sum(*cage_cells[a]) > Sum(*cage_cells[b]))
def cage_sum_equal(a, b):
sg.solver.add(Sum(*cage_cells[a]) == Sum(*cage_cells[b]))
cage_sum_equal("C", "G")
cage_sum_greater("J", "E")
cage_sum_greater("E", "K")
cage_sum_greater("W", "c")
cage_sum_greater("c", "b")
cage_sum_greater("f", "d")
cage_sum_greater("X", "f")
if sg.solve():
sg.print()
print()
if sg.is_unique():
print("Unique solution")
else:
print("Alternate solution")
sg.print()
else:
print("No solution")
def main():
"""Outflight Entertainment sudoku solver example."""
cages = [
"AABCCD",
"EFBCGD",
"EFFHGI",
"EJJHGI",
"EKJHGL",
"KKJLLL",
]
peaks = [
(0, 1),
(0, 2),
(1, 3),
(1, 4),
(1, 5),
(2, 1),
(2, 3),
(3, 0),
(3, 5),
(4, 2),
(5, 1),
(5, 4),
]
extract = {
"A": (5, 2),
"B": (0, 3),
"C": (3, 2),
"D": (1, 4),
"E": (0, 1),
"F": (1, 5),
"G": (4, 2),
"H": (3, 5),
"I": (1, 0),
"J": (5, 5),
"K": (3, 4),
"L": (5, 1),
"M": (0, 5),
"N": (4, 4),
}
def answer(sg):
solved_grid = sg.solved_grid()
s = ""
s += chr(64 + solved_grid[extract["A"]] + solved_grid[extract["B"]])
s += chr(64 + solved_grid[extract["C"]] + solved_grid[extract["D"]])
s += chr(64 + solved_grid[extract["E"]] + solved_grid[extract["F"]] +
solved_grid[extract["G"]])
s += chr(64 + solved_grid[extract["H"]] + solved_grid[extract["I"]])
s += chr(64 + solved_grid[extract["J"]] + solved_grid[extract["K"]] +
solved_grid[extract["L"]])
s += chr(64 + solved_grid[extract["M"]] + solved_grid[extract["N"]])
return s
sym = grilops.make_number_range_symbol_set(1, 6)
lattice = grilops.get_square_lattice(6)
sg = grilops.SymbolGrid(lattice, sym)
add_sudoku_constraints(sg)
# Constrain regions to match the cages and be rooted at the peaks.
cage_label_to_region_id = {}
for py, px in peaks:
cage_label_to_region_id[cages[py][px]] = lattice.point_to_index(
(py, px))
rc = grilops.regions.RegionConstrainer(lattice, sg.solver)
for y, x in lattice.points:
sg.solver.add(
rc.region_id_grid[(y, x)] == cage_label_to_region_id[cages[y][x]])
# Within each region, a parent cell must have a greater value than a child
# cell, so that the values increase as you approach the root cell (the peak).
for p in lattice.points:
for n in sg.edge_sharing_neighbors(p):
sg.solver.add(
Implies(
rc.edge_sharing_direction_to_index(
n.direction) == rc.parent_grid[p],
n.symbol > sg.grid[p]))
if sg.solve():
sg.print()
print()
print(answer(sg))
while not sg.is_unique():
print()
print("Alternate solution")
sg.print()
print()
print(answer(sg))
else:
print("No solution")