def find_triangles(radius):
radius2 = radius ** 2
# generate points in circle, except origin
r = range(-radius + 1, radius)
points = set()
for x in r:
for y in r:
if mag2(x, y) < radius2:
points.add((x, y))
points.remove((0,0))
#print(points)
triangles = set()
for p1, p2 in all_pairs(points):
# test if p1-p2 crosses the origin
if cmp_line(p1, p2, (0,0)) == 0:
continue
# get the set of points p3 that is on the side of the line p1-(0,0)
# farthest from p2, and likewise for p2-(0,0). each of these points
# completes a triangle (p1, p2, p3) that contains the origin.
#TODO prevent duplication of triangles, reducing reliance on sets
p3_set = (partition(points, p1, (0,0), p2)[1] &
partition(points, p2, (0,0), p1)[1])
for p3 in p3_set:
triangles.add(frozenset((p1, p2, p3)))
return len(triangles)
def find_arithmetic_sequences(numbers, length):
"""
Find arithmetic sequences in the given numbers of the specified length or longer.
>>> list(find_arithmetic_sequences([1, 2, 3], 3))
[(1, 2, 3)]
>>> list(find_arithmetic_sequences([1, 3, 2], 3))
[(1, 2, 3)]
>>> list(find_arithmetic_sequences([1, 4, 5, 9, 10], 3))
[(1, 5, 9)]
"""
numbers = sorted(numbers)
numbers_set = set(numbers)
for a, b in all_pairs(numbers):
assert a < b
sequence = [a, b]
diff = b - a
c = b + diff
while c in numbers_set:
sequence.append(c)
c = c + diff
if len(sequence) >= length:
yield tuple(sequence)
break
def all_cuts(points, size):
cuts = set()
for a, b in all_pairs(points):
cuts.add(tuple(sorted((a, b))))
return set(c for c in cuts if is_legal_cut(*c, size=size))
