def universe_repeats(moons):
# find when there is a cycle in the x position and velocity of all moons,
# and also for the y and z coordinates too
seenx, seeny, seenz = set(), set(), set()
findx, findy, findz = True, True, True
while findx or findy or findz:
x = tuple([(m[0], m[3]) for m in moons])
y = tuple([(m[1], m[4]) for m in moons])
z = tuple([(m[2], m[5]) for m in moons])
if x in seenx: findx = False
if y in seeny: findy = False
if z in seenz: findz = False
seenx.add(x); seeny.add(y); seenz.add(z)
update(moons)
# now we can say that the first cycle of all three elements is
# the lowest common multiple of the three independent cycles
return lcm(lcm(len(seenx), len(seeny)), len(seenz))
def earliest(ids):
time, interval = ids[0]
for offset, period in ids[1:]:
while True:
if (time + offset) % period == 0:
break
time += interval
interval = lcm(interval, period)
return time
def cycle_product(m1: Monomial, m2: Monomial) -> Monomial:
"""
Given two monomials (from the
cycle index of a symmetry group),
compute the resultant monomial
in the cartesian product
corresponding to their merging.
"""
assert isinstance(m1, Monomial) and isinstance(m2, Monomial)
A = m1.variables
B = m2.variables
result_variables = dict()
for i in A:
for j in B:
k = lcm(i, j)
g = (i * j) // k
if k in result_variables:
result_variables[k] += A[i] * B[j] * g
else:
result_variables[k] = A[i] * B[j] * g
return Monomial(result_variables, Fraction(m1.coeff * m2.coeff, 1))
def test_lcm(self):
self.assertEqual(24, lcm(8, 12))