def second_part(instructions):
earliest_timestamp, bus_ids = instructions
all_bus_cycles = {
int(bus): -i % int(bus)
for i, bus in enumerate(bus_ids) if bus != "x"
}
# un bus a une vitesse (son numéro) et une vitesse de cycle son numéro -i (le temps de départ)... mais comme on veux un cycle c'est un modulo : -i%int(bus)
sorted_bus_by_relative_speed = list(reversed(sorted(all_bus_cycles)))
print(sorted_bus_by_relative_speed)
bus_synchro_cycle = all_bus_cycles[sorted_bus_by_relative_speed[
0]] # optim on choisi le bus au cycle le plus rapide
lcm = math.lcm(sorted_bus_by_relative_speed[0])
for bus_index, bus_cycle in enumerate(sorted_bus_by_relative_speed[1:]):
while (
bus_synchro_cycle % bus_cycle != all_bus_cycles[bus_cycle]
): # on s'attend a ce que le temps de sychro / cycle du bus/superbus courrant soit égale au temps de syncro du bus vis à vis de l'ensemble des bus
bus_synchro_cycle += lcm # on bouge a la vitesse la plus "rapide" de l'ensemble des bus (superbus) déjà traité , hormis celui à sychroniser
# at the end of this loop n bus are in sync at bus_synchro_cycle according to their cycle
# for loop 1 :
# bus_synchro_cycle % bus_1_relative_speed == bus_1_cycle
# bus_synchro_cycle % bus_2_relavive_speed == bus_2_cycle
# so each bus is at the start point at bus_synchro_cycle
# let's move faster using the lcm! PPCM en francais didious!
lcm = math.lcm(
lcm, bus_cycle
) # on recalcul la vitesse la plus "rapide" de l'ensemble des bus avec le bus courrant pour le suivant
print(bus_synchro_cycle)
def find_smallest_sequence_departures(departures: dict):
previous = None
offset = 0
lcm_previous = None
lcm_current = None
departure_offset = None
departure_offset_previous = 0
for current in departures:
if not previous:
previous = current
elif current == 'x':
pass
elif not lcm_current:
lcm_current = math.lcm(previous, current)
departure_offset = arrow_alignment(previous, current, -offset)
previous = current
else:
lcm_previous = lcm_current
departure_offset_previous += departure_offset
lcm_current = math.lcm(lcm_current, current)
departure_offset = arrow_alignment(
lcm_previous, current, -departure_offset_previous - offset)
previous = current
offset += 1
print(
f"The definite answer is here: {departure_offset + departure_offset_previous}\n\n\n"
)
def lcm(*integers):
'''Return least common multiple of a and b'''
try:
_math.lcm(*integers)
except Exception:
pass
return _reduce(_lcm1, integers)
def part2() -> int:
# It assumes that the first step will be the one to repeat.
# It's a heuristic, but works for this problem.
x_cords = [(moon.position[0], 0) for moon in moons_list]
y_cords = [(moon.position[1], 0) for moon in moons_list]
z_cords = [(moon.position[2], 0) for moon in moons_list]
return lcm(get_repeat_steps(x_cords),
lcm(get_repeat_steps(y_cords), get_repeat_steps(z_cords)))
def part2(self):
lcms = []
for axis in reversed(range(3)):
moons = self.get_moons2(axis)
while not all([moon.stabilised for moon in moons]):
for moon, other_moon in itertools.permutations(moons, 2):
moon.add_velocity(other_moon)
for moon in moons:
moon.apply_velocity()
lcms.append(lcm(*[moon.steps for moon in moons]))
return lcm(*lcms)
def get_change(m): #m==money
#write your code here
LCM = lcm(1,3,4) #LCM of denominations available
''' Safe whole number and it's least coins is with biggest denomination'''
safeCoins = (m//LCM)*3
print(f'safeCoins {safeCoins} of [Rs.4]')
remaining = m%LCM #remaining < LCM
print('FillingCoins')
coins = [0,0]
for i in [0,1]:
r = remaining
if i==0:
add4 = r//4
rem = r%4
add3 = rem//3 if rem%3==0 else 0
add1 = rem%3
coins[i] = add4+add3+add1
print(f'<1> {add4}[Rs.4] + {add3}[Rs.3] + {add1}[Rs.1] = {coins[i]} Coins')
else:
add3 = r//3
add1 = r%3
add4 = 0
while add1>0 and add3>0:
for j in range(1,add1+1):
add4 = add4 + 1
add3 = add3 - 1
add1 = add1 - 1
coins[i] = add4 + add3 + add1
print(f'<2> {add4}[Rs.4] + {add3}[Rs.3] + {add1}[Rs.1] = {coins[i]} Coins')
print('Best Choice is Minimum of',coins)
return safeCoins + min(coins)
def get_period(self):
"""
Return a number P such that G(x*exp(I*P)) == G(x).
>>> meijerg([1], [], [], [], z).get_period()
2*pi
>>> meijerg([pi], [], [], [], z).get_period()
oo
>>> meijerg([1, 2], [], [], [], z).get_period()
oo
>>> meijerg([1, 1], [2], [1, Rational(1, 2), Rational(1, 3)], [1], z).get_period()
12*pi
"""
# This follows from slater's theorem.
def compute(l):
# first check that no two differ by an integer
for i, b in enumerate(l):
if not b.is_Rational:
return oo
for j in range(i + 1, len(l)):
if not Mod((b - l[j]).simplify(), 1):
return oo
return functools.reduce(math.lcm, (x.denominator for x in l), 1)
beta = compute(self.bm)
alpha = compute(self.an)
p, q = len(self.ap), len(self.bq)
if p == q:
if oo in (beta, alpha):
return oo
return 2*pi*math.lcm(alpha, beta)
elif p < q:
return 2*pi*beta
else:
return 2*pi*alpha
def func():
n, m = map(int, input().split())
a = list(map(int, input().split()))
# a -> a'
for av in a:
if av % 2 == 1:
print(0)
return
av /= 2
# a -> a''
t = f(a[0])
for av in a:
if f(av) != t:
print(0)
return
av >>= t # 2でT回割る av /= 2^t
m >>= t
l = 1
for av in a:
l = math.lcm(l, av)
if l > m:
print(0)
return
m /= l
ans = (m + 1) / 2
print(ans)
return
def part1(moons, velos):
og_moons = [[x for x in row] for row in moons]
og_velos = [[x for x in row] for row in velos]
dones = [0,0,0]
print(moons, og_moons)
iterations = 3000
ret = 1
while 0 in dones:
for index, m in enumerate(moons):
for m2 in moons:
for i in range(3):
if m[i] < m2[i]:
velos[index][i] += 1
elif m[i] > m2[i]:
velos[index][i] -= 1
for i, m in enumerate(moons):
for j in range(3):
m[j] += velos[i][j]
for axis in range(3):
if dones[axis] != 0: continue
if compare_og(axis, moons, og_moons) and sum([velos[k][axis] for k in range(3)]) == 0:
print(dones)
dones[axis] = ret
if ret % 100000 == 0:
print("GOLYY")
ret += 1
print("Res: ",dones)
print(math.lcm(dones))
return "this is part one"
def find_earliest_timestamp(bus_ids):
# List buses and their time delta from the first bus
buses = [(int(bus), delta) for delta, bus in enumerate(bus_ids)
if bus != 'x']
i = 1
max_matches = 1
bus, delta = max(buses, key=lambda x: x[
0]) # Start with the largest bus value - can't be any smaller
t = bus - delta # Set initial time to first time the very first bus arrives
increment_size = bus # Start increments at the largest bus value
while True:
t += increment_size
matches = set(
) # Keep track of how many valid matches we have this run
for bus, delta in buses:
if int(t + delta) % bus != 0:
# Not a valid match, jump ahead and try the next time
i += increment_size
break
else:
matches.add(bus)
if len(matches) > max_matches:
max_matches = len(matches)
# We have found the next closest matching interval, set increment size to this interval
increment_size = lcm(*matches)
else:
# Matched all buses
return t
def part02(lines, verbose=False):
bus_ids = lines[1].split(',')
offsets = [(int(bus_ids[0]), 0)]
for i, x in enumerate(bus_ids[1:]):
if x == 'x':
continue
else:
offsets.append((int(x), (i + 1) % int(x)))
offsets = [offsets[0]] + sorted(offsets[1:], key=lambda x: -x[0])
step_width = offsets[0][0]
start_pos = 0
for i in range(1, len(offsets)):
if verbose:
print(f'outer iteration {i}: step_width', step_width,
', start pos', start_pos)
pos = start_pos
bus, offset = offsets[i]
while True:
if get_next_arrival(pos, bus) == offset + pos:
step_width = lcm(step_width, offsets[i][0])
start_pos = pos
break
else:
pos += step_width
print('part 02:', pos)
def get_time_and_multiplier(acc, new_bus_info):
multiplier, time = acc
bus, offset = new_bus_info
while (time + offset) % bus:
time += multiplier
multiplier = math.lcm(multiplier, bus)
return multiplier, time
def combine(a, b):
for n in count():
t = (a.mult * n) - a.offset
if (t - b.offset) % b.mult == 0:
mult = lcm(a.mult, b.mult)
off = mult - t
return route(mult, off)
def find_common_denom_start(num, num_new, offset):
x = 1
while True:
if 0 == ((x * num) + offset) % num_new:
first_occ = (x * num) + offset
common_denom = math.lcm(num, num_new)
return common_denom, first_occ
x += 1
def execute(self, table, tree):
super().execute(table, tree)
exp1 = self.exp1.execute(table, tree)
exp2 = self.exp2.execute(table, tree)
try:
return math.lcm(exp1, exp2)
except :
raise(Error(self.line, self.column, ErrorType.SEMANTIC, 'TypeError: Both arguments must be a integral number'))
def part2():
moons = list(values)
prev_x_states = set()
prev_y_states = set()
prev_z_states = set()
x_period = None
y_period = None
z_period = None
while None in [x_period, y_period, z_period]:
def check_axis(period, index, prev_states):
if period is not None:
return period
current_state = ''.join(
[f'{moon[0][index]}{moon[1][index]}' for moon in moons])
if current_state in prev_states:
return len(prev_states)
else:
prev_states.add(current_state)
x_period = check_axis(x_period, 0, prev_x_states)
y_period = check_axis(y_period, 1, prev_y_states)
z_period = check_axis(z_period, 2, prev_z_states)
new_moons = []
for moon in moons:
x, y, z = moon[0]
xv, yv, zv = moon[1]
inner_moons = list(moons)
inner_moons.remove(moon)
for inner_moon in inner_moons:
if x < inner_moon[0][0]:
xv += 1
elif x > inner_moon[0][0]:
xv -= 1
if y < inner_moon[0][1]:
yv += 1
elif y > inner_moon[0][1]:
yv -= 1
if z < inner_moon[0][2]:
zv += 1
elif z > inner_moon[0][2]:
zv -= 1
x += xv
y += yv
z += zv
new_moons.append(((x, y, z), (xv, yv, zv)))
moons = new_moons
return lcm(x_period, y_period, z_period)
def get_lcm_prob_denominators(probs: Iterable[float]):
"""
Helper function to compute the LCM of the denominators of the probabilities
"""
prob_denominators = [
Fraction(prob).limit_denominator().denominator for prob in probs
]
lcm_prob_denominators = lcm(*prob_denominators)
return lcm_prob_denominators
def __sub__(self, other):
the_lcm = math.lcm(self.denominator, other.denominator)
self_denominator_lcm = the_lcm // self.denominator
other_denominator_lcm = the_lcm // other.denominator
return Fraction(
((self.num * self_denominator_lcm) -
(other.num * other_denominator_lcm)),
(the_lcm + the_lcm),
)
def __add__(self, other):
the_lcm = math.lcm(self.denominator, other.denominator)
self_denominator_lcm = the_lcm // self.denominator
other_denominator_lcm = the_lcm // other.denominator
return Fraction(
numerator=((self.num * self_denominator_lcm) +
(other.num * other_denominator_lcm)),
denominator=(the_lcm + the_lcm),
)
def main():
answer = 1
for i in range(1, 21):
# math.lcm is available in python3.9+
answer = math.lcm(answer, i)
# you can use this in python3.x
# answer *= i // math.gcd(i, answer)
return answer
def alignmenter(inp):
previous = None
for xdx, x in enumerate(inp[1:]):
if x is None:
continue
period = lcm(inp[0], x)
offset = arrow_alignment(inp[0], x, -(xdx + 1))
if previous:
inner_offset = arrow_alignment(previous[1], period,
offset - previous[0])
previous = (previous[0] + inner_offset, lcm(previous[1], period))
else:
previous = (offset, period)
return previous[0]
def timestamp_matching_list(bus_ids, bus_positions):
current_base = 1
current_num = 0
for i in range(len(bus_ids)):
while current_num % bus_ids[i] != (bus_ids[i] -
bus_positions[i]) % bus_ids[i]:
current_num += current_base
current_base = math.lcm(current_base, bus_ids[i])
return current_num
def crt(r1, q1, r2, q2):
q3 = math.lcm(q1, q2)
t, _, g = egcd(q1 + q2, q3)
r3 = (r1 * q2 + r2 * q1) * t
assert not r3 % g
r3 = r3 // g % q3
if (r3 < 0) != (q3 < 0):
r3 += q3
return r3, q3
def solve(a, b):
a, b = (b, a) if b[1] > a[1] else (a, b)
rep_a, mod_a = a
rep_b, mod_b = b
k = (rep_a - rep_b) // gcd(mod_a, mod_b)
x, y = euclide(mod_a, mod_b)
solution = rep_a - k * mod_a * x
mod_res = lcm(mod_a, mod_b)
return (solution % mod_res, mod_res)
def main():
text = (Path(__file__).parent /
"../../input/2019/input_12.txt").read_text()
moons = parse_moons(text)
# Part 1
for _ in range(1000):
step(moons, range(3))
print("Part 1:", sum(m.energy for m in moons))
tx, ty, tz = [find_period(parse_moons(text), dim) for dim in range(3)]
print("Part 2:", math.lcm(tx, ty, tz))
def nthMagicalNumber(self, n: int, a: int, b: int) -> int:
high = n * max(a, b)
low = min(a, b)
lcmab = lcm(a, b)
while (high >= low):
mid = (high + low) // 2
if mid // a + mid // b - mid // (lcmab) < n:
low = mid + 1
else:
high = mid - 1
return low % (1000000007)
def calc_student_sections(courses: list[Course],
course_total_sections: dict[Course, int],
courses_info: pd.DataFrame,
course_options: pd.DataFrame) -> dict[Course, list[Student]]:
year_sections = defaultdict(lambda: 1)
option_sections = defaultdict(int)
for course_code, total_sections in course_total_sections.items():
try:
student_year = courses_info.loc[course_code]['Year']
year_sections[student_year] = lcm(year_sections[student_year], total_sections)
except KeyError:
option = course_options.loc[course_code]['Option']
option_sections[option] += total_sections
for option, total_sections in option_sections.items():
student_year = courses_info.loc[option]['Year']
year_sections[student_year] = lcm(year_sections[student_year], total_sections)
section_cnt = defaultdict(lambda: 1)
res: dict[Course, list[Student]] = {}
for course_code, section in courses:
course = Course(course_code, section)
total_sections = course_total_sections[course_code]
try:
student_year = courses_info.loc[course_code]['Year']
no_of_student_sections = year_sections[student_year] // total_sections
res[course] = []
for i in range(no_of_student_sections):
res[course].append(Student(student_year, section_cnt[course_code]))
section_cnt[course_code] += 1
except KeyError:
option = course_options.loc[course_code]['Option']
student_year = courses_info.loc[option]['Year']
no_of_student_sections = year_sections[student_year] // option_sections[option]
res[course] = []
for i in range(no_of_student_sections):
res[course].append(Student(student_year, section_cnt[option]))
section_cnt[option] += 1
return res
def findTimestamp(fileName):
ids = readIDs(fileName)
buses = list(zip(*ids))[0]
candidate = ids[0][0] - ids[0][1]
increment = ids[0][0]
for bus, offset in ids[1:]:
while -candidate % bus != offset % bus:
oldOffset = -candidate % bus
candidate += increment
newOffset = -candidate % bus
increment = math.lcm(increment, bus)
return candidate
def find_first(bus_id_list):
_, step = bus_id_list.pop(0)
ts = 0
for offset, bus_id in bus_id_list:
# now using lcm from https://github.com/alasdairnicol/advent-of-code-2020/blob/master/day13b.py
# ts, step = find_step(ts, step, offset, bus_id)
while (ts + offset) % bus_id:
# print(f'{ts=} {offset=} {bus_id=} {step=}')
ts += step
step = lcm(step, bus_id)
return ts
def key_composer(*keywords: str) -> str:
keys = [convert_text_to_indices(keyword)
for keyword in keywords] # get each key as a list of indices
key_lengths = [len(key) for key in keys] # get the length of each key
new_key_length = lcm(*key_lengths) # calculate the length of the new key
long_keys = [
extend_keyword(keyword, new_key_length) for keyword in keywords
] # make keys the same length
new_key = long_keys[
0] # starting with the first key, encode the keys with each other
for i in range(len(long_keys) - 1):
new_key = vigenre_encoder(new_key, long_keys[i + 1])
return new_key.lower()