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# coding=utf-8
from fractions import Fraction
def solution(pegs):
# This function is based on the formulat I figured out on paper
# using what I could find online about gears, I used the
# (number of gears entraining one other) / (number of gears entraned by one other)
# Using the distances, I can express the sizes of all gears from the first one
# and then, using the formula, I can find x
# The developed formula shows that the signs + / - are present half of the time
# However, the x present in the last part changes if the number is event/pair
# (because of the signs, when n is pair, it adds up to the one present in the other branch
# of the equation, if n is impair, it removes 2x from the single x on the other branch, resulting
# in a negative x in the other branch
sm = 0
# Avoid .append by creating a sized array
distances = [pegs[0]] + [0] * (len(pegs) - 1)
# We calculate the points between the pegs, (except the first one)
for i, v in enumerate(pegs):
if i != 0:
distances[i] = pegs[i] - pegs[i - 1]
# Derived from the formula
for i, v in enumerate(distances[1:]):
sm += (-1) ** (i) * (2 * v)
denom = 1
# If the number of pegs is pair, we divide by 3
if len(pegs) % 2 == 0:
denom = 3
# Avoid .append by creating a sized array
sizes = [sm / denom] + [0] * (len(pegs) - 1)
# calculate all the gear sizes
for i, v in enumerate(distances):
if v == distances[0]:
continue
size = v - sizes[i - 1]
sizes[i] = size
# all gears must be >=
if size < 1:
return [-1, -1]
# used to simplify the fraction
f = Fraction(sm, denom)
# the first gear must be >= 1
if f < 1:
return [-1, -1]
return [f.numerator, f.denominator]
assert solution([4, 17, 50]) == [-1, -1], "invalid"
assert solution([4, 30, 50]) == [12, 1], "invalid"
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