#!/usr/bin/pytnon2

## You are given a list of "pegs" with their location (positive integers, asc order).
## these pegs can accept gears with radius > 1
## the goal is the last gear to be spinning twice as fast as the 1st one for any given list
## if it's possible return the first gear's radius in a list [a,b] where the ratio is a/b
## if not return [-1,-1]

from fractions import gcd
from functools import reduce

# test
from random import seed
from random import random

def simplify(ratio_nums):
    return list(map(reduce(gcd, ratio_nums).__rfloordiv__, ratio_nums))

def solution(pegs):
    if len(pegs) == 2:
        r1_by_three = 2 * (pegs[1] - pegs[0])
        nums = [r1_by_three, 3]
        return simplify(nums)
    S = pegs[-1] - pegs[0]
    if len(pegs) % 2 == 0:
        C = 0
        for i in range (1, len(pegs) - 1):
            C += (-1)**i * pegs[i]
        r1_by_three = 2 * S - 4 * C

        r = [r1_by_three/3]
        for i in range(len(pegs) -1):
          r.append(pegs[i+1]-pegs[i]-r[i])
          if r[-1] < 1:
            return [-1, -1]
        if (r[0]) < 1 or (r[0]) > pegs[1]-pegs[0]-1:
            return [-1, -1]
        return simplify([r1_by_three, 3])
    else:
        K = 0
        for i in range (1, len(pegs) - 2):
            K += (-1)**i * pegs[i]
        r1 = 2*S - 4*K - 4*pegs[-1] + 4*pegs[-2]
        if r1 < 1:
            return [-1, -1] 
        return [r1, 1]

seed(1)

def ran():
  return int(random() * 1000)

for i in range(1,1000):
  a=[ran(),ran(),ran(),ran(),ran(),ran()]
  a.sort()
  print(a)
  print(solution(a))
  print("")