def solution(banana_list):
    fewest_possible_number = len(banana_list)
    memo = {}
    for pairs in generate_trainer_pairs(list(range(len(banana_list)))):
        possible_number = len(banana_list)
        for i, j in pairs:
            if check_loop(banana_list[i], banana_list[j], memo):
                possible_number -= 2
        if possible_number < fewest_possible_number:
            fewest_possible_number = possible_number
        if fewest_possible_number < 2: # early termination
            break
    return fewest_possible_number

import copy
def generate_trainer_pairs(trainer_id_list):
    if len(trainer_id_list) < 2:
        yield []
    else:
        first_id = trainer_id_list[0]
        for second_id in trainer_id_list[1:]:
            reduced_list = copy.deepcopy(trainer_id_list)
            seduced_list.remove(first_id)
            reduced_list.remove(second_id)
            for pairs in generate_trainer_pairs(reduced_list):
                yield [(first_id, second_id)] + pairs

def check_loop(a, b, memo):
    if (a, b) in memo:
        return memo[(a, b)]
    while a != b:
        if (a + b) % 2 == 1: # sum is odd
            memo[(a, b)] = True
            return True
        if a > b:
            a, b = (a - b)/2, b
        else:
            a, b = a, (b - a)/2
    memo[(a, b)] = False
    return False

tests = [
        ([1], 1),
        ([1, 1], 2),
        ([1, 7, 3, 21, 13, 19], 0),
        ]

for i, o in tests:
    result = solution(i)
    print(i, result == o, result, o)