from typing import List, Optional
from math import sqrt
from functools import lru_cache


def to_int(digits: List[int]) -> int:
    return int("".join(map(str, digits)))


def int_len(n: int) -> int:
    return len(str(n))


@lru_cache
def digits_make_sum(digits, remainder: int) -> Optional[List]:
    if len(digits) == 0:
        if remainder == 0:
            return []
        return None

    remainder_digits_count = int_len(remainder)
    if remainder_digits_count > len(digits):
        return None

    if to_int(digits) < remainder:
        return None

    for group_size in range(1, remainder_digits_count + 1): 
        digits_list = list(digits)
        rest_value = remainder - to_int(digits_list[:group_size])
        rest_digits = tuple(digits_list[group_size:])
        r = digits_make_sum(rest_digits, rest_value)
        if type(r) is list:
            return [digits_list[:group_size]] + r
    return None


def digits_sum_to(digits, number: int):
    r = digits_make_sum(digits, number)
    if type(r) is list:
        # print(digits, r, number)
        return True
    return False


def t(limit: int) -> int:
    r = 0
    for base in range(4, int(sqrt(limit)) + 1):
        n = base * base
        n_digits = tuple(map(int, list(str(n))))
        if digits_sum_to(n_digits, base):
            r += n
    return r


def euler_719():
    assert(t(10**4) == 41333)
    assert(digits_sum_to((1, 0, 0), 10) == True)
    assert(digits_sum_to((1, 8), 9) == True)
    assert(digits_sum_to((2, 5), 5) == False)
    assert(digits_sum_to((8, 2, 8, 1), 91) == True)
    assert(t(10**6) == 10804656)
    return t(10**12)


if __name__ == "__main__":
    solution = euler_719()
    print("e719.py: " + str(solution))
    assert(solution == 128088830547982)