def is_leap_year(yyyy: int) -> bool:
    return not (yyyy % 4 or (yyyy % 400 and not yyyy % 100))


def sum_months(mm: int, yyyy: int) -> int:
    months = [
        31,
        29 if is_leap_year(yyyy) else 28,
        31,
        30,
        31,
        30,
        31,
        31,
        30,
        31,
        30,
        31,
    ]
    return sum(months[:mm])


def nth_day(s: str) -> int:
    """Calculate the nth day since the beginning of the year."""
    mm, dd, yyyy = map(int, [s[:2], s[2:4], s[4:]])
    day_since_new_year = dd + sum_months(mm - 1, yyyy)
    return yyyy * 1000 + day_since_new_year


print("we're testing the nth day function...", end="")
assert nth_day("02072016") == 2016038
assert nth_day("12201996") == 1996355
# assert(nth_day("abcdef") == None)
assert nth_day("12312020") == 2020366
assert nth_day("12312021") == 2021365
print("and it passes the example cases")