"""

Magic Number

A magic number is a number where recursively summing its digits eventually

yields 1. For example, 199 -> 1+9+9=19 -> 1+9=10 -> 1+0=1.

Reference: https://en.wikipedia.org/wiki/Digital_root

Complexity:

Time: O(log n) amortized

Space: O(1)

"""

from __future__ import annotations

def magic_number(n: int) -> bool:

"""Check if n is a magic number (digital root equals 1).

Args:

n: The integer to check.

Returns:

True if the digital root of n is 1, False otherwise.

Examples:

>>> magic_number(1234)

True

>>> magic_number(111)

False

"""

total_sum = 0

while n > 0 or total_sum > 9:

if n == 0:

n = total_sum

total_sum = 0

total_sum += n % 10

n //= 10

return total_sum == 1