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#!/usr/bin/env python3


def karatsuba_mul(m: int, n: int) -> int:
    """
    Implementation of recursive Karatsuba multiplication

    >>> karatsuba_mul(56, 12)
    672
    >>> karatsuba_mul(78, 34)

Per48edjes

"""
Inspired by Problem 3, Lab 09 from Berkeley's CS 61A course to investigate and 
implement the closed form solution via `catalan_nums` function
"""

from math import comb


def num_trees(n):
    """Returns the number of unique full binary trees with exactly n leaves. E.g.,

Per48edjes

#!/usr/bin/env python3

"""
Problem inspired by "Even Subsets" found here:
https://cs61a.org/assets/slides/28-Scheme_Lists_full.pdf

Playing around with recursion, generators, and decorators...of course, this
is completely for my own edification. I would not recommend writing code like
this in an applied setting... :)
"""

Per48edjes

def is_leap_year(n: int) -> bool:
    """
    Returns whether an year `n` AD is a leap year per Gregorian calendar.

    Definitionally, this function checks whether `n` is divisible by 400 or
    divisible by 4 and NOT 100.

    >>> is_leap_year(2000)
    True
    >>> is_leap_year(1896)

Per48edjes

#include <stdint.h>
#include <stdio.h>

double fast_exp(double base, uint64_t power)
{
    double result = 1;
    while (power > 0)
    {
        result *= (power & 1) ? base : 1;
        power >>= 1;

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def split_chocolate_bar(m: int, n: int) -> int:
    """
    Count the number of splits required to divide `m` by `n` chocolate bar into
    `mn` unit squares in O(mn) time complexity

    >>> split_chocolate_bar(1,1)
    0
    >>> split_chocolate_bar(2, 1)
    1
    >>> split_chocolate_bar(1, 2)

Per48edjes

from math import sqrt
from typing import List, Tuple, Union


def gcd(a: int, b: int) -> int:
    """
    Return the greatest common divisor of integers `a` and `b`

    >>> gcd(3, 6)
    3

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def digital_root(n: int, original=None) -> int:
    """
    Return the digital root of a non-negative integer `n`. Note that this is
    basically the same as `n % 9` except in the special case when `n` is a
    multiple of 9.

    @param n:  A non-negative integer
    @type  n:  int

    @return:  The digital root of `n`

Per48edjes

from functools import reduce


def merge(lst1, lst2):
    """Merges two sorted lists.
    >>> s1 = [1, 3, 5]
    >>> s2 = [2, 4, 6]
    >>> merge(s1, s2)
    [1, 2, 3, 4, 5, 6]
    >>> s1

Per48edjes

from typing import Tuple


def gcd(a: int, b: int) -> int:
    """
    Simple recursive implementation of Euclidean algorithm
    """
    a, b = abs(a), abs(b)
    if b == 0:
        return a