```python
from typing import List

def has_close_elements(numbers: List[float], threshold: float) -> bool:
    """ Check if in given list of numbers, are any two numbers closer to each other than
    given threshold.
    >>> has_close_elements([1.0, 2.0, 3.0], 0.5)
    False
    >>> has_close_elements([1.0, 2.8, 3.0, 4.0, 5.0, 2.0], 0.3)
    True
    """
    # Sort the list to compare adjacent elements
    sorted_numbers = sorted(numbers)

    # Check each adjacent pair
    for i in range(len(sorted_numbers) - 1):
        if sorted_numbers[i + 1] - sorted_numbers[i] < threshold:
            return True

    return False
```


```python
from typing import List

def separate_paren_groups(paren_string: str) -> List[str]:
    """ Input to this function is a string containing multiple groups of nested parentheses. Your goal is to
    separate those group into separate strings and return the list of those.
    Separate groups are balanced (each open brace is properly closed) and not nested within each other
    Ignore any spaces in the input string.
    >>> separate_paren_groups('( ) (( )) (( )( ))')
    ['()', '(())', '(()())']
    """
    # Remove all spaces from the input
    cleaned = paren_string.replace(' ', '')

    result = []
    current_group = ""
    balance = 0

    for ch in cleaned:
        current_group += ch
        if ch == '(':
            balance += 1
        elif ch == ')':
            balance -= 1

        # When balance reaches 0, we have a complete balanced group
        if balance == 0:
            result.append(current_group)
            current_group = ""

    return result
```


```python

def truncate_number(number: float) -> float:
    """ Given a positive floating point number, it can be decomposed into
    and integer part (largest integer smaller than given number) and decimals
    (leftover part always smaller than 1).

    Return the decimal part of the number.
    >>> truncate_number(3.5)
    0.5
    """
    return number - int(number)

```


```python

from typing import List

def below_zero(operations: List[int]) -> bool:
    balance = 0
    for op in operations:
        balance += op
        if balance < 0:
            return True
    return False

```


```python
from typing import List

def mean_absolute_deviation(numbers: List[float]) -> float:
    """ For a given list of input numbers, calculate Mean Absolute Deviation
    around the mean of this dataset.
    Mean Absolute Deviation is the average absolute difference between each
    element and a centerpoint (mean in this case):
    MAD = average | x - x_mean |
    >>> mean_absolute_deviation([1.0, 2.0, 3.0, 4.0])
    1.0
    """
    if not numbers:
        return 0.0
    mean = sum(numbers) / len(numbers)
    deviations = [abs(x - mean) for x in numbers]
    return sum(deviations) / len(numbers)
```


```python
from typing import List

def intersperse(numbers: List[int], delimeter: int) -> List[int]:
    """ Insert a number 'delimeter' between every two consecutive elements of input list `numbers'
    >>> intersperse([], 4)
    []
    >>> intersperse([1, 2, 3], 4)
    [1, 4, 2, 4, 3]
    """
    if not numbers:
        return []

    result = []
    for i, num in enumerate(numbers):
        result.append(num)
        if i < len(numbers) - 1:
            result.append(delimeter)
    return result
```


```python
from typing import List

def parse_nested_parens(paren_string: str) -> List[int]:
    """ Input to this function is a string represented multiple groups for nested parentheses separated by spaces.
    For each of the group, output the deepest level of nesting of parentheses.
    E.g. (()()) has maximum two levels of nesting while ((())) has three.

    >>> parse_nested_parens('(()()) ((())) () ((())()())')
    [2, 3, 1, 3]
    """
    groups = paren_string.split()
    result = []

    for group in groups:
        max_depth = 0
        current_depth = 0

        for char in group:
            if char == '(':
                current_depth += 1
                max_depth = max(max_depth, current_depth)
            elif char == ')':
                current_depth -= 1

        result.append(max_depth)

    return result
```


```python
from typing import List

def filter_by_substring(strings: List[str], substring: str) -> List[str]:
    """ Filter an input list of strings only for ones that contain given substring
    >>> filter_by_substring([], 'a')
    []
    >>> filter_by_substring(['abc', 'bacd', 'cde', 'array'], 'a')
    ['abc', 'bacd', 'array']
    """
    return [s for s in strings if substring in s]
```


```python
from typing import List, Tuple

def sum_product(numbers: List[int]) -> Tuple[int, int]:
    """ For a given list of integers, return a tuple consisting of a sum and a product of all the integers in a list.
    Empty sum should be equal to 0 and empty product should be equal to 1.
    >>> sum_product([])
    (0, 1)
    >>> sum_product([1, 2, 3, 4])
    (10, 24)
    """
    total_sum = 0
    total_product = 1
    for num in numbers:
        total_sum += num
        total_product *= num
    return (total_sum, total_product)
```


```python
from typing import List

def rolling_max(numbers: List[int]) -> List[int]:
    result = []
    current_max = float('-inf')
    for num in numbers:
        if num > current_max:
            current_max = num
        result.append(current_max)
    return result
```


```python

def is_palindrome(string: str) -> bool:
    """ Test if given string is a palindrome """
    return string == string[::-1]

def make_palindrome(string: str) -> str:
    """ Find the shortest palindrome that begins with a supplied string.
    Algorithm idea is simple:
    - Find the longest postfix of supplied string that is a palindrome.
    - Append to the end of the string reverse of a string prefix that comes before the palindromic suffix.
    >>> make_palindrome('')
    ''
    >>> make_palindrome('cat')
    'catac'
    >>> make_palindrome('cata')
    'catac'
    """
    if not string:
        return ""

    # Find the longest palindromic suffix
    for i in range(len(string), 0, -1):
        suffix = string[-i:]
        if is_palindrome(suffix):
            # The prefix before the palindromic suffix
            prefix = string[:-i]
            # Append the reverse of the prefix to make the whole string a palindrome
            return string + prefix[::-1]

    # Should never reach here because at least the last character is a palindrome
    return string

```


```python
from typing import List

def string_xor(a: str, b: str) -> str:
    """ Input are two strings a and b consisting only of 1s and 0s.
    Perform binary XOR on these inputs and return result also as a string.
    >>> string_xor('010', '110')
    '100'
    """
    result = []
    for i in range(len(a)):
        if a[i] == b[i]:
            result.append('0')
        else:
            result.append('1')
    return ''.join(result)
```


```python

from typing import List, Optional

def longest(strings: List[str]) -> Optional[str]:
    """ Out of list of strings, return the longest one. Return the first one in case of multiple
    strings of the same length. Return None in case the input list is empty.
    >>> longest([])

    >>> longest(['a', 'b', 'c'])
    'a'
    >>> longest(['a', 'bb', 'ccc'])
    'ccc'
    """
    if not strings:
        return None
    longest_str = strings[0]
    for s in strings[1:]:
        if len(s) > len(longest_str):
            longest_str = s
    return longest_str

```


```python

def greatest_common_divisor(a: int, b: int) -> int:
    """ Return a greatest common divisor of two integers a and b
    >>> greatest_common_divisor(3, 5)
    1
    >>> greatest_common_divisor(25, 15)
    5
    """
    a, b = abs(a), abs(b)
    while b:
        a, b = b, a % b
    return a

```


```python
from typing import List

def all_prefixes(string: str) -> List[str]:
    """ Return list of all prefixes from shortest to longest of the input string
    >>> all_prefixes('abc')
    ['a', 'ab', 'abc']
    """
    return [string[:i] for i in range(1, len(string) + 1)]
```


```python

def string_sequence(n: int) -> str:
    """ Return a string containing space-delimited numbers starting from 0 upto n inclusive.
    >>> string_sequence(0)
    '0'
    >>> string_sequence(5)
    '0 1 2 3 4 5'
    """
    return ' '.join(str(i) for i in range(n + 1))

```


```python

def count_distinct_characters(string: str) -> int:
    """ Given a string, find out how many distinct characters (regardless of case) does it consist of
    >>> count_distinct_characters('xyzXYZ')
    3
    >>> count_distinct_characters('Jerry')
    4
    """
    return len(set(string.lower()))

```


Here's the implementation for the `parse_music` function:

```python
from typing import List

def parse_music(music_string: str) -> List[int]:
    notes = music_string.split()
    result = []
    for note in notes:
        if note == 'o':
            result.append(4)
        elif note == 'o|':
            result.append(2)
        elif note == '.|':
            result.append(1)
    return result
```


```python

def how_many_times(string: str, substring: str) -> int:
    """ Find how many times a given substring can be found in the original string. Count overlaping cases.
    >>> how_many_times('', 'a')
    0
    >>> how_many_times('aaa', 'a')
    3
    >>> how_many_times('aaaa', 'aa')
    3
    """
    if not substring:
        return 0
    count = 0
    start = 0
    while True:
        pos = string.find(substring, start)
        if pos == -1:
            break
        count += 1
        start = pos + 1
    return count

```


```python
from typing import List

def sort_numbers(numbers: str) -> str:
    """ Input is a space-delimited string of numberals from 'zero' to 'nine'.
    Valid choices are 'zero', 'one', 'two', 'three', 'four', 'five', 'six', 'seven', 'eight' and 'nine'.
    Return the string with numbers sorted from smallest to largest
    >>> sort_numbers('three one five')
    'one three five'
    """
    numeral_to_value = {
        'zero': 0, 'one': 1, 'two': 2, 'three': 3, 'four': 4,
        'five': 5, 'six': 6, 'seven': 7, 'eight': 8, 'nine': 9
    }
    tokens = numbers.split()
    sorted_tokens = sorted(tokens, key=lambda x: numeral_to_value[x])
    return ' '.join(sorted_tokens)
```


```python

from typing import List, Tuple

def find_closest_elements(numbers: List[float]) -> Tuple[float, float]:
    # Sort the list to easily compare adjacent elements
    sorted_numbers = sorted(numbers)

    # Initialize with a large difference
    min_diff = float('inf')
    closest_pair = (0.0, 0.0)

    # Compare each adjacent pair
    for i in range(len(sorted_numbers) - 1):
        a = sorted_numbers[i]
        b = sorted_numbers[i + 1]
        diff = b - a
        if diff < min_diff:
            min_diff = diff
            closest_pair = (a, b)

    return closest_pair

```


```python
from typing import List

def rescale_to_unit(numbers: List[float]) -> List[float]:
    """ Given list of numbers (of at least two elements), apply a linear transform to that list,
    such that the smallest number will become 0 and the largest will become 1
    >>> rescale_to_unit([1.0, 2.0, 3.0, 4.0, 5.0])
    [0.0, 0.25, 0.5, 0.75, 1.0]
    """
    min_val = min(numbers)
    max_val = max(numbers)
    range_val = max_val - min_val
    if range_val == 0:
        # If all numbers are the same, return list of zeros (or handle as needed)
        return [0.0] * len(numbers)
    return [(x - min_val) / range_val for x in numbers]
```


```python
from typing import List, Any

def filter_integers(values: List[Any]) -> List[int]:
    """ Filter given list of any python values only for integers
    >>> filter_integers(['a', 3.14, 5])
    [5]
    >>> filter_integers([1, 2, 3, 'abc', {}, []])
    [1, 2, 3]
    """
    return [v for v in values if isinstance(v, int)]
```


```python

def strlen(string: str) -> int:
    """ Return length of given string
    >>> strlen('')
    0
    >>> strlen('abc')
    3
    """
    return len(string)

```


```python

def largest_divisor(n: int) -> int:
    """ For a given number n, find the largest number that divides n evenly, smaller than n
    >>> largest_divisor(15)
    5
    """
    for d in range(n // 2, 0, -1):
        if n % d == 0:
            return d
    return 1

```


```python

from typing import List

def factorize(n: int) -> List[int]:
    """ Return list of prime factors of given integer in the order from smallest to largest.
    Each of the factors should be listed number of times corresponding to how many times it appeares in factorization.
    Input number should be equal to the product of all factors
    >>> factorize(8)
    [2, 2, 2]
    >>> factorize(25)
    [5, 5]
    >>> factorize(70)
    [2, 5, 7]
    """
    factors = []
    d = 2
    while d * d <= n:
        while n % d == 0:
            factors.append(d)
            n //= d
        d += 1
    if n > 1:
        factors.append(n)
    return factors

```


```python
from typing import List

def remove_duplicates(numbers: List[int]) -> List[int]:
    """ From a list of integers, remove all elements that occur more than once.
    Keep order of elements left the same as in the input.
    >>> remove_duplicates([1, 2, 3, 2, 4])
    [1, 3, 4]
    """
    # Count occurrences of each number
    counts = {}
    for num in numbers:
        counts[num] = counts.get(num, 0) + 1

    # Keep only numbers that appear exactly once
    result = []
    for num in numbers:
        if counts[num] == 1:
            result.append(num)

    return result
```


```python

def flip_case(string: str) -> str:
    """ For a given string, flip lowercase characters to uppercase and uppercase to lowercase.
    >>> flip_case('Hello')
    'hELLO'
    """
    return string.swapcase()

```


```python
from typing import List

def concatenate(strings: List[str]) -> str:
    """ Concatenate list of strings into a single string
    >>> concatenate([])
    ''
    >>> concatenate(['a', 'b', 'c'])
    'abc'
    """
    return ''.join(strings)
```


```python
from typing import List

def filter_by_prefix(strings: List[str], prefix: str) -> List[str]:
    """ Filter an input list of strings only for ones that start with a given prefix.
    >>> filter_by_prefix([], 'a')
    []
    >>> filter_by_prefix(['abc', 'bcd', 'cde', 'array'], 'a')
    ['abc', 'array']
    """
    return [s for s in strings if s.startswith(prefix)]
```


```python

def get_positive(l: list):
    """Return only positive numbers in the list.
    >>> get_positive([-1, 2, -4, 5, 6])
    [2, 5, 6]
    >>> get_positive([5, 3, -5, 2, -3, 3, 9, 0, 123, 1, -10])
    [5, 3, 2, 3, 9, 123, 1]
    """
    return [x for x in l if x > 0]

```


```python

def is_prime(n):
    """Return true if a given number is prime, and false otherwise.
    >>> is_prime(6)
    False
    >>> is_prime(101)
    True
    >>> is_prime(11)
    True
    >>> is_prime(13441)
    True
    >>> is_prime(61)
    True
    >>> is_prime(4)
    False
    >>> is_prime(1)
    False
    """
    if n <= 1:
        return False
    if n <= 3:
        return True
    if n % 2 == 0 or n % 3 == 0:
        return False
    i = 5
    while i * i <= n:
        if n % i == 0 or n % (i + 2) == 0:
            return False
        i += 6
    return True

```


Traceback (most recent call last):
  File "<string>", line 76, in <module>
    check(find_zero)
    ~~~~~^^^^^^^^^^^
  File "<string>", line 73, in check
    assert math.fabs(poly(coeffs, solution)) < 1e-4
           ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
AssertionError

Here's the implementation for the `find_zero` function using the bisection method, assuming the polynomial has an even number of coefficients and a nonzero leading coefficient:

```python
import math

def poly(xs: list, x: float):
    """
    Evaluates polynomial with coefficients xs at point x.
    return xs[0] + xs[1] * x + xs[1] * x^2 + .... xs[n] * x^n
    """
    return sum([coeff * math.pow(x, i) for i, coeff in enumerate(xs)])

def find_zero(xs: list):
    """ xs are coefficients of a polynomial.
    find_zero find x such that poly(x) = 0.
    find_zero returns only only zero point, even if there are many.
    Moreover, find_zero only takes list xs having even number of coefficients
    and largest non zero coefficient as it guarantees
    a solution.
    >>> round(find_zero([1, 2]), 2) # f(x) = 1 + 2x
    -0.5
    >>> round(find_zero([-6, 11, -6, 1]), 2) # (x - 1) * (x - 2) * (x - 3) = -6 + 11x - 6x^2 + x^3
    1.0
    """
    # Find a starting interval where poly changes sign
    # Use a simple search: start from 0 and expand outward
    low = 0.0
    high = 1.0
    f_low = poly(xs, low)
    f_high = poly(xs, high)

    # Expand until we find a sign change
    while f_low * f_high > 0:
        low -= 1.0
        high += 1.0
        f_low = poly(xs, low)
        f_high = poly(xs, high)

    # Bisection method
    for _ in range(50):  # enough iterations for high precision
        mid = (low + high) / 2
        f_mid = poly(xs, mid)
        if f_mid == 0:
            return mid
        if f_low * f_mid < 0:
            high = mid
            f_high = f_mid
        else:
            low = mid
            f_low = f_mid

    return (low + high) / 2
```


Here's the implementation for the function you described:

```python
def sort_third(l: list):
    """This function takes a list l and returns a list l' such that
    l' is identical to l in the indices that are not divisible by three, while its values at the indices that are divisible by three are equal
    to the values of the corresponding indices of l, but sorted.
    >>> sort_third([1, 2, 3])
    [1, 2, 3]
    >>> sort_third([5, 6, 3, 4, 8, 9, 2])
    [2, 6, 3, 4, 8, 9, 5]
    """
    # Extract values at indices divisible by 3
    divisible_values = [l[i] for i in range(len(l)) if i % 3 == 0]
    # Sort them
    divisible_values.sort()

    # Build result list
    result = []
    sorted_index = 0
    for i in range(len(l)):
        if i % 3 == 0:
            result.append(divisible_values[sorted_index])
            sorted_index += 1
        else:
            result.append(l[i])

    return result
```


```python

def unique(l: list):
    """Return sorted unique elements in a list
    >>> unique([5, 3, 5, 2, 3, 3, 9, 0, 123])
    [0, 2, 3, 5, 9, 123]
    """
    return sorted(set(l))

```


```python

def max_element(l: list):
    """Return maximum element in the list.
    >>> max_element([1, 2, 3])
    3
    >>> max_element([5, 3, -5, 2, -3, 3, 9, 0, 123, 1, -10])
    123
    """
    if not l:
        raise ValueError("max_element() arg is an empty list")
    max_val = l[0]
    for x in l[1:]:
        if x > max_val:
            max_val = x
    return max_val

```


Here's the implementation for the function:

```python
def fizz_buzz(n: int):
    """Return the number of times the digit 7 appears in integers less than n which are divisible by 11 or 13.
    >>> fizz_buzz(50)
    0
    >>> fizz_buzz(78)
    2
    >>> fizz_buzz(79)
    3
    """
    count = 0
    for i in range(n):
        if i % 11 == 0 or i % 13 == 0:
            count += str(i).count('7')
    return count
```


```python

def sort_even(l: list):
    """This function takes a list l and returns a list l' such that
    l' is identical to l in the odd indicies, while its values at the even indicies are equal
    to the values of the even indicies of l, but sorted.
    >>> sort_even([1, 2, 3])
    [1, 2, 3]
    >>> sort_even([5, 6, 3, 4])
    [3, 6, 5, 4]
    """
    even_values = sorted([l[i] for i in range(0, len(l), 2)])
    result = []
    even_idx = 0
    for i in range(len(l)):
        if i % 2 == 0:
            result.append(even_values[even_idx])
            even_idx += 1
        else:
            result.append(l[i])
    return result

```


Traceback (most recent call last):
  File "<string>", line 30, in <module>
    check(decode_cyclic)
    ~~~~~^^^^^^^^^^^^^^^
  File "<string>", line 26, in check
    encoded_str = encode_cyclic(str)
                  ^^^^^^^^^^^^^
NameError: name 'encode_cyclic' is not defined. Did you mean: 'decode_cyclic'?

```python

def decode_cyclic(s: str):
    """
    takes as input string encoded with encode_cyclic function. Returns decoded string.
    """
    # split string into groups of three characters
    groups = [s[(3 * i):min((3 * i + 3), len(s))] for i in range((len(s) + 2) // 3)]
    # reverse the cycle: move last character to front for groups of length 3
    groups = [(group[-1:] + group[:-1]) if len(group) == 3 else group for group in groups]
    return "".join(groups)

```


```python

def prime_fib(n: int):
    """
    prime_fib returns n-th number that is a Fibonacci number and it's also prime.
    >>> prime_fib(1)
    2
    >>> prime_fib(2)
    3
    >>> prime_fib(3)
    5
    >>> prime_fib(4)
    13
    >>> prime_fib(5)
    89
    """
    def is_prime(num: int) -> bool:
        if num < 2:
            return False
        if num == 2:
            return True
        if num % 2 == 0:
            return False
        i = 3
        while i * i <= num:
            if num % i == 0:
                return False
            i += 2
        return True

    count = 0
    a, b = 1, 1
    while True:
        # next Fibonacci number
        fib = a + b
        a, b = b, fib
        if is_prime(fib):
            count += 1
            if count == n:
                return fib

```


```python

def triples_sum_to_zero(l: list):
    """
    triples_sum_to_zero takes a list of integers as an input.
    it returns True if there are three distinct elements in the list that
    sum to zero, and False otherwise.
    """
    n = len(l)
    for i in range(n):
        for j in range(i + 1, n):
            for k in range(j + 1, n):
                if l[i] + l[j] + l[k] == 0:
                    return True
    return False

```


```python

def car_race_collision(n: int) -> int:
    """
    Imagine a road that's a perfectly straight infinitely long line.
    n cars are driving left to right;  simultaneously, a different set of n cars
    are driving right to left.   The two sets of cars start out being very far from
    each other.  All cars move in the same speed.  Two cars are said to collide
    when a car that's moving left to right hits a car that's moving right to left.
    However, the cars are infinitely sturdy and strong; as a result, they continue moving
    in their trajectory as if they did not collide.

    This function outputs the number of such collisions.
    """
    # Each car from the left-to-right set will eventually meet each car
    # from the right-to-left set exactly once, because they all move at
    # the same speed and the road is infinite. So total collisions = n * n.
    return n * n

```


```python

def incr_list(l: list):
    """Return list with elements incremented by 1.
    >>> incr_list([1, 2, 3])
    [2, 3, 4]
    >>> incr_list([5, 3, 5, 2, 3, 3, 9, 0, 123])
    [6, 4, 6, 3, 4, 4, 10, 1, 124]
    """
    return [x + 1 for x in l]

```


```python

def pairs_sum_to_zero(l):
    """
    pairs_sum_to_zero takes a list of integers as an input.
    it returns True if there are two distinct elements in the list that
    sum to zero, and False otherwise.
    >>> pairs_sum_to_zero([1, 3, 5, 0])
    False
    >>> pairs_sum_to_zero([1, 3, -2, 1])
    False
    >>> pairs_sum_to_zero([1, 2, 3, 7])
    False
    >>> pairs_sum_to_zero([2, 4, -5, 3, 5, 7])
    True
    >>> pairs_sum_to_zero([1])
    False
    """
    seen = set()
    for num in l:
        if -num in seen:
            return True
        seen.add(num)
    return False

```


```python
def change_base(x: int, base: int):
    """Change numerical base of input number x to base.
    return string representation after the conversion.
    base numbers are less than 10.
    >>> change_base(8, 3)
    '22'
    >>> change_base(8, 2)
    '1000'
    >>> change_base(7, 2)
    '111'
    """
    if x == 0:
        return '0'

    digits = []
    while x > 0:
        digits.append(str(x % base))
        x //= base

    return ''.join(reversed(digits))
```


```python

def triangle_area(a, h):
    """Given length of a side and high return area for a triangle.
    >>> triangle_area(5, 3)
    7.5
    """
    return a * h / 2

```


Here's the efficient, iterative implementation for the fib4 sequence:

```python
def fib4(n: int):
    """The Fib4 number sequence is a sequence similar to the Fibbonacci sequnece that's defined as follows:
    fib4(0) -> 0
    fib4(1) -> 0
    fib4(2) -> 2
    fib4(3) -> 0
    fib4(n) -> fib4(n-1) + fib4(n-2) + fib4(n-3) + fib4(n-4).
    Please write a function to efficiently compute the n-th element of the fib4 number sequence.  Do not use recursion.
    >>> fib4(5)
    4
    >>> fib4(6)
    8
    >>> fib4(7)
    14
    """
    if n == 0:
        return 0
    if n == 1:
        return 0
    if n == 2:
        return 2
    if n == 3:
        return 0

    # Initial values for n=0,1,2,3
    a, b, c, d = 0, 0, 2, 0

    for _ in range(4, n + 1):
        next_val = a + b + c + d
        a, b, c, d = b, c, d, next_val

    return d
```


Here's the implementation for the `median` function:

```python
def median(l: list):
    """Return median of elements in the list l.
    >>> median([3, 1, 2, 4, 5])
    3
    >>> median([-10, 4, 6, 1000, 10, 20])
    15.0
    """
    sorted_l = sorted(l)
    n = len(sorted_l)
    if n % 2 == 1:
        return sorted_l[n // 2]
    else:
        left = sorted_l[n // 2 - 1]
        right = sorted_l[n // 2]
        return (left + right) / 2.0
```


```python
def is_palindrome(text: str):
    """
    Checks if given string is a palindrome
    >>> is_palindrome('')
    True
    >>> is_palindrome('aba')
    True
    >>> is_palindrome('aaaaa')
    True
    >>> is_palindrome('zbcd')
    False
    """
    cleaned = ''.join(c.lower() for c in text if c.isalnum())
    return cleaned == cleaned[::-1]
```


Here's the implementation using fast modular exponentiation to handle large values efficiently:

```python
def modp(n: int, p: int):
    """Return 2^n modulo p (be aware of numerics).
    >>> modp(3, 5)
    3
    >>> modp(1101, 101)
    2
    >>> modp(0, 101)
    1
    >>> modp(3, 11)
    8
    >>> modp(100, 101)
    1
    """
    if p == 0:
        raise ValueError("p must be non-zero")
    if n == 0:
        return 1 % p

    result = 1
    base = 2 % p
    exp = n

    while exp > 0:
        if exp & 1:
            result = (result * base) % p
        base = (base * base) % p
        exp >>= 1

    return result
```


```python

def encode_shift(s: str):
    """
    returns encoded string by shifting every character by 5 in the alphabet.
    """
    return "".join([chr(((ord(ch) + 5 - ord("a")) % 26) + ord("a")) for ch in s])

def decode_shift(s: str):
    """
    takes as input string encoded with encode_shift function. Returns decoded string.
    """
    return "".join([chr(((ord(ch) - 5 - ord("a")) % 26) + ord("a")) for ch in s])

```


```python

def remove_vowels(text):
    """
    remove_vowels is a function that takes string and returns string without vowels.
    >>> remove_vowels('')
    ''
    >>> remove_vowels("abcdef\nghijklm")
    'bcdf\nghjklm'
    >>> remove_vowels('abcdef')
    'bcdf'
    >>> remove_vowels('aaaaa')
    ''
    >>> remove_vowels('aaBAA')
    'B'
    >>> remove_vowels('zbcd')
    'zbcd'
    """
    vowels = set('aeiouAEIOU')
    return ''.join(ch for ch in text if ch not in vowels)

```


```python
def below_threshold(l: list, t: int):
    """Return True if all numbers in the list l are below threshold t.
    >>> below_threshold([1, 2, 4, 10], 100)
    True
    >>> below_threshold([1, 20, 4, 10], 5)
    False
    """
    for num in l:
        if num >= t:
            return False
    return True
```


```python
def add(x: int, y: int):
    """Add two numbers x and y
    >>> add(2, 3)
    5
    >>> add(5, 7)
    12
    """
    return x + y
```


```python

def same_chars(s0: str, s1: str):
    """
    Check if two words have the same characters.
    >>> same_chars('eabcdzzzz', 'dddzzzzzzzddeddabc')
    True
    >>> same_chars('abcd', 'dddddddabc')
    True
    >>> same_chars('dddddddabc', 'abcd')
    True
    >>> same_chars('eabcd', 'dddddddabc')
    False
    >>> same_chars('abcd', 'dddddddabce')
    False
    >>> same_chars('eabcdzzzz', 'dddzzzzzzzddddabc')
    False
    """
    # Remove duplicates by converting to set and compare sorted characters
    return sorted(set(s0)) == sorted(set(s1))

```


```python

def fib(n: int):
    """Return n-th Fibonacci number.
    >>> fib(10)
    55
    >>> fib(1)
    1
    >>> fib(8)
    21
    """
    if n <= 0:
        return 0
    elif n == 1:
        return 1
    a, b = 0, 1
    for _ in range(2, n + 1):
        a, b = b, a + b
    return b

```


```python
def correct_bracketing(brackets: str):
    """ brackets is a string of "<" and ">".
    return True if every opening bracket has a corresponding closing bracket.

    >>> correct_bracketing("<")
    False
    >>> correct_bracketing("<>")
    True
    >>> correct_bracketing("<<><>>")
    True
    >>> correct_bracketing("><<>")
    False
    """
    count = 0
    for b in brackets:
        if b == "<":
            count += 1
        elif b == ">":
            if count == 0:
                return False
            count -= 1
    return count == 0
```


```python

def monotonic(l: list):
    """Return True if list elements are monotonically increasing or decreasing."""
    if len(l) <= 1:
        return True
    # Determine direction from first non-equal pair
    increasing = None
    for i in range(1, len(l)):
        if l[i] > l[i-1]:
            if increasing is None:
                increasing = True
            elif not increasing:
                return False
        elif l[i] < l[i-1]:
            if increasing is None:
                increasing = False
            elif increasing:
                return False
        # if equal, continue without changing direction
    return True

```


```python

def common(l1: list, l2: list):
    """Return sorted unique common elements for two lists.
    >>> common([1, 4, 3, 34, 653, 2, 5], [5, 7, 1, 5, 9, 653, 121])
    [1, 5, 653]
    >>> common([5, 3, 2, 8], [3, 2])
    [2, 3]

    """
    # Convert both lists to sets to remove duplicates
    set1 = set(l1)
    set2 = set(l2)

    # Find intersection of the two sets
    common_elements = set1 & set2

    # Return sorted list of the common elements
    return sorted(common_elements)

```


```python

def largest_prime_factor(n: int):
    """Return the largest prime factor of n. Assume n > 1 and is not a prime.
    >>> largest_prime_factor(13195)
    29
    >>> largest_prime_factor(2048)
    2
    """
    factor = 2
    while factor * factor <= n:
        if n % factor == 0:
            n //= factor
        else:
            factor += 1 if factor == 2 else 2  # after 2, check only odd numbers
    return n

```


```python
def sum_to_n(n: int):
    """sum_to_n is a function that sums numbers from 1 to n.
    >>> sum_to_n(30)
    465
    >>> sum_to_n(100)
    5050
    >>> sum_to_n(5)
    15
    >>> sum_to_n(10)
    55
    >>> sum_to_n(1)
    1
    """
    return n * (n + 1) // 2
```


```python
def correct_bracketing(brackets: str):
    """ brackets is a string of "(" and ")".
    return True if every opening bracket has a corresponding closing bracket.

    >>> correct_bracketing("(")
    False
    >>> correct_bracketing("()")
    True
    >>> correct_bracketing("(()())")
    True
    >>> correct_bracketing(")(()")
    False
    """
    balance = 0
    for b in brackets:
        if b == '(':
            balance += 1
        elif b == ')':
            balance -= 1
        if balance < 0:
            return False
    return balance == 0
```


```python

def derivative(xs: list):
    """ xs represent coefficients of a polynomial.
    xs[0] + xs[1] * x + xs[2] * x^2 + ....
     Return derivative of this polynomial in the same form.
    >>> derivative([3, 1, 2, 4, 5])
    [1, 4, 12, 20]
    >>> derivative([1, 2, 3])
    [2, 6]
    """
    if len(xs) <= 1:
        return []
    result = []
    for i in range(1, len(xs)):
        result.append(i * xs[i])
    return result

```


Here's the efficient implementation using iterative computation with three variables:

```python
def fibfib(n: int):
    """The FibFib number sequence is a sequence similar to the Fibbonacci sequnece that's defined as follows:
    fibfib(0) == 0
    fibfib(1) == 0
    fibfib(2) == 1
    fibfib(n) == fibfib(n-1) + fibfib(n-2) + fibfib(n-3).
    Please write a function to efficiently compute the n-th element of the fibfib number sequence.
    >>> fibfib(1)
    0
    >>> fibfib(5)
    4
    >>> fibfib(8)
    24
    """
    if n == 0 or n == 1:
        return 0
    if n == 2:
        return 1

    a, b, c = 0, 0, 1  # representing fibfib(0), fibfib(1), fibfib(2)

    for _ in range(3, n + 1):
        next_val = a + b + c
        a, b, c = b, c, next_val

    return c
```


Here's the implementation for the `vowels_count` function:

```python
def vowels_count(s):
    """Write a function vowels_count which takes a string representing
    a word as input and returns the number of vowels in the string.
    Vowels in this case are 'a', 'e', 'i', 'o', 'u'. Here, 'y' is also a
    vowel, but only when it is at the end of the given word.

    Example:
    >>> vowels_count("abcde")
    2
    >>> vowels_count("ACEDY")
    3
    """
    vowels = set('aeiou')
    count = 0
    s_lower = s.lower()
    for i, ch in enumerate(s_lower):
        if ch in vowels:
            count += 1
        elif ch == 'y' and i == len(s) - 1:
            count += 1
    return count
```


Here's the implementation for the `circular_shift` function:

```python
def circular_shift(x, shift):
    """Circular shift the digits of the integer x, shift the digits right by shift
    and return the result as a string.
    If shift > number of digits, return digits reversed.
    >>> circular_shift(12, 1)
    "21"
    >>> circular_shift(12, 2)
    "12"
    """
    digits = str(x)
    if shift > len(digits):
        return digits[::-1]
    return digits[-shift:] + digits[:-shift]
```


```python

def digitSum(s):
    """Task
    Write a function that takes a string as input and returns the sum of the upper characters only'
    ASCII codes.

    Examples:
        digitSum("") => 0
        digitSum("abAB") => 131
        digitSum("abcCd") => 67
        digitSum("helloE") => 69
        digitSum("woArBld") => 131
        digitSum("aAaaaXa") => 153
    """
    total = 0
    for ch in s:
        if 'A' <= ch <= 'Z':
            total += ord(ch)
    return total

```


Here's the implementation for the function you described:

```python
import re

def fruit_distribution(s: str, n: int) -> int:
    """
    Given a string representing the number of apples and oranges and an integer
    representing the total number of fruits in the basket, returns the number of
    mango fruits in the basket.
    """
    # Extract numbers for apples and oranges from the string
    numbers = list(map(int, re.findall(r'\d+', s)))

    # The string should contain exactly two numbers: apples and oranges
    apples, oranges = numbers[0], numbers[1]

    # Calculate mango fruits
    mangoes = n - apples - oranges

    return mangoes
```


Here's the implementation for the `pluck` function:

```python
def pluck(arr):
    """
    Given an array representing a branch of a tree that has non-negative integer nodes
    your task is to pluck one of the nodes and return it.
    The plucked node should be the node with the smallest even value.
    If multiple nodes with the same smallest even value are found return the node that has smallest index.

    The plucked node should be returned in a list, [ smallest_value, its index ],
    If there are no even values or the given array is empty, return [].
    """
    if not arr:
        return []

    smallest_even = None
    smallest_index = None

    for i, val in enumerate(arr):
        if val % 2 == 0:
            if smallest_even is None or val < smallest_even:
                smallest_even = val
                smallest_index = i
            elif val == smallest_even and i < smallest_index:
                smallest_index = i

    if smallest_even is None:
        return []

    return [smallest_even, smallest_index]
```


Here's the implementation for the function you described:

```python
def search(lst):
    """
    Returns the greatest integer that is greater than zero and has a frequency
    greater than or equal to the integer itself. If no such value exists, returns -1.
    """
    freq = {}
    for num in lst:
        freq[num] = freq.get(num, 0) + 1

    result = -1
    for num, count in freq.items():
        if num > 0 and count >= num:
            if result < num:
                result = num

    return result
```


Here's the implementation for the `strange_sort_list` function:

```python
def strange_sort_list(lst):
    """
    Given list of integers, return list in strange order.
    Strange sorting, is when you start with the minimum value,
    then maximum of the remaining integers, then minimum and so on.

    Examples:
    strange_sort_list([1, 2, 3, 4]) == [1, 4, 2, 3]
    strange_sort_list([5, 5, 5, 5]) == [5, 5, 5, 5]
    strange_sort_list([]) == []
    """
    if not lst:
        return []

    result = []
    remaining = sorted(lst)

    # We alternate picking from the front (min) and back (max)
    # of the sorted list
    i, j = 0, len(remaining) - 1
    take_min = True

    while i <= j:
        if take_min:
            result.append(remaining[i])
            i += 1
        else:
            result.append(remaining[j])
            j -= 1
        take_min = not take_min

    return result
```


Here's the implementation using Heron's formula:

```python
import math

def triangle_area(a, b, c):
    '''
    Given the lengths of the three sides of a triangle. Return the area of
    the triangle rounded to 2 decimal points if the three sides form a valid triangle.
    Otherwise return -1
    Three sides make a valid triangle when the sum of any two sides is greater
    than the third side.
    Example:
    triangle_area(3, 4, 5) == 6.00
    triangle_area(1, 2, 10) == -1
    '''
    # Check if the sides form a valid triangle
    if a + b > c and a + c > b and b + c > a:
        # Calculate semi-perimeter
        s = (a + b + c) / 2
        # Calculate area using Heron's formula
        area = math.sqrt(s * (s - a) * (s - b) * (s - c))
        # Round to 2 decimal places
        return round(area, 2)
    else:
        return -1
```


Here's the implementation for the function you described:

```python
def will_it_fly(q, w):
    # Check if the list is a palindrome (balanced)
    if q != q[::-1]:
        return False
    # Check if the sum of elements is less than or equal to w
    if sum(q) > w:
        return False
    return True
```


Here's the implementation for the `smallest_change` function:

```python
def smallest_change(arr):
    """
    Given an array arr of integers, find the minimum number of elements that
    need to be changed to make the array palindromic. A palindromic array is an array that
    is read the same backwards and forwards. In one change, you can change one element to any other element.
    """
    n = len(arr)
    changes = 0
    left = 0
    right = n - 1

    while left < right:
        if arr[left] != arr[right]:
            changes += 1
        left += 1
        right -= 1

    return changes
```


Here's the implementation for the function you described:

```python
def total_match(lst1, lst2):
    '''
    Write a function that accepts two lists of strings and returns the list that has
    total number of chars in the all strings of the list less than the other list.

    if the two lists have the same number of chars, return the first list.

    Examples
    total_match([], []) ➞ []
    total_match(['hi', 'admin'], ['hI', 'Hi']) ➞ ['hI', 'Hi']
    total_match(['hi', 'admin'], ['hi', 'hi', 'admin', 'project']) ➞ ['hi', 'admin']
    total_match(['hi', 'admin'], ['hI', 'hi', 'hi']) ➞ ['hI', 'hi', 'hi']
    total_match(['4'], ['1', '2', '3', '4', '5']) ➞ ['4']
    '''
    total1 = sum(len(s) for s in lst1)
    total2 = sum(len(s) for s in lst2)

    if total1 <= total2:
        return lst1
    else:
        return lst2
```


Here's the implementation for the function:

```python
def is_multiply_prime(a):
    """Write a function that returns true if the given number is the multiplication of 3 prime numbers
    and false otherwise.
    Knowing that (a) is less then 100.
    Example:
    is_multiply_prime(30) == True
    30 = 2 * 3 * 5
    """
    if a < 2:
        return False

    # Find prime factors
    n = a
    factors = []
    d = 2
    while d * d <= n:
        while n % d == 0:
            factors.append(d)
            n //= d
        d += 1
    if n > 1:
        factors.append(n)

    # Check if exactly 3 prime factors (allowing repeats like 2*2*2 = 8)
    return len(factors) == 3
```


```python

def is_simple_power(x, n):
    """Returns True if x is a simple power of n (n**int == x), otherwise False."""
    if n == 0 or n == 1:
        # 0**positive = 0, 0**0 = 1, 1**anything = 1
        return x == n or x == 1
    if x < 1:
        return False
    # Repeatedly divide x by n
    while x >= n:
        if x % n != 0:
            return False
        x //= n
    return x == 1

```


```python
def iscube(a):
    '''
    Write a function that takes an integer a and returns True
    if this integer is a cube of some integer number.
    Note: you may assume the input is always valid.
    Examples:
    iscube(1) ==> True
    iscube(2) ==> False
    iscube(-1) ==> True
    iscube(64) ==> True
    iscube(0) ==> True
    iscube(180) ==> False
    '''
    if a < 0:
        # Handle negative numbers by working with absolute value
        a = -a
        # Check if the cube root of the absolute value is an integer
        root = round(a ** (1/3))
        return root * root * root == a
    else:
        # For non-negative numbers
        root = round(a ** (1/3))
        return root * root * root == a
```


```python

def hex_key(num):
    """Counts the number of hexadecimal digits that are primes.
    Prime hex digits: 2, 3, 5, 7, B (11), D (13).
    """
    prime_digits = {'2', '3', '5', '7', 'B', 'D'}
    count = 0
    for digit in num:
        if digit in prime_digits:
            count += 1
    return count

```


```python

def decimal_to_binary(decimal):
    """You will be given a number in decimal form and your task is to convert it to
    binary format. The function should return a string, with each character representing a binary
    number. Each character in the string will be '0' or '1'.

    There will be an extra couple of characters 'db' at the beginning and at the end of the string.
    The extra characters are there to help with the format.

    Examples:
    decimal_to_binary(15)   # returns "db1111db"
    decimal_to_binary(32)   # returns "db100000db"
    """
    if decimal == 0:
        return "db0db"
    binary = ""
    n = decimal
    while n > 0:
        binary = str(n % 2) + binary
        n //= 2
    return "db" + binary + "db"

```


```python

def is_happy(s):
    """You are given a string s.
    Your task is to check if the string is happy or not.
    A string is happy if its length is at least 3 and every 3 consecutive letters are distinct
    For example:
    is_happy(a) => False
    is_happy(aa) => False
    is_happy(abcd) => True
    is_happy(aabb) => False
    is_happy(adb) => True
    is_happy(xyy) => False
    """
    if len(s) < 3:
        return False
    for i in range(len(s) - 2):
        if s[i] == s[i+1] or s[i] == s[i+2] or s[i+1] == s[i+2]:
            return False
    return True

```


Here's the implementation for the function you described:

```python
def numerical_letter_grade(grades):
    letter_grades = []
    for gpa in grades:
        if gpa == 4.0:
            letter_grades.append('A+')
        elif gpa > 3.7:
            letter_grades.append('A')
        elif gpa > 3.3:
            letter_grades.append('A-')
        elif gpa > 3.0:
            letter_grades.append('B+')
        elif gpa > 2.7:
            letter_grades.append('B')
        elif gpa > 2.3:
            letter_grades.append('B-')
        elif gpa > 2.0:
            letter_grades.append('C+')
        elif gpa > 1.7:
            letter_grades.append('C')
        elif gpa > 1.3:
            letter_grades.append('C-')
        elif gpa > 1.0:
            letter_grades.append('D+')
        elif gpa > 0.7:
            letter_grades.append('D')
        elif gpa > 0.0:
            letter_grades.append('D-')
        else:
            letter_grades.append('E')
    return letter_grades
```


Here's the implementation:

```python
def prime_length(string):
    """Write a function that takes a string and returns True if the string
    length is a prime number or False otherwise
    Examples
    prime_length('Hello') == True
    prime_length('abcdcba') == True
    prime_length('kittens') == True
    prime_length('orange') == False
    """
    length = len(string)
    if length < 2:
        return False
    for i in range(2, int(length ** 0.5) + 1):
        if length % i == 0:
            return False
    return True
```


```python

def starts_one_ends(n):
    """
    Given a positive integer n, return the count of the numbers of n-digit
    positive integers that start or end with 1.
    """
    if n == 1:
        # For 1-digit numbers: 1-9, those starting or ending with 1 are just [1]
        return 1

    # Total n-digit numbers: 9 * 10^(n-1)
    # Numbers that start with 1: 10^(n-1)
    # Numbers that end with 1: 9 * 10^(n-2) (first digit 1-9, last digit fixed to 1)
    # Numbers that both start and end with 1: 10^(n-2) (first digit 1, last digit 1, middle digits any)
    # By inclusion-exclusion:
    # count = start_with_1 + end_with_1 - both_start_and_end_with_1
    start_with_1 = 10 ** (n - 1)
    end_with_1 = 9 * 10 ** (n - 2)
    both = 10 ** (n - 2)
    return start_with_1 + end_with_1 - both

```


```python

def solve(N):
    """Given a positive integer N, return the total sum of its digits in binary.

    Example
        For N = 1000, the sum of digits will be 1 the output should be "1".
        For N = 150, the sum of digits will be 6 the output should be "110".
        For N = 147, the sum of digits will be 12 the output should be "1100".

    Variables:
        @N integer
             Constraints: 0 ≤ N ≤ 10000.
    Output:
         a string of binary number
    """
    digit_sum = sum(int(d) for d in str(N))
    return bin(digit_sum)[2:]

```


```python

from typing import List

def add(lst: List[int]) -> int:
    """
    Given a non-empty list of integers lst, add the even elements that are at odd indices.

    Examples:
        add([4, 2, 6, 7]) ==> 2
    """
    total = 0
    for i in range(1, len(lst), 2):  # odd indices only
        if lst[i] % 2 == 0:
            total += lst[i]
    return total

```


Here's the implementation for the `anti_shuffle` function:

```python
def anti_shuffle(s):
    """
    Write a function that takes a string and returns an ordered version of it.
    Ordered version of string, is a string where all words (separated by space)
    are replaced by a new word where all the characters arranged in
    ascending order based on ascii value.
    Note: You should keep the order of words and blank spaces in the sentence.

    For example:
    anti_shuffle('Hi') returns 'Hi'
    anti_shuffle('hello') returns 'ehllo'
    anti_shuffle('Hello World!!!') returns 'Hello !!!Wdlor'
    """
    words = s.split(' ')
    ordered_words = []
    for word in words:
        ordered_word = ''.join(sorted(word))
        ordered_words.append(ordered_word)
    return ' '.join(ordered_words)
```


Here's the implementation for the function you described:

```python
def get_row(lst, x):
    result = []
    for row_idx, row in enumerate(lst):
        for col_idx, value in enumerate(row):
            if value == x:
                result.append((row_idx, col_idx))

    # Sort by row ascending, then by column descending
    result.sort(key=lambda coord: (coord[0], -coord[1]))
    return result
```


```python

def sort_array(array):
    """
    Given an array of non-negative integers, return a copy of the given array after sorting,
    you will sort the given array in ascending order if the sum( first index value, last index value) is odd,
    or sort it in descending order if the sum( first index value, last index value) is even.

    Note:
    * don't change the given array.

    Examples:
    * sort_array([]) => []
    * sort_array([5]) => [5]
    * sort_array([2, 4, 3, 0, 1, 5]) => [0, 1, 2, 3, 4, 5]
    * sort_array([2, 4, 3, 0, 1, 5, 6]) => [6, 5, 4, 3, 2, 1, 0]
    """
    if len(array) == 0:
        return []
    if len(array) == 1:
        return [array[0]]

    first = array[0]
    last = array[-1]
    total = first + last

    result = array.copy()

    if total % 2 == 1:  # odd sum -> ascending
        result.sort()
    else:               # even sum -> descending
        result.sort(reverse=True)

    return result

```


```python

def encrypt(s):
    """Create a function encrypt that takes a string as an argument and
    returns a string encrypted with the alphabet being rotated.
    The alphabet should be rotated in a manner such that the letters
    shift down by two multiplied to two places.
    For example:
    encrypt('hi') returns 'lm'
    encrypt('asdfghjkl') returns 'ewhjklnop'
    encrypt('gf') returns 'kj'
    encrypt('et') returns 'ix'
    """
    result = []
    for ch in s:
        if 'a' <= ch <= 'z':
            # shift by 4 positions (2 * 2)
            shifted = chr((ord(ch) - ord('a') + 4) % 26 + ord('a'))
            result.append(shifted)
        elif 'A' <= ch <= 'Z':
            shifted = chr((ord(ch) - ord('A') + 4) % 26 + ord('A'))
            result.append(shifted)
        else:
            result.append(ch)
    return ''.join(result)

```


```python

def next_smallest(lst):
    """
    You are given a list of integers.
    Write a function next_smallest() that returns the 2nd smallest element of the list.
    Return None if there is no such element.

    next_smallest([1, 2, 3, 4, 5]) == 2
    next_smallest([5, 1, 4, 3, 2]) == 2
    next_smallest([]) == None
    next_smallest([1, 1]) == None
    """
    # Remove duplicates by converting to a set, then sort unique values
    unique_sorted = sorted(set(lst))
    # If there are fewer than 2 unique elements, return None
    if len(unique_sorted) < 2:
        return None
    # Return the second smallest unique element
    return unique_sorted[1]

```


Here's the implementation:

```python
def is_bored(S):
    """
    You'll be given a string of words, and your task is to count the number
    of boredoms. A boredom is a sentence that starts with the word "I".
    Sentences are delimited by '.', '?' or '!'.

    For example:
    >>> is_bored("Hello world")
    0
    >>> is_bored("The sky is blue. The sun is shining. I love this weather")
    1
    """
    import re
    sentences = re.split(r'[.!?]', S)
    count = 0
    for sentence in sentences:
        stripped = sentence.strip()
        if stripped.startswith("I ") or stripped == "I":
            count += 1
    return count
```


Here's the implementation for the function you described:

```python
def any_int(x, y, z):
    '''
    Create a function that takes 3 numbers.
    Returns true if one of the numbers is equal to the sum of the other two, and all numbers are integers.
    Returns false in any other cases.

    Examples
    any_int(5, 2, 7) ➞ True

    any_int(3, 2, 2) ➞ False

    any_int(3, -2, 1) ➞ True

    any_int(3.6, -2.2, 2) ➞ False
    '''
    # Check if all inputs are integers
    if not (isinstance(x, int) and isinstance(y, int) and isinstance(z, int)):
        return False

    # Check if one number equals the sum of the other two
    return (x == y + z) or (y == x + z) or (z == x + y)
```


Traceback (most recent call last):
  File "<string>", line 35, in <module>
    check(encode)
    ~~~~~^^^^^^^^
  File "<string>", line 26, in check
    assert candidate('TEST') == 'tgst', "This prints if this assert fails 1 (good for debugging!)"
           ^^^^^^^^^^^^^^^^^^^^^^^^^^^
AssertionError: This prints if this assert fails 1 (good for debugging!)

Here's the implementation for the `encode` function:

```python
def encode(message):
    vowels = 'aeiouAEIOU'
    result = []
    for char in message:
        if char in vowels:
            # Shift vowel 2 places ahead in the alphabet
            shifted = chr(ord(char) + 2)
            result.append(shifted)
        elif char.isalpha():
            # Swap case for non-vowel letters
            if char.islower():
                result.append(char.upper())
            else:
                result.append(char.lower())
        else:
            # Keep non-letter characters as is
            result.append(char)
    return ''.join(result)
```


Here's the implementation:

```python
def skjkasdkd(lst):
    def is_prime(n):
        if n < 2:
            return False
        for i in range(2, int(n ** 0.5) + 1):
            if n % i == 0:
                return False
        return True

    largest_prime = 0
    for num in lst:
        if is_prime(num) and num > largest_prime:
            largest_prime = num

    if largest_prime == 0:
        return 0

    return sum(int(digit) for digit in str(largest_prime))
```


```python

def check_dict_case(dict):
    """
    Given a dictionary, return True if all keys are strings in lower
    case or all keys are strings in upper case, else return False.
    The function should return False is the given dictionary is empty.
    """
    if not dict:
        return False

    lower_case = True
    upper_case = True

    for key in dict.keys():
        if not isinstance(key, str):
            lower_case = False
            upper_case = False
            break
        if not key.islower():
            lower_case = False
        if not key.isupper():
            upper_case = False
        # If both are already False, no need to continue
        if not lower_case and not upper_case:
            break

    return lower_case or upper_case

```


```python

def count_up_to(n):
    """Implement a function that takes an non-negative integer and returns an array of the first n
    integers that are prime numbers and less than n.
    for example:
    count_up_to(5) => [2,3]
    count_up_to(11) => [2,3,5,7]
    count_up_to(0) => []
    count_up_to(20) => [2,3,5,7,11,13,17,19]
    count_up_to(1) => []
    count_up_to(18) => [2,3,5,7,11,13,17]
    """
    def is_prime(num):
        if num < 2:
            return False
        for i in range(2, int(num ** 0.5) + 1):
            if num % i == 0:
                return False
        return True

    result = []
    for candidate in range(2, n):
        if is_prime(candidate):
            result.append(candidate)
    return result

```


```python
def multiply(a, b):
    """Complete the function that takes two integers and returns
    the product of their unit digits.
    Assume the input is always valid.
    Examples:
    multiply(148, 412) should return 16.
    multiply(19, 28) should return 72.
    multiply(2020, 1851) should return 0.
    multiply(14,-15) should return 20.
    """
    # Get the unit digit of a (absolute value to handle negatives)
    unit_a = abs(a) % 10
    unit_b = abs(b) % 10
    return unit_a * unit_b
```


Here's the implementation:

```python
def count_upper(s):
    """
    Given a string s, count the number of uppercase vowels in even indices.

    For example:
    count_upper('aBCdEf') returns 1
    count_upper('abcdefg') returns 0
    count_upper('dBBE') returns 0
    """
    vowels = set('AEIOU')
    count = 0
    for i, ch in enumerate(s):
        if i % 2 == 0 and ch in vowels:
            count += 1
    return count
```


Here's the implementation for the function you described:

```python
def closest_integer(value):
    '''
    Create a function that takes a value (string) representing a number
    and returns the closest integer to it. If the number is equidistant
    from two integers, round it away from zero.

    Examples
    >>> closest_integer("10")
    10
    >>> closest_integer("15.3")
    15

    Note:
    Rounding away from zero means that if the given number is equidistant
    from two integers, the one you should return is the one that is the
    farthest from zero. For example closest_integer("14.5") should
    return 15 and closest_integer("-14.5") should return -15.
    '''
    from decimal import Decimal, ROUND_HALF_UP
    num = Decimal(value)
    rounded = num.to_integral_value(rounding=ROUND_HALF_UP)
    return int(rounded)
```


Here's the implementation for the function you described:

```python
def make_a_pile(n):
    """
    Given a positive integer n, you have to make a pile of n levels of stones.
    The first level has n stones.
    The number of stones in the next level is:
        - the next odd number if n is odd.
        - the next even number if n is even.
    Return the number of stones in each level in a list, where element at index
    i represents the number of stones in the level (i+1).

    Examples:
    >>> make_a_pile(3)
    [3, 5, 7]
    """
    result = []
    current = n
    for _ in range(n):
        result.append(current)
        # Determine parity of the original n to decide increment
        if n % 2 == 0:
            current += 2
        else:
            current += 2
    return result
```


```python

import re

def words_string(s):
    """
    Splits a string of words separated by commas or spaces into a list of words.
    """
    # Split on commas or spaces (one or more)
    words = re.split(r'[,\s]+', s.strip())
    # Remove any empty strings that might result from leading/trailing separators
    return [w for w in words if w]

```


```python

def choose_num(x, y):
    """This function takes two positive numbers x and y and returns the
    biggest even integer number that is in the range [x, y] inclusive. If
    there's no such number, then the function should return -1.

    For example:
    choose_num(12, 15) = 14
    choose_num(13, 12) = -1
    """
    if x > y:
        return -1
    # Start from the largest number in the range and go downwards
    for num in range(y, x - 1, -1):
        if num % 2 == 0:
            return num
    return -1

```


Here's the implementation for the function you described:

```python
def rounded_avg(n, m):
    """You are given two positive integers n and m, and your task is to compute the
    average of the integers from n through m (including n and m).
    Round the answer to the nearest integer and convert that to binary.
    If n is greater than m, return -1.
    Example:
    rounded_avg(1, 5) => "0b11"
    rounded_avg(7, 5) => -1
    rounded_avg(10, 20) => "0b1111"
    rounded_avg(20, 33) => "0b11010"
    """
    if n > m:
        return -1

    total = 0
    count = 0
    for i in range(n, m + 1):
        total += i
        count += 1

    avg = total / count
    rounded = round(avg)
    return bin(rounded)
```


```python

def unique_digits(x):
    """Given a list of positive integers x. return a sorted list of all
    elements that hasn't any even digit.

    Note: Returned list should be sorted in increasing order.

    For example:
    >>> unique_digits([15, 33, 1422, 1])
    [1, 15, 33]
    >>> unique_digits([152, 323, 1422, 10])
    []
    """
    result = []
    for num in x:
        if all(int(d) % 2 != 0 for d in str(num)):
            result.append(num)
    return sorted(result)

```


Here's the implementation for the function you described:

```python
def by_length(arr):
    """
    Given an array of integers, sort the integers that are between 1 and 9 inclusive,
    reverse the resulting array, and then replace each digit by its corresponding name from
    "One", "Two", "Three", "Four", "Five", "Six", "Seven", "Eight", "Nine".
    """
    # Filter numbers between 1 and 9 inclusive
    filtered = [x for x in arr if 1 <= x <= 9]

    # Sort the filtered list
    filtered.sort()

    # Reverse the sorted list
    filtered.reverse()

    # Map numbers to their word equivalents
    num_to_word = {
        1: "One",
        2: "Two",
        3: "Three",
        4: "Four",
        5: "Five",
        6: "Six",
        7: "Seven",
        8: "Eight",
        9: "Nine"
    }

    # Convert each number to its word form
    result = [num_to_word[num] for num in filtered]

    return result
```


Traceback (most recent call last):
  File "<string>", line 32, in <module>
    check(f)
    ~~~~~^^^
  File "<string>", line 27, in check
    assert candidate(5) == [1, 2, 6, 24, 15]
           ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
AssertionError

```python

def f(n):
    def factorial(k):
        result = 1
        for i in range(1, k + 1):
            result *= i
        return result

    def sum_to(k):
        total = 0
        for i in range(1, k + 1):
            total += i
        return total

    result = []
    for i in range(1, n + 1):
        if i % 2 == 0:
            result.append(sum_to(i))
        else:
            result.append(factorial(i))
    return result

```


```python

def even_odd_palindrome(n):
    """
    Given a positive integer n, return a tuple that has the number of even and odd
    integer palindromes that fall within the range(1, n), inclusive.
    """
    even_count = 0
    odd_count = 0
    for num in range(1, n + 1):
        s = str(num)
        if s == s[::-1]:
            if num % 2 == 0:
                even_count += 1
            else:
                odd_count += 1
    return (even_count, odd_count)

```


Here's the implementation for the function you described:

```python
def count_nums(arr):
    """
    Write a function count_nums which takes an array of integers and returns
    the number of elements which has a sum of digits > 0.
    If a number is negative, then its first signed digit will be negative:
    e.g. -123 has signed digits -1, 2, and 3.
    >>> count_nums([]) == 0
    >>> count_nums([-1, 11, -11]) == 1
    >>> count_nums([1, 1, 2]) == 3
    """
    count = 0
    for num in arr:
        if num == 0:
            continue
        if num > 0:
            digits = list(map(int, str(num)))
            if sum(digits) > 0:
                count += 1
        else:
            # Negative number: treat first digit as negative
            num_str = str(num)[1:]  # remove the minus sign
            digits = list(map(int, num_str))
            digits[0] = -digits[0]
            if sum(digits) > 0:
                count += 1
    return count
```


```python

def move_one_ball(arr):
    """
    Determines if the array can be sorted in non-decreasing order by performing
    any number of right shift operations.
    """
    if not arr:
        return True

    n = len(arr)
    # Find the number of "break points" where arr[i] > arr[(i+1) % n]
    # In a sorted array after some rotations, there should be at most one such break.
    breaks = 0
    for i in range(n):
        if arr[i] > arr[(i + 1) % n]:
            breaks += 1
            if breaks > 1:
                return False
    return True

```


```python

def exchange(lst1, lst2):
    """In this problem, you will implement a function that takes two lists of numbers,
    and determines whether it is possible to perform an exchange of elements
    between them to make lst1 a list of only even numbers.
    There is no limit on the number of exchanged elements between lst1 and lst2.
    If it is possible to exchange elements between the lst1 and lst2 to make
    all the elements of lst1 to be even, return "YES".
    Otherwise, return "NO".
    For example:
    exchange([1, 2, 3, 4], [1, 2, 3, 4]) => "YES"
    exchange([1, 2, 3, 4], [1, 5, 3, 4]) => "NO"
    It is assumed that the input lists will be non-empty.
    """
    odd_in_lst1 = [x for x in lst1 if x % 2 != 0]
    even_in_lst2 = [x for x in lst2 if x % 2 == 0]

    if len(odd_in_lst1) <= len(even_in_lst2):
        return "YES"
    else:
        return "NO"

```


Here's the implementation for the `histogram` function:

```python
def histogram(test):
    """Given a string representing a space separated lowercase letters, return a dictionary
    of the letter with the most repetition and containing the corresponding count.
    If several letters have the same occurrence, return all of them.

    Example:
    histogram('a b c') == {'a': 1, 'b': 1, 'c': 1}
    histogram('a b b a') == {'a': 2, 'b': 2}
    histogram('a b c a b') == {'a': 2, 'b': 2}
    histogram('b b b b a') == {'b': 4}
    histogram('') == {}

    """
    if not test:
        return {}

    counts = {}
    for char in test.split():
        counts[char] = counts.get(char, 0) + 1

    max_count = max(counts.values())
    return {char: count for char, count in counts.items() if count == max_count}
```


Here's the implementation for the function:

```python
def reverse_delete(s, c):
    # Remove characters from s that are in c
    result = ''.join(ch for ch in s if ch not in c)
    # Check if the result string is a palindrome
    is_palindrome = result == result[::-1]
    return (result, is_palindrome)
```


Here's the implementation for the function you described:

```python
def odd_count(lst):
    """Given a list of strings, where each string consists of only digits, return a list.
    Each element i of the output should be "the number of odd elements in the
    string i of the input." where all the i's should be replaced by the number
    of odd digits in the i'th string of the input.

    >>> odd_count(['1234567'])
    ["the number of odd elements 4n the str4ng 4 of the 4nput."]
    >>> odd_count(['3',"11111111"])
    ["the number of odd elements 1n the str1ng 1 of the 1nput.",
     "the number of odd elements 8n the str8ng 8 of the 8nput."]
    """
    result = []
    for s in lst:
        count = sum(1 for ch in s if int(ch) % 2 != 0)
        result.append(
            f"the number of odd elements {count}n the str{count}ng {count} of the {count}nput."
        )
    return result
```


```python

def minSubArraySum(nums):
    """
    Given an array of integers nums, find the minimum sum of any non-empty sub-array
    of nums.
    Example
    minSubArraySum([2, 3, 4, 1, 2, 4]) == 1
    minSubArraySum([-1, -2, -3]) == -6
    """
    min_ending_here = nums[0]
    min_so_far = nums[0]
    for i in range(1, len(nums)):
        min_ending_here = min(nums[i], min_ending_here + nums[i])
        min_so_far = min(min_so_far, min_ending_here)
    return min_so_far

```


```python

def max_fill(grid, capacity):
    import math
    """
    You are given a rectangular grid of wells. Each row represents a single well,
    and each 1 in a row represents a single unit of water.
    Each well has a corresponding bucket that can be used to extract water from it,
    and all buckets have the same capacity.
    Your task is to use the buckets to empty the wells.
    Output the number of times you need to lower the buckets.

    Example 1:
        Input:
            grid : [[0,0,1,0], [0,1,0,0], [1,1,1,1]]
            bucket_capacity : 1
        Output: 6

    Example 2:
        Input:
            grid : [[0,0,1,1], [0,0,0,0], [1,1,1,1], [0,1,1,1]]
            bucket_capacity : 2
        Output: 5

    Example 3:
        Input:
            grid : [[0,0,0], [0,0,0]]
            bucket_capacity : 5
        Output: 0

    Constraints:
        * all wells have the same length
        * 1 <= grid.length <= 10^2
        * 1 <= grid[:,1].length <= 10^2
        * grid[i][j] -> 0 | 1
        * 1 <= capacity <= 10
    """
    total_bucket_lowerings = 0
    for row in grid:
        water_units = sum(row)
        if water_units > 0:
            total_bucket_lowerings += math.ceil(water_units / capacity)
    return total_bucket_lowerings

```


```python

def sort_array(arr):
    """
    Sort an array of integers according to number of ones in their binary
    representation in ascending order. For similar number of ones, sort based
    on decimal value.
    """
    def count_ones(x):
        # Count ones in binary representation for non-negative numbers
        # For negative numbers, use the two's complement representation
        # (32-bit signed integer)
        if x >= 0:
            return bin(x).count('1')
        else:
            # For negative numbers, treat as 32-bit two's complement
            return bin(x & 0xFFFFFFFF).count('1')

    return sorted(arr, key=lambda x: (count_ones(x), x))

```


Here's the implementation for the function you described:

```python
def select_words(s, n):
    """Given a string s and a natural number n, you have been tasked to implement
    a function that returns a list of all words from string s that contain exactly
    n consonants, in order these words appear in the string s.
    If the string s is empty then the function should return an empty list.
    Note: you may assume the input string contains only letters and spaces.
    Examples:
    select_words("Mary had a little lamb", 4) ==> ["little"]
    select_words("Mary had a little lamb", 3) ==> ["Mary", "lamb"]
    select_words("simple white space", 2) ==> []
    select_words("Hello world", 4) ==> ["world"]
    select_words("Uncle sam", 3) ==> ["Uncle"]
    """
    vowels = set("aeiouAEIOU")
    result = []
    words = s.split()
    for word in words:
        consonant_count = sum(1 for ch in word if ch.isalpha() and ch not in vowels)
        if consonant_count == n:
            result.append(word)
    return result
```


Here's the implementation for the function you described:

```python
def get_closest_vowel(word):
    vowels = set("aeiouAEIOU")

    # Find the rightmost vowel that is between two consonants
    for i in range(len(word) - 2, 0, -1):
        if word[i] in vowels:
            # Check if the character before and after are consonants
            if word[i-1].lower() not in "aeiou" and word[i+1].lower() not in "aeiou":
                return word[i]

    return ""
```


Here's the implementation for the function you described:

```python
def match_parens(lst):
    '''
    You are given a list of two strings, both strings consist of open
    parentheses '(' or close parentheses ')' only.
    Your job is to check if it is possible to concatenate the two strings in
    some order, that the resulting string will be good.
    A string S is considered to be good if and only if all parentheses in S
    are balanced. For example: the string '(())()' is good, while the string
    '())' is not.
    Return 'Yes' if there's a way to make a good string, and return 'No' otherwise.

    Examples:
    match_parens(['()(', ')']) == 'Yes'
    match_parens([')', ')']) == 'No'
    '''
    def is_balanced(s):
        bal = 0
        for ch in s:
            if ch == '(':
                bal += 1
            else:
                bal -= 1
            if bal < 0:
                return False
        return bal == 0

    # Try both possible concatenations
    for order in [(0, 1), (1, 0)]:
        combined = lst[order[0]] + lst[order[1]]
        if is_balanced(combined):
            return 'Yes'
    return 'No'
```


Here's the implementation for the `maximum` function:

```python
def maximum(arr, k):
    """
    Given an array arr of integers and a positive integer k, return a sorted list
    of length k with the maximum k numbers in arr.
    """
    arr.sort()
    return arr[-k:] if k > 0 else []
```


```python
def solution(lst):
    """Given a non-empty list of integers, return the sum of all of the odd elements that are in even positions."""
    total = 0
    for i in range(len(lst)):
        if i % 2 == 0 and lst[i] % 2 != 0:
            total += lst[i]
    return total
```


```python

def add_elements(arr, k):
    """
    Given a non-empty array of integers arr and an integer k, return
    the sum of the elements with at most two digits from the first k elements of arr.
    """
    total = 0
    for i in range(k):
        # Check if the number has at most two digits (i.e., between -99 and 99 inclusive)
        if -99 <= arr[i] <= 99:
            total += arr[i]
    return total

```


Here's the implementation for the function you described:

```python
def get_odd_collatz(n):
    """
    Given a positive integer n, return a sorted list that has the odd numbers in collatz sequence.

    The Collatz conjecture is a conjecture in mathematics that concerns a sequence defined
    as follows: start with any positive integer n. Then each term is obtained from the
    previous term as follows: if the previous term is even, the next term is one half of
    the previous term. If the previous term is odd, the next term is 3 times the previous
    term plus 1. The conjecture is that no matter what value of n, the sequence will always reach 1.

    Note:
        1. Collatz(1) is [1].
        2. returned list sorted in increasing order.

    For example:
    get_odd_collatz(5) returns [1, 5] # The collatz sequence for 5 is [5, 16, 8, 4, 2, 1], so the odd numbers are only 1, and 5.
    """
    sequence = []
    current = n
    while current != 1:
        if current % 2 == 1:
            sequence.append(current)
        if current % 2 == 0:
            current //= 2
        else:
            current = 3 * current + 1
    sequence.append(1)  # include the final 1
    sequence.sort()
    return sequence
```


Here's the complete implementation for the `valid_date` function:

```python
import re

def valid_date(date):
    """You have to write a function which validates a given date string and
    returns True if the date is valid otherwise False.
    The date is valid if all of the following rules are satisfied:
    1. The date string is not empty.
    2. The number of days is not less than 1 or higher than 31 days for months 1,3,5,7,8,10,12. And the number of days is not less than 1 or higher than 30 days for months 4,6,9,11. And, the number of days is not less than 1 or higher than 29 for the month 2.
    3. The months should not be less than 1 or higher than 12.
    4. The date should be in the format: mm-dd-yyyy

    for example:
    valid_date('03-11-2000') => True

    valid_date('15-01-2012') => False

    valid_date('04-0-2040') => False

    valid_date('06-04-2020') => True

    valid_date('06/04/2020') => False
    """
    if not date:
        return False

    # Check format: mm-dd-yyyy
    if not re.match(r'^\d{2}-\d{2}-\d{4}$', date):
        return False

    parts = date.split('-')
    month = int(parts[0])
    day = int(parts[1])
    year = int(parts[2])

    # Month validation
    if month < 1 or month > 12:
        return False

    # Day validation based on month
    days_in_month = {
        1: 31, 2: 29, 3: 31, 4: 30, 5: 31, 6: 30,
        7: 31, 8: 31, 9: 30, 10: 31, 11: 30, 12: 31
    }

    if day < 1 or day > days_in_month.get(month, 0):
        return False

    return True
```


Here's the implementation for the function you described:

```python
def split_words(txt):
    """
    Given a string of words, return a list of words split on whitespace, if no whitespace exists in the text you
    should split on commas ',' if no commas exist you should return the number of lower-case letters with odd order in the
    alphabet, ord('a') = 0, ord('b') = 1, ... ord('z') = 25
    """
    if ' ' in txt:
        return txt.split()
    if ',' in txt:
        return txt.split(',')
    # No whitespace and no commas: count lowercase letters with odd alphabetical order
    count = 0
    for ch in txt:
        if 'a' <= ch <= 'z':
            if (ord(ch) - ord('a')) % 2 == 1:
                count += 1
    return count
```


Traceback (most recent call last):
  File "<string>", line 46, in <module>
    check(is_sorted)
    ~~~~~^^^^^^^^^^^
  File "<string>", line 42, in check
    assert candidate([1, 2, 2, 3, 3, 4]) == True, "This prints if this assert fails 7 (good for debugging!)"
           ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
AssertionError: This prints if this assert fails 7 (good for debugging!)

Here's the implementation for the function you described:

```python
def is_sorted(lst):
    """
    Given a list of numbers, return whether or not they are sorted
    in ascending order. If list has more than 1 duplicate of the same
    number, return False. Assume no negative numbers and only integers.
    """
    if len(lst) <= 1:
        return True

    for i in range(1, len(lst)):
        if lst[i] < lst[i - 1]:
            return False

    # Check for duplicates appearing more than once
    seen = set()
    for num in lst:
        if num in seen:
            return False
        seen.add(num)

    return True
```


Traceback (most recent call last):
  File "<string>", line 41, in <module>
    check(intersection)
    ~~~~~^^^^^^^^^^^^^^
  File "<string>", line 30, in check
    assert candidate((-1, 1), (0, 4)) == "NO"
           ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
AssertionError

Here's the implementation:

```python
def intersection(interval1, interval2):
    # Determine the intersection range
    start = max(interval1[0], interval2[0])
    end = min(interval1[1], interval2[1])

    # If no overlap, return "NO"
    if start > end:
        return "NO"

    # Length of the intersection
    length = end - start + 1

    # Check if length is prime
    if length < 2:
        return "NO"

    for i in range(2, int(length ** 0.5) + 1):
        if length % i == 0:
            return "NO"

    return "YES"
```


```python

def prod_signs(arr):
    """
    You are given an array arr of integers and you need to return
    sum of magnitudes of integers multiplied by product of all signs
    of each number in the array, represented by 1, -1 or 0.
    Note: return None for empty arr.

    Example:
    >>> prod_signs([1, 2, 2, -4]) == -9
    >>> prod_signs([0, 1]) == 0
    >>> prod_signs([]) == None
    """
    if not arr:
        return None

    sum_magnitudes = 0
    sign_product = 1

    for num in arr:
        sum_magnitudes += abs(num)
        if num > 0:
            sign_product *= 1
        elif num < 0:
            sign_product *= -1
        else:
            sign_product *= 0

    return sum_magnitudes * sign_product

```


```python

import heapq

def minPath(grid, k):
    n = len(grid)
    # Directions: up, down, left, right
    dirs = [(-1, 0), (1, 0), (0, -1), (0, 1)]

    # For each cell, we store the smallest lexicographic path of length up to k
    # dp[r][c][l] = list of values for the best path of length l ending at (r,c)
    # We'll build incrementally.

    # Initialize dp for length 1: just the cell itself
    dp = [[[None for _ in range(k+1)] for _ in range(n)] for _ in range(n)]
    for r in range(n):
        for c in range(n):
            dp[r][c][1] = [grid[r][c]]

    # Build for lengths 2..k
    for length in range(2, k+1):
        for r in range(n):
            for c in range(n):
                best = None
                for dr, dc in dirs:
                    nr, nc = r + dr, c + dc
                    if 0 <= nr < n and 0 <= nc < n:
                        prev_path = dp[nr][nc][length-1]
                        if prev_path is not None:
                            candidate = prev_path + [grid[r][c]]
                            if best is None or candidate < best:
                                best = candidate
                dp[r][c][length] = best

    # Find the overall best path of length k
    overall_best = None
    for r in range(n):
        for c in range(n):
            path = dp[r][c][k]
            if path is not None:
                if overall_best is None or path < overall_best:
                    overall_best = path
    return overall_best

```


Traceback (most recent call last):
  File "<string>", line 39, in <module>
    check(tri)
    ~~~~~^^^^^
  File "<string>", line 26, in check
    assert candidate(3) == [1, 3, 2.0, 8.0]
           ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
AssertionError

Here's the implementation for the Tribonacci sequence as described:

```python
def tri(n):
    def compute_tri(k, memo):
        if k in memo:
            return memo[k]
        if k == 1:
            memo[k] = 3
        elif k % 2 == 0:
            memo[k] = 1 + k // 2
        else:  # k is odd
            memo[k] = compute_tri(k - 1, memo) + compute_tri(k - 2, memo) + compute_tri(k + 1, memo)
        return memo[k]

    memo = {}
    result = []
    for i in range(1, n + 1):
        result.append(compute_tri(i, memo))
    return result
```


```python

def digits(n):
    """Given a positive integer n, return the product of the odd digits.
    Return 0 if all digits are even.
    For example:
    digits(1)  == 1
    digits(4)  == 0
    digits(235) == 15
    """
    product = 1
    found_odd = False
    while n > 0:
        digit = n % 10
        if digit % 2 == 1:
            product *= digit
            found_odd = True
        n //= 10
    return product if found_odd else 0

```


Traceback (most recent call last):
  File "<string>", line 55, in <module>
    check(is_nested)
    ~~~~~^^^^^^^^^^^
  File "<string>", line 38, in check
    assert candidate('[]]]]]]][[[[[]') == False
           ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
AssertionError

Here's the implementation for the function you described:

```python

def is_nested(string):
    '''
    Create a function that takes a string as input which contains only square brackets.
    The function should return True if and only if there is a valid subsequence of brackets
    where at least one bracket in the subsequence is nested.

    is_nested('[[]]') ➞ True
    is_nested('[]]]]]]][[[[[]') ➞ False
    is_nested('[][]') ➞ False
    is_nested('[]') ➞ False
    is_nested('[[][]]') ➞ True
    is_nested('[[]][[') ➞ True
    '''
    # Track the depth of valid bracket subsequences
    depth = 0
    max_depth = 0

    for char in string:
        if char == '[':
            depth += 1
            max_depth = max(max_depth, depth)
        elif char == ']':
            if depth > 0:
                depth -= 1

    # A valid subsequence with nesting means at least one '[' is matched
    # with a later ']' while another '[' is opened inside before closing.
    # This is equivalent to having a maximum depth >= 2 at some point.
    return max_depth >= 2

```


Here's the implementation:

```python
import math

def sum_squares(lst):
    """You are given a list of numbers.
    You need to return the sum of squared numbers in the given list,
    round each element in the list to the upper int(Ceiling) first.
    Examples:
    For lst = [1,2,3] the output should be 14
    For lst = [1,4,9] the output should be 98
    For lst = [1,3,5,7] the output should be 84
    For lst = [1.4,4.2,0] the output should be 29
    For lst = [-2.4,1,1] the output should be 6
    """
    total = 0
    for num in lst:
        ceil_val = math.ceil(num)
        total += ceil_val * ceil_val
    return total
```


Here's the implementation for the function:

```python
def check_if_last_char_is_a_letter(txt):
    """
    Returns True if the last character of a given string is an alphabetical character
    and is not part of a word (i.e., it is a standalone letter separated by spaces),
    and False otherwise.
    """
    if not txt or txt[-1] == ' ':
        return False

    # Check if the last character is an alphabetical letter
    if not txt[-1].isalpha():
        return False

    # Check if it's part of a word: if there's a preceding character that is not a space,
    # then it's part of a word.
    # If the string has length 1, it's a standalone letter.
    if len(txt) == 1:
        return True

    # If the character before the last is a space, then the last char is standalone
    return txt[-2] == ' '
```


```python
def can_arrange(arr):
    """Create a function which returns the largest index of an element which
    is not greater than or equal to the element immediately preceding it. If
    no such element exists then return -1. The given array will not contain
    duplicate values.

    Examples:
    can_arrange([1,2,4,3,5]) = 3
    can_arrange([1,2,3]) = -1
    """
    if len(arr) < 2:
        return -1

    result = -1
    for i in range(1, len(arr)):
        if arr[i] < arr[i-1]:
            result = i
    return result
```


Here's the implementation for the function you described:

```python
def largest_smallest_integers(lst):
    '''
    Create a function that returns a tuple (a, b), where 'a' is
    the largest of negative integers, and 'b' is the smallest
    of positive integers in a list.
    If there is no negative or positive integers, return them as None.

    Examples:
    largest_smallest_integers([2, 4, 1, 3, 5, 7]) == (None, 1)
    largest_smallest_integers([]) == (None, None)
    largest_smallest_integers([0]) == (None, None)
    '''
    largest_negative = None
    smallest_positive = None

    for num in lst:
        if num < 0:
            if largest_negative is None or num > largest_negative:
                largest_negative = num
        elif num > 0:
            if smallest_positive is None or num < smallest_positive:
                smallest_positive = num

    return (largest_negative, smallest_positive)
```


Here's the implementation for the function you described:

```python
def compare_one(a, b):
    """
    Create a function that takes integers, floats, or strings representing
    real numbers, and returns the larger variable in its given variable type.
    Return None if the values are equal.
    Note: If a real number is represented as a string, the floating point might be . or ,

    compare_one(1, 2.5) ➞ 2.5
    compare_one(1, "2,3") ➞ "2,3"
    compare_one("5,1", "6") ➞ "6"
    compare_one("1", 1) ➞ None
    """
    def to_number(val):
        if isinstance(val, (int, float)):
            return float(val)
        if isinstance(val, str):
            # Replace comma with dot for parsing
            normalized = val.replace(',', '.')
            return float(normalized)
        return None

    num_a = to_number(a)
    num_b = to_number(b)

    if num_a is None or num_b is None:
        return None

    if abs(num_a - num_b) < 1e-12:
        return None

    if num_a > num_b:
        return a
    else:
        return b
```


Here's the implementation:

```python
def is_equal_to_sum_even(n):
    """Evaluate whether the given number n can be written as the sum of exactly 4 positive even numbers
    Example
    is_equal_to_sum_even(4) == False
    is_equal_to_sum_even(6) == False
    is_equal_to_sum_even(8) == True
    """
    # The smallest sum of 4 positive even numbers is 2+2+2+2 = 8
    # Any even number >= 8 can be written as sum of 4 positive even numbers
    # Because we can adjust by adding 2 to one term and subtracting 2 from another
    # So condition: n must be even and >= 8
    return n >= 8 and n % 2 == 0
```


```python

def special_factorial(n):
    """The Brazilian factorial is defined as:
    brazilian_factorial(n) = n! * (n-1)! * (n-2)! * ... * 1!
    where n > 0

    For example:
    >>> special_factorial(4)
    288

    The function will receive an integer as input and should return the special
    factorial of this integer.
    """
    # Compute factorial iteratively and accumulate product
    result = 1
    current_factorial = 1
    for i in range(1, n + 1):
        current_factorial *= i
        result *= current_factorial
    return result

```


Here's the implementation for the `fix_spaces` function:

```python
def fix_spaces(text):
    """
    Given a string text, replace all spaces in it with underscores,
    and if a string has more than 2 consecutive spaces,
    then replace all consecutive spaces with -

    fix_spaces("Example") == "Example"
    fix_spaces("Example 1") == "Example_1"
    fix_spaces(" Example 2") == "_Example_2"
    fix_spaces(" Example   3") == "_Example-3"
    """
    result = []
    i = 0
    while i < len(text):
        if text[i] == ' ':
            # Count consecutive spaces
            j = i
            while j < len(text) and text[j] == ' ':
                j += 1
            space_count = j - i
            if space_count > 2:
                result.append('-')
            else:
                for _ in range(space_count):
                    result.append('_')
            i = j
        else:
            result.append(text[i])
            i += 1
    return ''.join(result)
```


Here's the implementation for the function you described:

```python
def file_name_check(file_name):
    """Create a function which takes a string representing a file's name, and returns
    'Yes' if the the file's name is valid, and returns 'No' otherwise.
    A file's name is considered to be valid if and only if all the following conditions
    are met:
    - There should not be more than three digits ('0'-'9') in the file's name.
    - The file's name contains exactly one dot '.'
    - The substring before the dot should not be empty, and it starts with a letter from
    the latin alphapet ('a'-'z' and 'A'-'Z').
    - The substring after the dot should be one of these: ['txt', 'exe', 'dll']
    Examples:
    file_name_check("example.txt") # => 'Yes'
    file_name_check("1example.dll") # => 'No' (the name should start with a latin alphapet letter)
    """
    # Count digits
    digit_count = sum(1 for ch in file_name if ch.isdigit())
    if digit_count > 3:
        return 'No'

    # Check exactly one dot
    if file_name.count('.') != 1:
        return 'No'

    # Split into parts
    parts = file_name.split('.')
    before_dot = parts[0]
    after_dot = parts[1]

    # Check before dot is not empty and starts with a letter
    if not before_dot or not before_dot[0].isalpha():
        return 'No'

    # Check after dot is one of the allowed extensions
    if after_dot not in ['txt', 'exe', 'dll']:
        return 'No'

    return 'Yes'
```


Here's the implementation for the function you described:

```python
def sum_squares(lst):
    """
    This function will take a list of integers. For all entries in the list, the function shall square the integer entry if its index is a
    multiple of 3 and will cube the integer entry if its index is a multiple of 4 and not a multiple of 3. The function will not
    change the entries in the list whose indexes are not a multiple of 3 or 4. The function shall then return the sum of all entries.

    Examples:
    For lst = [1,2,3] the output should be 6
    For lst = []  the output should be 0
    For lst = [-1,-5,2,-1,-5]  the output should be -126
    """
    total = 0
    for i, val in enumerate(lst):
        if i % 3 == 0:
            total += val * val
        elif i % 4 == 0:
            total += val * val * val
        else:
            total += val
    return total
```


Here's the implementation:

```python
def words_in_sentence(sentence):
    def is_prime(n):
        if n < 2:
            return False
        for i in range(2, int(n ** 0.5) + 1):
            if n % i == 0:
                return False
        return True

    words = sentence.split()
    result_words = [word for word in words if is_prime(len(word))]
    return " ".join(result_words)
```


Here's the implementation for the `simplify` function:

```python
from fractions import Fraction

def simplify(x, n):
    """Your task is to implement a function that will simplify the expression
    x * n. The function returns True if x * n evaluates to a whole number and False
    otherwise. Both x and n, are string representation of a fraction, and have the following format,
    <numerator>/<denominator> where both numerator and denominator are positive whole numbers.

    You can assume that x, and n are valid fractions, and do not have zero as denominator.

    simplify("1/5", "5/1") = True
    simplify("1/6", "2/1") = False
    simplify("7/10", "10/2") = False
    """
    frac_x = Fraction(x)
    frac_n = Fraction(n)
    product = frac_x * frac_n
    return product.denominator == 1
```


Here's the implementation for the function you described:

```python
def order_by_points(nums):
    def digit_sum(n):
        return sum(int(d) for d in str(abs(n)))

    # Create list of (digit_sum, original_index, value) tuples
    decorated = [(digit_sum(nums[i]), i, nums[i]) for i in range(len(nums))]

    # Sort by digit_sum first, then by original index
    decorated.sort(key=lambda x: (x[0], x[1]))

    # Extract values in sorted order
    return [x[2] for x in decorated]
```


```python

def specialFilter(nums):
    """
    Returns the count of numbers in the list that are greater than 10 and
    whose first and last digits are odd.
    """
    count = 0
    for num in nums:
        if num > 10:
            # Convert to positive string to handle negative numbers
            s = str(abs(num))
            first_digit = int(s[0])
            last_digit = int(s[-1])
            if first_digit % 2 == 1 and last_digit % 2 == 1:
                count += 1
    return count

```


We need to compute the number of triples (i, j, k) with i < j < k such that the sum of the three values a[i] + a[j] + a[k] is divisible by 3, where a[i] = i*i - i + 1.

First, let’s analyze the expression modulo 3:

For i mod 3:
- If i ≡ 0 (mod 3):
  i² ≡ 0, -i ≡ 0, so a[i] ≡ 1 (mod 3)
- If i ≡ 1 (mod 3):
  i² ≡ 1, -i ≡ -1 ≡ 2, so a[i] ≡ 1 + 2 + 1 ≡ 4 ≡ 1 (mod 3)
- If i ≡ 2 (mod 3):
  i² ≡ 4 ≡ 1, -i ≡ -2 ≡ 1, so a[i] ≡ 1 + 1 + 1 ≡ 3 ≡ 0 (mod 3)

So:
- i ≡ 0 or 1 mod 3 → a[i] ≡ 1 mod 3
- i ≡ 2 mod 3 → a[i] ≡ 0 mod 3

We want triples where sum ≡ 0 mod 3.
Let’s count how many indices fall into each residue class for a given n.

Let:
- c0 = number of i in [1..n] with i ≡ 0 mod 3 → floor(n/3)
- c1 = number with i ≡ 1 mod 3 → ceil(n/3) or (n+2)//3
- c2 = number with i ≡ 2 mod 3 → floor((n+1)/3)

But careful: i starts at 1, so:
- i ≡ 1 mod 3: indices 1, 4, 7, ... → count = (n+2)//3
- i ≡ 2 mod 3: indices 2, 5, 8, ... → count = (n+1)//3
- i ≡ 0 mod 3: indices 3, 6, 9, ... → count = n//3

Now a[i] mod 3:
- i ≡ 0 or 1 → a ≡ 1
- i ≡ 2 → a ≡ 0

So we have two groups:
- Group A (a ≡ 1): count = c0 + c1
- Group B (a ≡ 0): count = c2

We need triples (i<j<k) with sum ≡ 0 mod 3.
Possible combinations of residues (1,1,1) → sum ≡ 0 mod 3? 1+1+1=3 ≡ 0 → yes.
(1,1,0) → 2 mod 3 → no.
(1,0,0) → 1 mod 3 → no.
(0,0,0) → 0 mod 3 → yes.

So only triples where all three are from group A, or all three from group B.

Number of triples from group A: choose 3 from countA = c0+c1.
Number from group B: choose 3 from countB = c2.

Total = C(countA, 3) + C(countB, 3).

We can compute using integer arithmetic.

Let’s implement:

```python
def get_max_triples(n):
    c0 = n // 3
    c1 = (n + 2) // 3
    c2 = (n + 1) // 3
    countA = c0 + c1
    countB = c2

    def choose3(x):
        if x < 3:
            return 0
        return x * (x-1) * (x-2) // 6

    return choose3(countA) + choose3(countB)
```


Here's the implementation:

```python
def bf(planet1, planet2):
    planets = ["Mercury", "Venus", "Earth", "Mars", "Jupiter", "Saturn", "Uranus", "Neptune"]

    if planet1 not in planets or planet2 not in planets:
        return ()

    idx1 = planets.index(planet1)
    idx2 = planets.index(planet2)

    if idx1 < idx2:
        start = idx1 + 1
        end = idx2
    else:
        start = idx2 + 1
        end = idx1

    return tuple(planets[start:end])
```


Here's the implementation for the function you described:

```python
def sorted_list_sum(lst):
    """
    Removes strings with odd lengths from the list, then sorts the remaining
    strings first by length (ascending) and then alphabetically.
    """
    # Filter out strings with odd lengths
    even_length_strings = [s for s in lst if len(s) % 2 == 0]

    # Sort by length first, then alphabetically
    even_length_strings.sort(key=lambda x: (len(x), x))

    return even_length_strings
```


Here's the implementation:

```python
def x_or_y(n, x, y):
    """A simple program which should return the value of x if n is
    a prime number and should return the value of y otherwise.

    Examples:
    for x_or_y(7, 34, 12) == 34
    for x_or_y(15, 8, 5) == 5

    """
    if n < 2:
        return y
    for i in range(2, int(n ** 0.5) + 1):
        if n % i == 0:
            return y
    return x
```


```python

def double_the_difference(lst):
    '''
    Given a list of numbers, return the sum of squares of the numbers
    in the list that are odd. Ignore numbers that are negative or not integers.

    double_the_difference([1, 3, 2, 0]) == 1 + 9 + 0 + 0 = 10
    double_the_difference([-1, -2, 0]) == 0
    double_the_difference([9, -2]) == 81
    double_the_difference([0]) == 0

    If the input list is empty, return 0.
    '''
    total = 0
    for num in lst:
        if isinstance(num, int) and num >= 0 and num % 2 == 1:
            total += num * num
    return total

```


Here's the implementation for the function you described:

```python
def compare(game, guess):
    """I think we all remember that feeling when the result of some long-awaited
    event is finally known. The feelings and thoughts you have at that moment are
    definitely worth noting down and comparing.
    Your task is to determine if a person correctly guessed the results of a number of matches.
    You are given two arrays of scores and guesses of equal length, where each index shows a match.
    Return an array of the same length denoting how far off each guess was. If they have guessed correctly,
    the value is 0, and if not, the value is the absolute difference between the guess and the score.

    example:

    compare([1,2,3,4,5,1],[1,2,3,4,2,-2]) -> [0,0,0,0,3,3]
    compare([0,5,0,0,0,4],[4,1,1,0,0,-2]) -> [4,4,1,0,0,6]
    """
    result = []
    for i in range(len(game)):
        if game[i] == guess[i]:
            result.append(0)
        else:
            result.append(abs(game[i] - guess[i]))
    return result
```


Here's the implementation for the function you described:

```python
def Strongest_Extension(class_name, extensions):
    def strength(ext):
        cap = sum(1 for c in ext if c.isupper())
        sm = sum(1 for c in ext if c.islower())
        return cap - sm

    strongest_ext = None
    strongest_strength = None

    for ext in extensions:
        s = strength(ext)
        if strongest_ext is None or s > strongest_strength:
            strongest_ext = ext
            strongest_strength = s

    return f"{class_name}.{strongest_ext}"
```


Here's the implementation for the function you described:

```python
def cycpattern_check(a, b):
    """You are given 2 words. You need to return True if the second word or any of its rotations is a substring in the first word
    cycpattern_check("abcd","abd") => False
    cycpattern_check("hello","ell") => True
    cycpattern_check("whassup","psus") => False
    cycpattern_check("abab","baa") => True
    cycpattern_check("efef","eeff") => False
    cycpattern_check("himenss","simen") => True
    """
    if not b:
        return False
    n = len(b)
    # Generate all rotations of b
    for i in range(n):
        rotated = b[i:] + b[:i]
        if rotated in a:
            return True
    return False
```