**π‘ Problem Formulation:** Imagine you have a list of numbers representing a subset of integers from 1 to n. Your task is to find all the missing numbers in that range which are not present in your list. For example, given the list [1, 2, 4, 6] and n = 6, the output should be [3, 5] since these numbers are absent from the list but are within the desired range.

## Method 1: Using set operations

This method utilizes the concept of sets in Python to find missing numbers. By converting the input list to a set and subtracting it from another set containing all numbers from 1 to n, we can efficiently identify the missing numbers.

Here’s an example:

def find_missing_numbers(nums, n): return list(set(range(1, n + 1)) - set(nums)) missing_numbers = find_missing_numbers([1, 2, 4, 6], 6) print(missing_numbers)

Output:

[3, 5]

This code snippet defines a function `find_missing_numbers()`

that takes a list of numbers and the value n as its parameters. It returns the missing numbers by subtracting the set of input numbers from another set containing all numbers from 1 to n, providing an easy-to-understand and concise way to solve the problem.

## Method 2: Iterative search

The iterative search method iterates through the range from 1 to n and appends any number not found in the input list to the result list. It’s a straightforward brute-force approach.

Here’s an example:

def find_missing_numbers(nums, n): missing_numbers = [] for i in range(1, n + 1): if i not in nums: missing_numbers.append(i) return missing_numbers missing_numbers = find_missing_numbers([1, 2, 4, 6], 6) print(missing_numbers)

Output:

[3, 5]

In this example, the function `find_missing_numbers()`

creates an empty list and checks each number in the range 1 to n. If a number is not present in the input list, it is added to the missing numbers list. This method is simple but may be less efficient for large n or lists.

## Method 3: Using a boolean array

The boolean array method initializes an array of False values, and then marks the indices corresponding to the values in the input list as True. The indices with False remain are the missing numbers.

Here’s an example:

def find_missing_numbers(nums, n): presence = [False] * n missing_numbers = [] for num in nums: presence[num - 1] = True for i, is_present in enumerate(presence): if not is_present: missing_numbers.append(i + 1) return missing_numbers missing_numbers = find_missing_numbers([1, 2, 4, 6], 6) print(missing_numbers)

Output:

[3, 5]

This code defines a function `find_missing_numbers()`

which uses a temporary boolean array to track the presence of each number from 1 to n. Non-present numbers are identified by their corresponding False values in the array and then appended to the result list.

## Method 4: Mathematical computation

This method involves calculating the sum of numbers from 1 to n using the formula n*(n+1)/2 and then subtracting the sum of the input list. The difference gives the sum of the missing numbers. However, this method only works when there is exactly one missing number.

Here’s an example:

def find_missing_number(nums, n): expected_sum = n * (n + 1) // 2 actual_sum = sum(nums) return expected_sum - actual_sum missing_number = find_missing_number([1, 2, 4, 6], 6) print(missing_number)

Output:

3

This snippet presents the function `find_missing_number()`

, which calculates the expected sum of numbers from 1 to n and subtracts the sum of numbers present in the list. The result is the missing number. It is an efficient method, but only suitable for finding a single missing number.

## Bonus One-Liner Method 5: Using list comprehension and set operations

A concise one-liner method combines list comprehension and set operations to quickly derive the missing numbers. It is short and harnesses the expressiveness of Python’s list comprehension.

Here’s an example:

missing_numbers = list(set(range(1, 7)) - set([1, 2, 4, 6])) print(missing_numbers)

Output:

[3, 5]

This one-liner code takes advantage of Python’s set operations and list comprehensions to find the missing numbers from the range 1 to n. It’s essentially a condensed version of Method 1.

## Summary/Discussion

**Method 1:**Set operations. Efficient for small to medium-sized lists. Will not handle duplicate entries in input.**Method 2:**Iterative search. Intuitive but may be slow for large lists or ranges. Does not require any additional space.**Method 3:**Boolean array. Faster than the iterative search. However, it requires additional space proportional to n.**Method 4:**Mathematical computation. Highly efficient but only applicable when there is one missing number in the sequence.**Method 5:**One-liner using list comprehension and set operations. Quick and readable, best suited for situations where brevity is desired and readability is not compromised.