**π‘ Problem Formulation:** We encounter a problem where we need to find missing numbers in a sequence from 1 to n. Consider an array of integers with a length that’s exactly four less than n. The array is supposed to contain a sequence of numbers from 1 to n, but four numbers are missing. Our goal is to identify the missing numbers efficiently. For instance, given an array [1, 3, 4, 7], with n=7, the desired output is [2, 5, 6].

## Method 1: Using Set Operations

Set operations in Python allow us to find the difference between two collections of items. By creating a set of all numbers from 1 to n and another set from the given array, we can use set subtraction to find the missing numbers. This method is efficient and has a time complexity of O(n).

Here’s an example:

def find_missing_numbers(arr, n): return list(set(range(1, n + 1)) - set(arr)) array = [1, 3, 4, 7] missing_numbers = find_missing_numbers(array, 7) print(missing_numbers)

Output: [2, 5, 6]

This code defines a function `find_missing_numbers()`

which takes an array and the number n as arguments. It returns the difference between the set of numbers from 1 to n and the set obtained from the array. When we print the result for the given sample array, it outputs the missing numbers correctly.

## Method 2: Using a Boolean Array

The Boolean array method initializes an array of size n with False values and marks the indices corresponding to the numbers in the given array as True. Then, the indices that remain False indicate the missing numbers. This method has linear complexity and is simple to implement.

Here’s an example:

def find_missing_numbers(arr, n): present = [False] * (n + 1) for num in arr: present[num] = True return [num for num, is_present in enumerate(present) if not is_present and num != 0] array = [1, 3, 4, 7] missing_numbers = find_missing_numbers(array, 7) print(missing_numbers)

Output: [2, 5, 6]

In this snippet, we define a function `find_missing_numbers()`

that creates a Boolean array, `present`

, to keep track of which numbers are present in the input array. By iterating over this Boolean array, we collect all indices that are marked False (excluding the dummy 0th index) and return them as the missing numbers.

## Method 3: Mathematical Summation

The Mathematical Summation method leverages the property that the sum of a sequence of integers from 1 to n can be calculated with the formula n(n+1)/2. By calculating this sum and subtracting the sum of the array elements, we can determine the total sum of the missing numbers. Additional steps are required to find the individual numbers. This method uses constant space and has linear complexity.

Here’s an example:

# This example assumes that you have implemented additional logic to find individual missing numbers # after finding the total sum of missing numbers. It is a simplified example. def find_sum_of_missing_numbers(arr, n): total = n * (n + 1) // 2 return total - sum(arr) array = [1, 3, 4, 7] sum_of_missing_numbers = find_sum_of_missing_numbers(array, 7) print(sum_of_missing_numbers)

Output: 13

The code snippet provides a simplified version of the Mathematical Summation method for finding the sum of the missing numbers rather than the individual missing numbers. It uses the formula for the sum of the first n natural numbers and subtracts the sum of array elements from it. Additional logic is required to split this sum into individual missing numbers.

## Method 4: Using Sorting

Sorting the given array allows us to compare neighboring elements and find gaps indicating the missing numbers. The time complexity is O(nlogn) due to the sorting process. This method is straightforward but not the most efficient for large arrays due to the sorting overhead.

Here’s an example:

def find_missing_numbers(arr, n): arr.sort() missing_numbers = [] expected_num = 1 for num in arr: while expected_num < num: missing_numbers.append(expected_num) expected_num += 1 expected_num += 1 while expected_num <= n: missing_numbers.append(expected_num) expected_num += 1 return missing_numbers array = [1, 3, 4, 7] missing_numbers = find_missing_numbers(array, 7) print(missing_numbers)

Output: [2, 5, 6]

The example code defines `find_missing_numbers()`

function that sorts the array and iteratively checks for the expected sequence of numbers. Whenever a number in the expected sequence isnβt found in the array, it is added to the missing numbers list until all numbers from 1 to n have been checked.

## Bonus One-Liner Method 5: List Comprehension

List comprehensions in Python can be used for concise expressions that generate lists. By creating a list comprehension that iterates over the range from 1 to n and selects numbers not present in the array, we can find the missing numbers in a single line. However, this method can be less readable for those unfamiliar with comprehensions.

Here’s an example:

array = [1, 3, 4, 7] missing_numbers = [num for num in range(1, 8) if num not in array] print(missing_numbers)

Output: [2, 5, 6]

The code utilizes a list comprehension to generate a list of missing numbers by checking for each number in the specified range if it is not present in the array. It’s a compact way to find missing elements but may not be efficient for large arrays as it checks for the presence of each number individually.

## Summary/Discussion

**Method 1:**Using Set Operations. Efficient and simple. Best for small to medium-sized arrays.**Method 2:**Using a Boolean Array. Easy to understand and implement. Space complexity increases with the value of n.**Method 3:**Mathematical Summation. Memory-efficient, avoids any data structure. Requires additional logic for separate missing numbers.**Method 4:**Using Sorting. Straightforward method. Not optimal for large arrays due to sorting overhead.**Method 5:**List Comprehension. Concise expression. Might be inefficient and less readable for large arrays or beginners.