**π‘ Problem Formulation:** Imagine you are given a sorted array of consecutive numbers, but one number is missing. Your task is to efficiently find this missing element. For example, if the input is `[1, 2, 4, 5]`

, the desired output is `3`

because it is the missing number in the sequence.

## Method 1: Summation Formula

The summation formula method leverages the properties of arithmetic series to find the missing element. By calculating the expected sum of the full sequence (using the formula `n*(n+1)/2`

where `n`

is the highest number) and subtracting the actual sum of the current elements, you can identify the missing number.

Here’s an example:

def find_missing_summation(arr): n = arr[-1] expected_sum = n*(n+1)//2 actual_sum = sum(arr) return expected_sum - actual_sum print(find_missing_summation([1, 2, 4, 5]))

**Output:** `3`

This code calculates the expected sum of a complete sequence from 1 to the largest number in the array. Then, it computes the actual sum of the array’s elements. The difference between these two sums gives us the missing element.

## Method 2: Binary Search

Binary Search is an efficient algorithm for finding an element in a sorted array. It can be adapted to find the missing element by comparing the index of the middle element to its value and deciding which half of the array should contain the missing number.

Here’s an example:

def find_missing_binary_search(arr): low, high = 0, len(arr) - 1 while low <= high: mid = (low + high) // 2 if arr[mid] != mid + arr[0]: high = mid - 1 else: low = mid + 1 return low + arr[0] print(find_missing_binary_search([1, 2, 3, 5]))

**Output:** `4`

The code performs a binary search to locate the gap between elements in a sorted array. We know there is a missing element if an element does not match the expected value (current index plus the first element in the array).

## Method 3: XOR Method

The XOR method uses the bitwise XOR operation to find the missing element. XORing all the numbers in the array with all the numbers from 1 to n (inclusive) will leave the missing number, as XORing a number with itself results in 0 and XORing with 0 results in the number itself.

Here’s an example:

def find_missing_xor(arr): xor_sum = 0 for num in arr: xor_sum ^= num for i in range(1, arr[-1] + 1): xor_sum ^= i return xor_sum print(find_missing_xor([1, 2, 3, 5]))

**Output:** `4`

This snippet XORs all elements in the array, then XORs the result with each number from 1 to n. The final result is the missing number, since all other elements cancel out.

## Method 4: Iterative Comparison

The iterative comparison method checks each consecutive pair of elements in the array. If the difference between any two consecutive elements is more than 1, the missing element is found.

Here’s an example:

def find_missing_iterative(arr): for i in range(len(arr) - 1): if arr[i+1] - arr[i] > 1: return arr[i] + 1 print(find_missing_iterative([1, 2, 4, 5]))

**Output:** `3`

This method iterates through the array and checks if the difference between consecutive elements is more than 1, which would indicate a missing element. The missing number is the current element plus one.

## Bonus One-Liner Method 5: List Comprehension

With list comprehension, you can create a list of expected numbers and subtract the original array to find the missing number. This one-liner method is highly readable and concise.

Here’s an example:

arr = [1, 2, 4, 5] print(next(i for i in range(arr[0], arr[-1]) if i not in arr))

**Output:** `3`

This code snippet generates a range of numbers from the first to the last element in the array and checks which one is not in the array. The first such number is the missing element.

## Summary/Discussion

**Method 1: Summation Formula.**Simple and efficient for small ranges. Not suitable for large numbers due to the risk of integer overflow.**Method 2: Binary Search.**Highly efficient for large arrays. Requires a sorted array and does not work as efficiently if there are multiple missing elements.**Method 3: XOR Method.**A clever use of bitwise operations that is efficient and handles large numbers well. However, it may be less intuitive to understand at first.**Method 4: Iterative Comparison.**Straightforward and easy to understand. Inefficient for large arrays as it has a linear runtime complexity.**Method 5: List Comprehension.**Pythonic and concise. It’s less efficient due to the need to iterate over the range and check membership in the array.