Missing Number

You can find the missing number in the range by using a mathematical approach. The sum of the first ( n ) natural numbers is given by:

[ \text{sum} = \frac{n \cdot (n + 1)}{2} ]

Now, you can subtract the sum of the numbers in the given array from this sum to find the missing number.

Here’s the code to implement this approach:

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class Solution:
    def missingNumber(self, nums: List[int]) -> int:
        n = len(nums)
        total_sum = (n * (n + 1)) // 2
        actual_sum = sum(nums)
        return total_sum - actual_sum

Key Takeaways

  • The solution leverages the sum of the first ( n ) natural numbers to find the missing number.
  • This algorithm runs in ( O(n) ) time complexity and ( O(1) ) space complexity, where ( n ) is the length of the input array.
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class Solution:
    def missingNumber(self, nums: List[int]) -> int:
        return (len(nums) * (len(nums) + 1))//2 - sum(nums)
  1. The missing number can be within the range or just outside.
  2. 0 is always present in the input
  3. We numbers may not be in sorted order
  4. The numbers cannot be negative
   0 1 2
  [3,0,1]
  
  exchange 3 with 1
   0 1 2
  [1,0,3]
  
   0 1 2 
  [0,1,3]
  
   0 1
  [0,1]

When the numbers match with the index, the answer 2 is the exclusive range

  [0]

Zero is at zeroth index, just add 1 to 0 to get the answer

   0 1 2 3 4 5 6 7 8
  [9,6,4,2,3,5,7,0,1]
  
  [1,6,4,2,3,5,7,0,9]
  
  [1,5,4,2,3,6,7,0,9]

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def missing_number(nums)
    length = nums.length
    
    sum = length * (length+1)/2
    
    nums.each do |n|
       sum -= n 
    end

    sum
end

I looked at the examples and made observations. It describes a process for solving a problem where you are given an array of unique integers, and you need to find the “missing” integer in the range from 0 to the length of the array. Here’s a simpler explanation of the steps:

  1. The missing number could be any number in the range from 0 to the length of the array, inclusive. That is, it could be any of the numbers that “should” be in the array based on its length, or it could be the next number just outside the range.

  2. The number 0 is always included in the array. This is a useful starting point for the process.

  3. The numbers in the array may not be sorted. This means you can’t rely on the order of the numbers to find the missing one.

  4. The numbers are all non-negative. This means you don’t need to worry about negative numbers when looking for the missing number.

The process you’ve described involves swapping numbers in the array so that each number is at the index that corresponds to its value (if possible). For example, the number 3 should be at index 3, the number 1 should be at index 1, and so on.

You keep swapping numbers until you find a number that can’t be swapped to its correct position because it’s out of the range of the indices. This number is the “missing” number. In your examples, this is the number 2 in the first array and the number 8 in the second array.

If you successfully swap all numbers to their correct positions, then the missing number is the next number outside the range, which is equal to the length of the array. This is the case in your second example, where the array is [0], and the missing number is 1.

This process works because if a number is at its correct index, you know it’s not missing. The first number you find that can’t be put at its correct index (or the absence of such a number, indicating that the missing number is outside the range) tells you which number is missing.

Problem Classification

The problem can be classified into a few categories:

  1. Arrays/Lists: The problem provides an array or list of numbers, so we can categorize this under array or list manipulation.

  2. Mathematical/Arithmetic: The problem involves finding a missing number in a range, which falls under arithmetic progressions, indicating it involves basic mathematical concepts.

  3. Searching: We’re tasked to find a particular element (the missing number) in the given list, which can be seen as a searching problem.

  4. Set Theory: The concept of a missing element can also be related to set theory where we are dealing with a complete set [0, n] and an incomplete set (given array) and we are required to find the missing elements.

Language Agnostic Coding Drills

  1. Understanding the Problem: The first thing we need to do is understand the problem. We are given an array with distinct integers ranging from 0 to n and we are tasked with finding the missing number in the array.

  2. Mathematical Concept - Arithmetic Progression: This problem is based on the concept of Arithmetic Progression. In an Arithmetic Progression sequence, the difference of any two successive members is a constant. A sequence like 1, 2, 3, 4, 5, 6…n has a common difference of 1. The sum of this sequence is given by the formula n*(n+1)/2. We can use this formula to find the sum of numbers from 0 to n.

  3. Calculation of Expected Sum: Apply the formula n*(n+1)/2 to calculate the expected sum for the given range. This would be the sum of the array if it had all the numbers from 0 to n.

  4. Calculation of Actual Sum: Next, we need to calculate the actual sum of the given array. This sum will be less than the expected sum due to the missing number.

  5. Finding the Missing Number: To find the missing number, we simply subtract the actual sum from the expected sum. The result of this subtraction is the missing number in the array.

By implementing and practicing these individual components, we can combine them into the final solution to solve the problem. Understanding the problem, the underlying mathematical concepts, and the process of breaking down the problem into smaller manageable parts is a valuable skill for any programming language or problem.

Targeted Drills in Python

Drill 1: Understanding Arrays in Python

In Python, an array can be created using a list. Practice creating lists of different lengths and accessing elements from them.

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nums = [0, 1, 2, 3, 4]  # create a list
print(len(nums))  # prints: 5
print(nums[2])  # prints: 2

Drill 2: Using Arithmetic Progression to Compute Sum

Implement the formula for the sum of an arithmetic progression to compute the sum from 0 to n.

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def arithmetic_sum(n):
    return n * (n+1) // 2

print(arithmetic_sum(5))  # prints: 15

Drill 3: Calculating the Actual Sum of an Array

Practice summing all elements of an array using the sum function.

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nums = [0, 1, 2, 3, 4]
print(sum(nums))  # prints: 10

Drill 4: Subtracting Two Numbers

Subtraction in Python can be done using the - operator.

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print(15 - 10)  # prints: 5

Integrated Drill: Finding the Missing Number

Now combine all these skills to solve the problem: find the missing number in an array.

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def missing_number(nums):
    expected_sum = arithmetic_sum(len(nums))
    actual_sum = sum(nums)
    return expected_sum - actual_sum

print(missing_number([0, 1, 2, 3, 4, 6]))  # prints: 5

This drill illustrates the process of breaking down a problem into its constituent parts, implementing each part separately, and then combining them into a final solution. Practicing this approach will make you a better problem solver and improve your coding skills.