Find the Integer Added to Array I - Solution & Explanation

Problem Overview: You receive two integer arrays nums1 and nums2. Every element in nums1 had the same integer added to it, producing nums2. Your job is to determine that integer. The order of elements may differ, so the solution relies on properties that remain unchanged when a constant value is added to every element.

Approach 1: Difference Calculation Approach (O(n) time, O(1) space)

The key observation: adding the same integer x to every element shifts the entire array by the same amount. The smallest value in nums1 becomes the smallest value in nums2 after adding x. Compute min(nums1) and min(nums2), then return x = min(nums2) - min(nums1). This works even if the arrays are reordered because the relative ordering of elements does not change after adding a constant. You only need one pass to compute the minimum of each array, making this a clean linear-time solution using basic array traversal.

Approach 2: Average Difference Approach (O(n) time, O(1) space)

Another property of adding a constant to all elements is that the total sum increases by n * x, where n is the array length. Compute sum(nums1) and sum(nums2), then derive the value using x = (sum(nums2) - sum(nums1)) / n. This approach relies on simple arithmetic and avoids sorting or element comparisons. It’s essentially a math-based observation applied during a single pass through the arrays.

Recommended for interviews: The Difference Calculation approach is typically preferred because it is simple, intuitive, and avoids division. Interviewers expect you to recognize that adding a constant shifts the minimum value by the same amount. The Average Difference method also works and demonstrates a good understanding of array sums, but the minimum-difference observation usually surfaces faster during interviews.