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Assuming the problem refers to LeetCode Problem 16: 3Sum Closest, here's the step-by-step solution:

Problem Statement

Given an integer array nums and an integer target, find three integers in nums such that their sum is closest to target. Return the sum of these three integers (each input has exactly one solution).

Approach

The optimal approach uses sorting + two-pointer technique to reduce the time complexity:

1. Sort the Array: Sorting allows efficient adjustment of the sum using two pointers.
2. Iterate with Fixed First Element: For each element as the first in the triplet, use two pointers (left = i+1, right = end of array).
3. Adjust Pointers:
   • If the current sum < target → move left pointer right to increase sum.
   • If current sum > target → move right pointer left to decrease sum.
   • If sum == target → return immediately (exact match).
4. Track Closest Sum: Keep updating the closest sum as you find better matches.

Solution Code

def threeSumClosest(nums, target):
    nums.sort()
    n = len(nums)
    closest_sum = nums[0] + nums[1] + nums[2]  # Initial guess

    for i in range(n - 2):
        left = i + 1
        right = n - 1

        while left < right:
            current_sum = nums[i] + nums[left] + nums[right]

            # Update closest sum if current is better
            if abs(current_sum - target) < abs(closest_sum - target):
                closest_sum = current_sum

            # Adjust pointers
            if current_sum < target:
                left += 1
            elif current_sum > target:
                right -= 1
            else:
                return current_sum  # Exact match

    return closest_sum

Explanation

• Sorting: Takes O(n log n) time.
• Two-Pointer Technique: For each element (O(n)), the two pointers traverse the array once (O(n)), leading to O(n2) time.
• Space Complexity: O(1) (auxiliary space; sorting uses O(log n) stack space, but it's negligible).

Example Test Case

For nums = [-1,2,1,-4] and target =1:

• Sorted array → [-4,-1,1,2].
• The closest sum is 2 (from triplet -1,1,2), which is 1 unit away from the target.

This solution efficiently finds the closest sum and handles all edge cases (like exact matches or all elements being the same).

If this isn't the intended problem, please provide more details about the image content!

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