Find N Unique Integers Sum up to Zero - Practice Coding Problems

Find N Unique Integers Sum up to Zero - Problem

Array Math Easy

You're tasked with creating a balanced array of unique integers! Given an integer n, you need to return an array containing exactly n unique integers that sum to zero.

This is a creative construction problem where you have complete freedom in choosing the integers, as long as they meet two requirements:

All n integers must be unique (no duplicates)
Their sum must equal 0

Example: If n = 3, you could return [-1, 0, 1] since all numbers are unique and -1 + 0 + 1 = 0.

The beauty of this problem is that there are multiple valid solutions - you just need to find any valid array that satisfies the conditions!

Input & Output

example_1.py — Basic Case

$ Input: n = 5

› Output: [0, 1, -1, 2, -2]

💡 Note: We have 5 unique integers that sum to zero. Since n is odd, we include 0, then add two pairs (1,-1) and (2,-2). Total: 0 + 1 + (-1) + 2 + (-2) = 0

example_2.py — Even Case

$ Input: n = 4

› Output: [1, -1, 2, -2]

💡 Note: Since n is even, we don't need zero. We use two pairs: (1,-1) and (2,-2). Total: 1 + (-1) + 2 + (-2) = 0

example_3.py — Edge Case

$ Input: n = 1

› Output: [0]

💡 Note: The only way to have 1 unique integer that sums to zero is to use [0]

Constraints

1 ≤ n ≤ 1000
The returned array must contain exactly n unique integers
The sum of all integers must equal 0
Any valid solution is acceptable

Visualization

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Understanding the Visualization

Identify the Pattern

For n elements, we need pairs that cancel out

Handle Odd Numbers

If n is odd, use 0 as the 'neutral' weight

Create Pairs

Add pairs (1,-1), (2,-2), etc. until we reach n elements

Verify Balance

All pairs sum to 0, ensuring perfect balance

Key Takeaway

🎯 Key Insight: Instead of searching for a solution, we construct one systematically using the mathematical property that opposite numbers cancel out!

Asked in

G Google 15 ⊞ Microsoft 12 a Amazon 8 f Meta 6

This problem has an elegant mathematical solution using pairs of opposite numbers. The key insight is to use pairs like (1,-1), (2,-2) which always sum to zero, and handle odd cases by including 0. This gives us O(n) time and O(1) extra space - optimal for a construction problem!

Common Approaches

Approach	Time	Space	Notes
✓ Systematic Construction	O(n)	O(1)	Build the array step by step using mathematical insight

Systematic Construction — Algorithm Steps

If n is odd, include 0 in the result to handle the extra element
Create pairs of positive and negative numbers: (1,-1), (2,-2), etc.
Add these pairs until we have exactly n elements
The sum will automatically be zero due to the pairing strategy

Visualization

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Step-by-Step Walkthrough

Check if Odd

If n is odd, start with 0

Create Pairs

Add positive-negative pairs (1,-1), (2,-2)...

Complete Array

Continue until we have exactly n elements

Verify Sum

All pairs cancel out, sum equals zero

Code -

solution.c — C

#include <stdio.h>
#include <stdlib.h>

int* sumZero(int n, int* returnSize) {
    *returnSize = n;
    int* result = (int*)malloc(n * sizeof(int));
    int index = 0;
    
    // If n is odd, add 0
    if (n % 2 == 1) {
        result[index++] = 0;
    }
    
    // Add pairs of positive and negative numbers
    for (int i = 1; i <= n / 2; i++) {
        result[index++] = i;
        result[index++] = -i;
    }
    
    return result;
}

int main() {
    int n;
    scanf("%d", &n);
    
    int returnSize;
    int* result = sumZero(n, &returnSize);
    
    printf("[");
    for (int i = 0; i < returnSize; i++) {
        printf("%d", result[i]);
        if (i < returnSize - 1) printf(",");
    }
    printf("]\n");
    
    free(result);
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(n)

We need to generate exactly n numbers, so O(n) is optimal

✓ Linear Growth

Space Complexity

O(1)

Only using constant extra space besides the output array

✓ Linear Space

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Medium Frequency

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Smart Actions

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Input & Output

Constraints

Visualization

Related Problems

Common Approaches

Systematic Construction — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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