Find N Unique Integers Sum up to Zero

The goal is to create a list of n unique integers that add up to 0. This can be achieved by adding pairs of negative and positive numbers, and if n is odd, adding a zero to the list.

Here is how the code:

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class Solution:
    def sumZero(self, n: int) -> List[int]:
        result = []
        for i in range(1, n // 2 + 1):
            result.append(i)
            result.append(-i)
        if n % 2 == 1:
            result.append(0)
        return result

Let’s break it down:

  1. Initialize an empty list, result.
  2. Iterate over a range from 1 to n // 2 + 1. This range is chosen because we’re going to add both positive and negative numbers to the list in each iteration.
  3. For each number i in the range, add i and -i to result. These are a pair of positive and negative numbers, so they add up to 0.
  4. After the loop, if n is odd (n % 2 == 1), add a zero to result. This makes sure result has n elements, as required.
  5. Return the result list.

Use an Equation

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class Solution:
    def sumZero(self, n: int) -> List[int]:
        result = []
        for i in range(n):
            result.append(i * 2 - n + 1)
        return result

An arithmetic sequence is a sequence of numbers such that the difference between any two consecutive terms is constant. This constant difference is known as the “common difference”.

In this problem, we want to construct an arithmetic sequence of n unique numbers that add up to 0.

Let’s denote the first term of the sequence as a, and the common difference as d. Then, the n terms of the sequence can be expressed as a, a + d, a + 2d, …, a + (n-1)d.

The sum of an arithmetic sequence is given by the formula (n/2) * (first_term + last_term). As we want the sum of the sequence to be 0, we have (n/2) * (a + a + (n-1)d) = 0.

This implies that a + (n-1)d = 0 (since the sum of the sequence is 0), or a = -(n-1)d.

Now, we need to ensure that all numbers in the sequence are distinct. The easiest way to do this is to set d = 2, which guarantees that the increments between consecutive terms in the sequence are not 0.

Therefore, a = -(n-1) * 2 = 2 - n.

Substituting a + kd (where k varies from 0 to n-1) with the derived values of a and d, we get (2 - n) + 2k = 2k + 2 - n = 2k - n + 2 = 2(k - n/2 + 1).

So the formula i * 2 - n + 1 is derived, where i is the index of the term in the sequence (0-indexed). Each term calculated by this formula is unique and the sum of these terms is 0. This is exactly what we need to solve this problem.

Abstract Representation of the Problem

The problem is about generating a sequence of n distinct numbers where the sum of all the numbers in this sequence is zero.

We need a function generateZeroSumSequence(n) which takes an integer n as input, and outputs an array of length n where:

  1. All elements in the array are unique.
  2. The sum of all elements in the array equals zero.

The problem does not have any specific real-world constraints or details, and the integers in the output array do not need to follow any particular order. The output can be one of many possible solutions, as long as it meets the conditions above.

Therefore, this problem can be abstracted to finding a sequence of n distinct numbers that add up to zero. It is a mathematical problem which can be solved using concepts from number theory and arithmetic sequences.