LeetCode #2221 — MEDIUM

Find Triangular Sum of an Array

Move from brute-force thinking to an efficient approach using array strategy.

The Problem

Problem Statement

You are given a 0-indexed integer array nums, where nums[i] is a digit between 0 and 9 (inclusive).

The triangular sum of nums is the value of the only element present in nums after the following process terminates:

Let nums comprise of n elements. If n == 1, end the process. Otherwise, create a new 0-indexed integer array newNums of length n - 1.
For each index i, where 0 <= i < n - 1, assign the value of newNums[i] as (nums[i] + nums[i+1]) % 10, where % denotes modulo operator.
Replace the array nums with newNums.
Repeat the entire process starting from step 1.

Return the triangular sum of nums.

Example 1:

Input: nums = [1,2,3,4,5]
Output: 8
Explanation:
The above diagram depicts the process from which we obtain the triangular sum of the array.

Example 2:

Input: nums = [5]
Output: 5
Explanation:
Since there is only one element in nums, the triangular sum is the value of that element itself.

Constraints:

1 <= nums.length <= 1000
0 <= nums[i] <= 9

Step 01

Brute Force Baseline

Problem summary: You are given a 0-indexed integer array nums, where nums[i] is a digit between 0 and 9 (inclusive). The triangular sum of nums is the value of the only element present in nums after the following process terminates: Let nums comprise of n elements. If n == 1, end the process. Otherwise, create a new 0-indexed integer array newNums of length n - 1. For each index i, where 0 <= i < n - 1, assign the value of newNums[i] as (nums[i] + nums[i+1]) % 10, where % denotes modulo operator. Replace the array nums with newNums. Repeat the entire process starting from step 1. Return the triangular sum of nums.

Baseline thinking

Start with the most direct exhaustive search. That gives a correctness anchor before optimizing.

Pattern signal: Array · Math

Example 1

[1,2,3,4,5]

Example 2

[5]

Core Insight

What unlocks the optimal approach

Try simulating the entire process.
To reduce space, use a temporary array to update nums in every step instead of creating a new array at each step.

Interview move: turn each hint into an invariant you can check after every iteration/recursion step.

Step 03

Algorithm Walkthrough

Iteration Checklist

Define state (indices, window, stack, map, DP cell, or recursion frame).
Apply one transition step and update the invariant.
Record answer candidate when condition is met.
Continue until all input is consumed.

Use the first example testcase as your mental trace to verify each transition.

Step 04

Edge Cases

Minimum Input

Single element / shortest valid input

Validate boundary behavior before entering the main loop or recursion.

Duplicates & Repeats

Repeated values / repeated states

Decide whether duplicates should be merged, skipped, or counted explicitly.

Extreme Constraints

Upper-end input sizes

Re-check complexity target against constraints to avoid time-limit issues.

Invalid / Corner Shape

Empty collections, zeros, or disconnected structures

Handle special-case structure before the core algorithm path.

Step 05

Full Annotated Code

Source-backed implementations are provided below for direct study and interview prep.

// Accepted solution for LeetCode #2221: Find Triangular Sum of an Array
class Solution {
    public int triangularSum(int[] nums) {
        for (int k = nums.length - 1; k > 0; --k) {
            for (int i = 0; i < k; ++i) {
                nums[i] = (nums[i] + nums[i + 1]) % 10;
            }
        }
        return nums[0];
    }
}

// Accepted solution for LeetCode #2221: Find Triangular Sum of an Array
func triangularSum(nums []int) int {
	for k := len(nums) - 1; k > 0; k-- {
		for i := 0; i < k; i++ {
			nums[i] = (nums[i] + nums[i+1]) % 10
		}
	}
	return nums[0]
}

# Accepted solution for LeetCode #2221: Find Triangular Sum of an Array
class Solution:
    def triangularSum(self, nums: List[int]) -> int:
        for k in range(len(nums) - 1, 0, -1):
            for i in range(k):
                nums[i] = (nums[i] + nums[i + 1]) % 10
        return nums[0]

// Accepted solution for LeetCode #2221: Find Triangular Sum of an Array
impl Solution {
    pub fn triangular_sum(mut nums: Vec<i32>) -> i32 {
        let mut k = nums.len() as i32 - 1;
        while k > 0 {
            for i in 0..k as usize {
                nums[i] = (nums[i] + nums[i + 1]) % 10;
            }
            k -= 1;
        }
        nums[0]
    }
}

// Accepted solution for LeetCode #2221: Find Triangular Sum of an Array
function triangularSum(nums: number[]): number {
    for (let k = nums.length - 1; k; --k) {
        for (let i = 0; i < k; ++i) {
            nums[i] = (nums[i] + nums[i + 1]) % 10;
        }
    }
    return nums[0];
}

Step 06

Interactive Study Demo

Use this to step through a reusable interview workflow for this problem.

Custom checkpoints (one per line)

Press Step or Run All to begin.

Step 07

Complexity Analysis

Time

O(n)

Space

O(1)

Approach Breakdown

BRUTE FORCE

O(n²) time

O(1) space

Two nested loops check every pair or subarray. The outer loop fixes a starting point, the inner loop extends or searches. For n elements this gives up to n²/2 operations. No extra space, but the quadratic time is prohibitive for large inputs.

OPTIMIZED

O(n) time

O(1) space

Most array problems have an O(n²) brute force (nested loops) and an O(n) optimal (single pass with clever state tracking). The key is identifying what information to maintain as you scan: a running max, a prefix sum, a hash map of seen values, or two pointers.

Shortcut: If you are using nested loops on an array, there is almost always an O(n) solution. Look for the right auxiliary state.

Coach Notes

Common Mistakes

Review these before coding to avoid predictable interview regressions.

Off-by-one on range boundaries

Wrong move: Loop endpoints miss first/last candidate.

Usually fails on: Fails on minimal arrays and exact-boundary answers.

Fix: Re-derive loops from inclusive/exclusive ranges before coding.

Overflow in intermediate arithmetic

Wrong move: Temporary multiplications exceed integer bounds.

Usually fails on: Large inputs wrap around unexpectedly.

Fix: Use wider types, modular arithmetic, or rearranged operations.