Find Triangular Sum of an Array

Find Triangular Sum of an Array

1. What is this problem about?

The Find Triangular Sum of an Array interview question is a simulation based on Pascal's Triangle logic. You are given an array of integers. In each step, you create a new array of length $n-1$, where the $i^{th}$ element is (arr[i] + arr[i+1]) % 10. You repeat this process until only one integer remains. The final result is this single integer.

2. Why is this asked in interviews?

Companies like Microsoft and Zoho use the Find Triangular Sum coding problem to test a candidate's ability to implement an iterative simulation. It evaluates your skills in array manipulation and loop control. It also opens up a discussion about mathematical shortcuts—since this process is identical to calculating a value in Pascal's triangle, it can be solved using Combinatorics in $O(N)$ time.

3. Algorithmic pattern used

This problem can be solved with Simulation or Math (Pascal's Triangle).

1. Simulation Pattern ($O(N^2)$):
   • While the array length is $> 1$:
   • Create a temporary array.
   • Fill it with adjacent sums modulo 10.
   • Replace the old array with the new one.
2. Math Pattern ($O(N)$):
   • The final value is $\sum_{i=0}^{n-1} \binom{n-1}{i} imes nums[i] \pmod{10}$.
   • Note that $\binom{n}{k} \pmod{10}$ requires careful modulo math since 10 is not prime.

4. Example explanation

Array: [1, 2, 3, 4, 5]

1. Sum: [(1+2)%10, (2+3)%10, (3+4)%10, (4+5)%10] = [3, 5, 7, 9]
2. Sum: [(3+5)%10, (5+7)%10, (7+9)%10] = [8, 2, 6]
3. Sum: [(8+2)%10, (2+6)%10] = [0, 8]
4. Sum: [(0+8)%10] = 8 Result: 8.

5. Common mistakes candidates make

• Inefficient Space: Creating a new array in every iteration of the simulation instead of modifying the existing array in-place or using two arrays to swap.
• Modulo: Forgetting to apply % 10 at each addition step.
• Combinatorial Complexity: Trying to implement the $O(N)$ solution without properly handling the Lucas theorem or prime factorization for the modulo 10 case.

6. Interview preparation tip

Simulation is often the expected answer for $N \leq 1000$. However, mentioning the connection to binomial coefficients and Pascal's Triangle shows the interviewer that you have strong mathematical foundations and can recognize higher-level patterns in Array interview patterns.