Find the Pivot Integer

Find the Pivot Integer

1. What is this problem about?

The Find the Pivot Integer interview question is a mathematical search task. You are given a positive integer $n$. You need to find a "pivot" integer $x$ (between 1 and $n$) such that the sum of all integers from 1 to $x$ is equal to the sum of all integers from $x$ to $n$. If no such $x$ exists, return -1.

2. Why is this asked in interviews?

Companies like Apple and Google ask the Find the Pivot Integer coding problem to see if a candidate can optimize a search using math. While you can iterate through all numbers ($O(N)$), there is an $O(1)$ solution using algebra. It evaluations your knowledge of Arithmetic Series and your ability to simplify equations before coding.

Algorithmic pattern used

This problem follows the Math interview pattern and Prefix Sum logic.

• Series Sum Formula: The sum of integers from 1 to $k$ is $S(k) = \frac{k(k+1)}{2}$.
• Equation: We need $S(x) = S(n) - S(x-1)$.
• Simplification: $\frac{x(x+1)}{2} = \frac{n(n+1)}{2} - \frac{(x-1)x}{2}$ $x^2 + x = n^2 + n - x^2 + x$ $2x^2 = n^2 + n$ $x = \sqrt{\frac{n(n+1)}{2}}$
• Check: If $x$ is an integer (i.e., its square is exactly $\frac{n(n+1)}{2}$), then $x$ is the pivot.

4. Example explanation

$n = 8$.

1. Calculate total sum $S(8) = \frac{8 imes 9}{2} = 36$.
2. Search for $x$:
   • Try $x=6: S(6) = 21$. Sum from 6 to 8: $6+7+8 = 21$.
   • Since $21 = 21$, $x=6$ is the pivot.

• Using the formula: $\sqrt{36} = 6$. Since 6 is an integer, it's the answer.

5. Common mistakes candidates make

• Off-by-one in series: Forgetting to include $x$ in both the left and right sums.
• Floating point precision: Relying on sqrt without checking if the result is a perfect integer squared.
• Inefficiency: Using a nested loop to calculate sums repeatedly ($O(N^2)$) instead of using the formula or a running sum ($O(N)$).

6. Interview preparation tip

Master the Gauss summation formula ($1 \dots n$). It is the foundation for many Math interview patterns. Also, always check if a search problem can be solved in $O(1)$ by solving for $x$ algebraically.