Find the Pivot Integer - Solution & Explanation

This approach involves iterating through each number from 1 to n and checking if it satisfies the given condition of a pivot integer. For each candidate pivot integer x, calculate the sum of numbers from 1 to x using the formula for the sum of an arithmetic series. Similarly, calculate the sum from x to n. Compare these sums to determine if x is a pivot integer.

The function find_pivot iterates through numbers from 1 to n. For each number, it calculates the sum of numbers from 1 to x (sum1) and from x to n (sum2). It returns x if the two sums are equal, otherwise returns -1 at the end.

Instead of iterating through each possible x, we can use mathematical formulation. As the sum from 1 to n is fixed, let's denote the total sum as S. Then, for pivot x, the condition becomes:

(x * (x + 1)) / 2 = (S - ((x - 1) * x) / 2)

From this, derive a formula to find x directly if it exists. Solving algebraically can help us identify the pivot integer without iteration.

This optimized solution uses algebra to directly verify if a number x is a pivot without recalculating sums within the loop.