You have a total of n coins that you want to form in a staircase shape, where every k-th row must have exactly k coins.

Given n, find the total number of full staircase rows that can be formed.

n is a non-negative integer and fits within the range of a 32-bit signed integer.

Example 1:

n = 5

The coins can form the following rows:
¤
¤ ¤
¤ ¤

Because the 3rd row is incomplete, we return 2.

Example 2:

n = 8

The coins can form the following rows:
¤
¤ ¤
¤ ¤ ¤
¤ ¤

Because the 4th row is incomplete, we return 3.

Math Problem: 1 + 2 +3....+x = (x+1)*x/2, now we know it equals to n

(x+1)*x/2 = n -> x² + x = 2n -> 4x² + 4x = 8n -> (2x+1)(2x+1) = 8n +1 -> x = (sqrt(8n+1) - 1)/2

need to convert int to long to calcuate the result, and convert it back to int as result

public class Solution {
    public int arrangeCoins(int n) {
        return (int)(Math.sqrt(8 * (long)n + 1) - 1)/2;
    }
}

[Leetcode + Lintcode] 441. Arranging Coins