題目：

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.

分析：

class Solution {
    public int arrangeCoins(int n) {
        //給定總的磚數，具體第k臺階滿足用k塊磚，求具體滿足了多少層臺階
        //思路：類似數列求和,求滿足k-1滿足小於n，k滿足n的數
        //暴力解法：TLM
        // if(n==0) return 0;
        // for(int i=1;i<=Math.sqrt(n);i++){
        //     if(((1+i)*i/2<=n)&&((i+1+1)*(i+1)/2>n)){
        //         return i;
        //     }
        // }
        // return 0;
        
        //巧妙解法
        //         `(x * ( x + 1)) / 2 <= n`
        // Using quadratic formula, `x` is evaluated to be,
        //利用二元一次方程求解 x=（-b+_sqrt(b^2-4ac)）/2a
        // `x = 1 / 2 * (-sqrt(8 * n + 1)-1)` (Inapplicable) or `x = 1 / 2 * (sqrt(8 * n + 1)-1)`
        // Negative root is ignored and positive root is used instead. Note that 8.0 * n is very important because it will cause Java to implicitly autoboxed the intermediate result into double data type. The code will not work if it is simply 8 * n.
        
        return (int)((Math.sqrt(1+8.0*n)-1)/2);
    }
}