441. Arranging Coins

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本文介绍了一种通过二分查找高效解决硬币排列成阶梯形状的问题。目标是确定最多可以形成多少行完整的阶梯，每行的硬币数递增。文章详细解释了使用二分搜索算法来寻找最大可行的阶梯行数，并利用高斯定理简化计算。

ou 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.

public class Solution {
    public int arrangeCoins(int n) {
        long nlong = (long)n;
        long start = 0;
        long end = nlong;
        
        while(start<=end){
            long mid = start+(end-start)/2;
            if(mid*(mid+1)<=2*nlong) start = mid+1;
            else end=mid-1;
        }
        return (int)(start-1);
    }
}

高斯定理：1+2+3.。。。+n=n*（n+1）/2

所以在这里应该要找到一个k使得k*（k+1）<=2n即可

其中要注意long的定义

还有mid的值是如何定义的