441. Arranging Coins

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

所以在這裏應該要找到一個k使得k*（k+1）<=2n即可

其中要注意long的定義

還有mid的值是如何定義的