441. Arranging Coins

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:
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¤ ¤
¤ ¤

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

 

Example 2:

n = 8

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

Because the 4th row is incomplete, we return 3.
题目含义：给定n个硬币构造等差数列，求能构成多少行 

方法一：直接遍历即可，从1开始，如果剩下是数不能构成一行则返回。注意要先判断剩下的数是否满足，而不是累加以后再判断，这样可能会导致溢出。

    public int arrangeCoins(int n) {
        int i=0;
        while (n>0)
        {
            i++;
            n-=i;
        }
        return n==0?i:i-1;      
    }

 

方法二：直接求根法

(x+1)*x/2 = n

x² + x = 2n 

4x² + 4x = 8n 

(2x+1)(2x+1) = 8n +1

x = (sqrt(8n+1) - 1)/2

 

    public int arrangeCoins(int n) {
        return (int)((-1.0 + Math.sqrt(1.0 + 8.0 * n)) / 2);
    }