# 441    Arranging Coins

## 441. [Arranging Coins](https://leetcode.com/problems/arranging-coins/description/)

## 1. Question

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.

nis 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.
```

## 2. Implementation

**(1) Binary Search**

思路：我们根据等差数列公式知道 coins的总数等于 row \* (row + 1) / 2， 所以我们可以根据这个公式找出row的值，使得 row \* (row + 1) / 2 小于等于n，即找row的右边界

```java
class Solution {
    public int arrangeCoins(int n) {
        long start = 1, end = n, mid = 0;
        long coins = (long)n;

        while (start + 1 < end) {
            mid = start + (end - start) / 2;

            if (isGreaterThanN(mid, coins)) {
                end = mid - 1;
            }
            else {
                start = mid;
            }
        }
        return isGreaterThanN(end, coins) ? (int)start : (int)end;
    }

    public boolean isGreaterThanN(long row, long coins) {
        return  (row * row  + row) > 2 * coins; 
    }
}
```

## 3. Time & Space Complexity

**Binary Search:** 时间复杂度O(logn), 空间复杂度O(1)