# 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:
¤
¤ ¤
¤ ¤

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.
```

## Tips

等差数列m(m+1)/2 ≤ n，m = sqrt(2 \* n + 0.25) - 0.5。或根据范围\[1, N]二分，找第一个满足m(m+1)/2 >= n的，它或者它前面的数为结果。

## Code

```python
class Solution:
    def arrangeCoins(self, n: int) -> int:
        return int(sqrt(2 * n + 0.25) - 0.5)
```

```python
class Solution:
    def arrangeCoins(self, n: int) -> int:
        start, end = 0, n
        while start < end:
            mid = start + (end - start) // 2
            if mid * (mid + 1) // 2 < n:
                start = mid + 1
            else:
                end = mid
        return start if start * (start + 1) // 2 <= n else start - 1
```

## Analysis

O(1).