441. Arranging Coins

https://leetcode.com/problems/arranging-coins/description/

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.

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

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

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).