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.

The Idea: There is a clear closed form solution to this. The mathematics is shown in the image below. The solution will always show a quadratic curve, but we are only ever interested in positive solutions. Also, floor ensures that a complete staircase can be made.

Complexity: O(1) time and space

class Solution: def arrangeCoins(self, n): """ :type n: int :rtype: int """

    return abs(math.floor((-1+math.sqrt(1+(4*n*2)))/2))