leetcode 441. Arranging Coins 等差数列求和

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JackZhangNJU

最新推荐文章于 2024-07-21 19:45:04 发布

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本文介绍了一种通过等比数列求和的方式解决硬币排列问题的方法，即如何使用给定数量的硬币形成完整的阶梯形状。核心在于利用数学公式快速求解能够形成的最大完整阶梯层数。

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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.

题意很简单，就是一个简单的等比数列求和问题

代码如下：

#include <iostream>
#include <vector>
#include <map>
#include <set>
#include <queue>
#include <stack>
#include <string>
#include <climits>
#include <algorithm>
#include <sstream>
#include <functional>
#include <bitset>
#include <cmath>

using namespace std;



class Solution 
{
public:
    /*
        1 + 2 + 3 + ... + x = n
        -> (1 + x)x / 2 = n
        ->x ^ 2 + x = 2n
        ->x ^ 2 + x + 1 / 4 = 2n + 1 / 4
        -> (x + 1 / 2) ^ 2 = 2n + 1 / 4
        -> (x + 0.5) = sqrt(2n + 0.25)
        ->x = -0.5 + sqrt(2n + 0.25)
    */

    int arrangeCoins(int n) 
    {
        return floor(-0.5 + sqrt((double)2 * n + 0.25));
    }
};