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.

 

 Approach #1: brup force(Time Limit Exceeded)

class Solution {
public:
    int arrangeCoins(int n) {
        int ans = 0;
        int sum = 1;
        int num = 1;
        while (sum <= n) {
            ans++;
            sum += ++num;
        }
        return ans;
    }
};

　　

Approach #2: math

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

Runtime: 28 ms, faster than 28.09% of C++ online submissions for Arranging Coins.

 

 Approach #3 Iteration

class Solution {
public:
    int arrangeCoins(int n) {
        int i = 1;
        while(n >= i) n -= i, i++;
        return i - 1;
    }
};

Runtime: 24 ms, faster than 39.92% of C++ online submissions for Arranging Coins.

 

Approach #4: Binary Search

class Solution {
public:
    int arrangeCoins(int n) {
        long l = 0;
        long r = n;
        while (l < r) {
            long m = l + (r - l + 1) / 2;
            if (judge(m, n)) 
                r = m - 1;
            else
                l = m;
        }
        return l;
    }
private:
    bool judge(long m, int n) {
        long sum = (m * (m + 1)) / 2;
        return sum > n;
    }
};

Runtime: 36 ms, faster than 11.69% of C++ online submissions for Arranging Coins.

 

Analysis:

1. using long type

2. m = l + (r - l + 1) / 2;