Formatted question description: https://leetcode.ca/all/441.html

# 441. Arranging Coins (Easy)

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.


Related Topics:
Math, Binary Search

## Solution 1. Brute Force

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

// Time: O(sqrt(N))
// Space: O(1)
class Solution {
public:
int arrangeCoins(int n) {
long i = 1, sum = 0;
while (sum + i <= n) sum += i++;
return i - 1;
}
};

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

// Time: O(logN)
// Space: O(1)
class Solution {
public:
int arrangeCoins(int n) {
int L = 1, R = n;
while (L <= R) {
long M = L + (R - L) / 2, sum = M * (1 + M) / 2;
if (sum == n) return M;
if (sum < n) L = M + 1;
else R = M - 1;
}
return R;
}
};


## Solution 3. Math

x * (x + 1) / 2 <= n

x^2 + x - 2n <= 0

x <= (sqrt(8n + 1) - 1) / 2

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

// Time: O(1)
// Space: O(1)
class Solution {
public:
int arrangeCoins(int n) {
return (sqrt((long)8 * n + 1) - 1) / 2;
}
};


Java

class Solution {
public int arrangeCoins(int n) {
int row = 0;
while (n > row) {
row++;
n -= row;
}
return row;
}
}