Arranging Coins (E)

题目

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.

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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.

题意

将n个硬币叠成阶梯状，求能达到的最大阶梯数。

思路

可以直接累减，或者二分查找，或者直接公式解答 (从 \(\frac{k*(k+1)}{2} \le n\) 推导)。

代码实现

Java

累减

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

二分

class Solution {
    public int arrangeCoins(int n) {
        long left = 0, right = n;
        while (left <= right) {
            long mid = (right - left) / 2 + left;
            long sum = (mid + 1) * mid / 2;
            if (sum < n) {
                left = mid + 1;
            } else if (sum > n) {
                right = mid - 1;
            } else {
                return (int) mid;
            }
        }
        return (int) right;
    }
}

公式计算

class Solution {
    public int arrangeCoins(int n) {
        return (int) (Math.sqrt(2 * (long) n + 0.25) - 0.5);
    }
}

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