What is the minimum number of cuts to divide a circle into n equal slices?

For even n, it is n/2 cuts. For odd n, it requires n cuts because each cut creates only one new slice.

Does the first cut always split the circle into two slices?

Not necessarily. For odd n greater than 1, the first cut may not produce equal slices, highlighting why additional cuts are needed.

Why does the solution differentiate between odd and even n?

Even n allows halving the circle efficiently, while odd n cannot be split symmetrically with fewer cuts, reflecting the problem's math-plus-geometry pattern.

Can you use diagonal or curved cuts instead of straight lines?

No, the problem requires straight cuts through the center. Non-straight cuts invalidate equal slice formation.

What is a common mistake when counting minimum cuts?

Counting total slices instead of actual cuts often leads to overestimation, especially for odd n values.

#2481

Easy

auto_awesome数学·结合·几何

LeetCode 题解工作台

分割圆的最少切割次数

圆内一个有效切割，符合以下二者之一：该切割是两个端点在圆上的线段，且该线段经过圆心。该切割是一端在圆心另一端在圆上的线段。一些有效和无效的切割如下图所示。给你一个整数 n ，请你返回将圆切割成相等的 n 等分的最少切割次数。示例 1：输入： n = 4 输出： 2 解释：上图展…

数学几何

题目描述

圆内一个 有效切割 ，符合以下二者之一：

该切割是两个端点在圆上的线段，且该线段经过圆心。
该切割是一端在圆心另一端在圆上的线段。

一些有效和无效的切割如下图所示。

给你一个整数 n ，请你返回将圆切割成相等的 n 等分的最少切割次数。

示例 1：

输入：n = 4
输出：2
解释：
上图展示了切割圆 2 次，得到四等分。

示例 2：

输入：n = 3
输出：3
解释：
最少需要切割 3 次，将圆切成三等分。
少于 3 次切割无法将圆切成大小相等面积相同的 3 等分。
同时可以观察到，第一次切割无法将圆切割开。

提示：

1 <= n <= 100

lightbulb

解题思路

方法一：分类讨论

当 $n=1$ 时，不需要切割，即切割次数为 $0$ ；
当 $n$ 为奇数时，不存在共线的情况，最少需要 $n$ 次切割；
当 $n$ 为偶数时，可以两两共线，最少需要 $\frac{n}{2}$ 次切割。

综上，可以得到：

\textit{ans} = \begin{cases} n, & n \gt 1 \textit{ 且 } n \textit{ 为奇数} \\ \frac{n}{2}, & n \textit{ 为其它} \\ \end{cases}

时间复杂度 $O(1)$ ，空间复杂度 $O(1)$ 。

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class Solution:
    def numberOfCuts(self, n: int) -> int:
        return n if (n > 1 and n & 1) else n >> 1

speed

复杂度分析

指标	值
时间	complexity is O(1) because the solution is a single conditional check based on parity. Space complexity is O(1) as no extra data structures are needed.
空间	Depends on the final approach

psychology

面试官常问的追问

外企场景

question_mark
Asks for the minimum number of cuts for small n like 4 or 5.
question_mark
Hints at separating odd and even n to reduce overcutting.
question_mark
Checks if candidate can explain why fewer cuts fail for odd n.

warning

常见陷阱

外企场景

error
Assuming each cut always doubles slices, which fails for odd n.
error
Ignoring the center-cross requirement, leading to invalid slices.
error
Confusing minimum number of cuts with total slices created.

swap_horiz

进阶变体

外企场景

arrow_right_alt
Determine cuts if slices must be non-overlapping sectors of different sizes.
arrow_right_alt
Find minimum cuts for a circle with concentric rings instead of one layer.
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Calculate cuts if the circle can only be cut along specific angles.

help

常见问题

外企场景

继续练习

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