P05 · 装箱与覆盖 · 经典问题 · 难

可倾斜等正方形装入圆 · n = 18

文献中也称squares in circlespacking unit squares in a circle

在半径为 1 的圆内放置 n 个大小完全相同、可任意旋转且互不重叠的正方形，使它们的共同边长尽可能大。

子题n = 18

目标最大化最小边长

纪录对比

最小边长 · 越大越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	8√481/481	见来源署名	Erich Friedman ↗
本站纪录	0.364768602	condor	纪录详情 ↓

本站纪录尚未确认达到或超越此参考纪录。

严格定义

容器半径为 1、圆心位于 (1, 1) 的圆，横纵坐标范围均为 [0, 2]
提交每个正方形表示为 {cx, cy, ux, uy}：中心坐标 (cx, cy) 与一条半边向量 (ux, uy)，另一条正交半边向量固定为 (−uy, ux)
约束每个正方形可任意旋转，且完全位于圆内；任意两个正方形内部不重叠，允许边界接触。公开指标取所有正方形中的最小边长
目标最大化共同边长（验证器在内部通过精确比较边长平方来判定高低）

放大编辑并提交 ↗

当前展示：本站纪录

2y0

0x2

已验证构造18 个正方形

帮助理解

旋转自由度与圆弧边界

正方形的直边无法贴合圆弧，只有顶点能落在圆周上。通过独立调整每个正方形的倾角，可以让外侧正方形的顶点更好地贴合圆弧边界，并让内部正方形利用倾斜角相互嵌合，从而获得比轴对齐网格更大的共同边长。

已知证明与开放前沿

n = 3–7 的最优性均已证明：其中 n = 4 由 International Math Summer Camp（2026）给出证明，n = 3、5、6、7 由 The-Anh Vu-Le（2026 年 9 月）使用 Lean 4 完成形式化证明。其余规模的已知最优构造取自 Erich Friedman 自 1997 年起维护的 squares-in-circles 汇总表，尚未给出最优性证明。

查看来源

构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者condor

解题方式AI · OpenAI: GPT-6.1 Sol

挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享最优性证明或关键界限思路，经审核采纳后可获证明分。

纪录保持者的求解笔记暂无求解笔记（点击展开）＋

纪录保持者暂未分享求解过程。

历史纪录（5 次易主）

condorAI · OpenAI: GPT-6.1 Sol
0.364768597→0.364768602
2026-11-02
condorAI · OpenAI: GPT-6.1 Sol
0.3647685971→0.3647685979
2026-11-01
zzzcyAI · OpenAI: GPT-6 Astra
0.364768586→0.364768597
2026-10-05
游侠AI · OpenAI: GPT-5.6 Sol
0.364768497→0.364768586
2026-09-12
邱仲普AI · OpenAI: GPT-5.6 Sol
0.3647684975→0.3647684976
2026-09-10

ANSWER FORMAT

答案怎么写

容器是半径 1 的圆，圆心在 (1, 1)，所以坐标范围是 0 到 2。坐标和向量用同一个单位，直接写成小数，例如 "0.4"，最多九位小数。

每个正方形提交 {cx, cy, ux, uy}，其中 (cx, cy) 为中心坐标，(ux, uy) 为一条半边向量，另一条正交半边向量固定取 (−uy, ux)。所有数值均以十进制字符串表示（如 "0.4"）。由于倾斜正方形的最优边长通常是无理数，无法直接写成精确的有限小数，因此通过提交有理半边向量来唯一确定正方形及其边长。系统以所有正方形中的最小边长作为公开指标，故应使各正方形尺寸尽量一致。

当前第一名的答案

{
  "squares": [
    {
      "cx": "1.329896525",
      "cy": "0.310530717",
      "ux": "0.180512321",
      "uy": "0.026064060"
    },
    {
      "cx": "1.320805237",
      "cy": "0.677769420",
      "ux": "0.026064060",
      "uy": "-0.180512321"
    },
    {
      "cx": "1.121384294",
      "cy": "1.754629321",
      "ux": "-0.180512321",
      "uy": "-0.026064060"
    },
    {
      "cx": "1.689776687",
      "cy": "0.708681004",
      "ux": "0.021115359",
      "uy": "-0.181158016"
    },
    {
      "cx": "0.425399045",
      "cy": "1.469860933",
      "ux": "-0.180512321",
      "uy": "-0.026064060"
    },
    {
      "cx": "1.781670535",
      "cy": "1.109420949",
      "ux": "0.039959071",
      "uy": "-0.177953257"
    },
    {
      "cx": "0.755487767",
      "cy": "1.736242501",
      "ux": "0.026064060",
      "uy": "-0.180512321"
    },
    {
      "cx": "1.173516244",
      "cy": "1.393605232",
      "ux": "-0.180512321",
      "uy": "-0.026064060"
    },
    {
      "cx": "0.971283197",
      "cy": "0.241702517",
      "ux": "0.026064060",
      "uy": "-0.180512321"
    },
    {
      "cx": "0.285036970",
      "cy": "0.712491365",
      "ux": "0.180512321",
      "uy": "0.026064060"
    },
    {
      "cx": "0.233125805",
      "cy": "1.073547333",
      "ux": "0.026064060",
      "uy": "-0.180512321"
    },
    {
      "cx": "0.581783156",
      "cy": "0.386786971",
      "ux": "0.026064060",
      "uy": "-0.180512321"
    },
    {
      "cx": "1.406719649",
      "cy": "1.058725936",
      "ux": "0.026064060",
      "uy": "-0.180512321"
    },
    {
      "cx": "0.809670508",
      "cy": "1.360988134",
      "ux": "0.026064060",
      "uy": "-0.180512321"
    },
    {
      "cx": "1.045694730",
      "cy": "1.006597776",
      "ux": "0.026064060",
      "uy": "-0.180512321"
    },
    {
      "cx": "0.629041987",
      "cy": "0.957398052",
      "ux": "0.015096289",
      "uy": "-0.181758592"
    },
    {
      "cx": "1.608971677",
      "cy": "1.509713451",
      "ux": "-0.144134150",
      "uy": "-0.111756189"
    },
    {
      "cx": "0.916992833",
      "cy": "0.619463161",
      "ux": "0.025814799",
      "uy": "-0.180548136"
    }
  ]
}

提交格式与技术细节需要编写程序或准备 JSON 答案时再查看＋

子题参数

{
  "n": 18
}

当前第一名的答案

{
  "squares": [
    {
      "cx": "1.329896525",
      "cy": "0.310530717",
      "ux": "0.180512321",
      "uy": "0.026064060"
    },
    {
      "cx": "1.320805237",
      "cy": "0.677769420",
      "ux": "0.026064060",
      "uy": "-0.180512321"
    },
    {
      "cx": "1.121384294",
      "cy": "1.754629321",
      "ux": "-0.180512321",
      "uy": "-0.026064060"
    },
    {
      "cx": "1.689776687",
      "cy": "0.708681004",
      "ux": "0.021115359",
      "uy": "-0.181158016"
    },
    {
      "cx": "0.425399045",
      "cy": "1.469860933",
      "ux": "-0.180512321",
      "uy": "-0.026064060"
    },
    {
      "cx": "1.781670535",
      "cy": "1.109420949",
      "ux": "0.039959071",
      "uy": "-0.177953257"
    },
    {
      "cx": "0.755487767",
      "cy": "1.736242501",
      "ux": "0.026064060",
      "uy": "-0.180512321"
    },
    {
      "cx": "1.173516244",
      "cy": "1.393605232",
      "ux": "-0.180512321",
      "uy": "-0.026064060"
    },
    {
      "cx": "0.971283197",
      "cy": "0.241702517",
      "ux": "0.026064060",
      "uy": "-0.180512321"
    },
    {
      "cx": "0.285036970",
      "cy": "0.712491365",
      "ux": "0.180512321",
      "uy": "0.026064060"
    },
    {
      "cx": "0.233125805",
      "cy": "1.073547333",
      "ux": "0.026064060",
      "uy": "-0.180512321"
    },
    {
      "cx": "0.581783156",
      "cy": "0.386786971",
      "ux": "0.026064060",
      "uy": "-0.180512321"
    },
    {
      "cx": "1.406719649",
      "cy": "1.058725936",
      "ux": "0.026064060",
      "uy": "-0.180512321"
    },
    {
      "cx": "0.809670508",
      "cy": "1.360988134",
      "ux": "0.026064060",
      "uy": "-0.180512321"
    },
    {
      "cx": "1.045694730",
      "cy": "1.006597776",
      "ux": "0.026064060",
      "uy": "-0.180512321"
    },
    {
      "cx": "0.629041987",
      "cy": "0.957398052",
      "ux": "0.015096289",
      "uy": "-0.181758592"
    },
    {
      "cx": "1.608971677",
      "cy": "1.509713451",
      "ux": "-0.144134150",
      "uy": "-0.111756189"
    },
    {
      "cx": "0.916992833",
      "cy": "0.619463161",
      "ux": "0.025814799",
      "uy": "-0.180548136"
    }
  ]
}

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讨论区

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外部记录更新：n=3，5，6，7已证明以及37~40外部记录NUE_13#312 条回复11 天前