P05 · 装箱与覆盖 · 经典问题 · 难

可倾斜等正方形装入圆 · n = 23

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 = 23

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	[0.3260100606704722, 0.3260111234995339]	见来源署名	Erich Friedman ↗
本站纪录	0.326009574	zzzcy	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23 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zzzcy

解题方式AI · OpenAI: GPT-6 Astra

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（2 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
0.326009563→0.326009574
2026-10-05
游侠AI · OpenAI: GPT-5.6 Sol
0.326009547→0.326009563
2026-09-12

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": "1.333021037",
      "cy": "1.706006012",
      "ux": "0.162996787",
      "uy": "-0.001615029"
    },
    {
      "cx": "1.072427817",
      "cy": "0.844866526",
      "ux": "0.162996786",
      "uy": "-0.001615048"
    },
    {
      "cx": "0.681033867",
      "cy": "1.712466227",
      "ux": "0.162996786",
      "uy": "-0.001615045"
    },
    {
      "cx": "1.004930341",
      "cy": "1.497586486",
      "ux": "0.162996786",
      "uy": "-0.001615050"
    },
    {
      "cx": "1.023208955",
      "cy": "1.171379790",
      "ux": "0.162996787",
      "uy": "-0.001615046"
    },
    {
      "cx": "0.243584900",
      "cy": "0.771885733",
      "ux": "0.162996765",
      "uy": "-0.001617195"
    },
    {
      "cx": "1.814139147",
      "cy": "1.058380476",
      "ux": "0.162985521",
      "uy": "0.002506146"
    },
    {
      "cx": "0.696082405",
      "cy": "1.060265959",
      "ux": "0.162996787",
      "uy": "-0.001615043"
    },
    {
      "cx": "1.413250369",
      "cy": "1.051074189",
      "ux": "0.162985521",
      "uy": "0.002506139"
    },
    {
      "cx": "0.886860906",
      "cy": "0.521086234",
      "ux": "0.036119829",
      "uy": "0.158952567"
    },
    {
      "cx": "1.469458176",
      "cy": "0.376336245",
      "ux": "0.159272269",
      "uy": "0.034682923"
    },
    {
      "cx": "0.788947742",
      "cy": "0.209015054",
      "ux": "0.036119827",
      "uy": "0.158952568"
    },
    {
      "cx": "0.499106943",
      "cy": "0.404757476",
      "ux": "0.036119829",
      "uy": "0.158952568"
    },
    {
      "cx": "0.597020107",
      "cy": "0.716828655",
      "ux": "0.036119828",
      "uy": "0.158952568"
    },
    {
      "cx": "1.741521640",
      "cy": "0.723234379",
      "ux": "0.011546090",
      "uy": "0.162595352"
    },
    {
      "cx": "1.175244096",
      "cy": "0.195239096",
      "ux": "0.159272267",
      "uy": "0.034682929"
    },
    {
      "cx": "1.350177647",
      "cy": "1.376152469",
      "ux": "0.162985521",
      "uy": "0.002506138"
    },
    {
      "cx": "1.401333120",
      "cy": "0.724842830",
      "ux": "0.162985521",
      "uy": "0.002506139"
    },
    {
      "cx": "1.676131141",
      "cy": "1.382306511",
      "ux": "0.162985521",
      "uy": "0.002506146"
    },
    {
      "cx": "0.352136508",
      "cy": "1.422860035",
      "ux": "0.162996765",
      "uy": "-0.001617271"
    },
    {
      "cx": "1.008160415",
      "cy": "1.823580062",
      "ux": "0.162996787",
      "uy": "-0.001615048"
    },
    {
      "cx": "0.677803782",
      "cy": "1.386472651",
      "ux": "0.162996786",
      "uy": "-0.001615044"
    },
    {
      "cx": "0.223928928",
      "cy": "1.098106449",
      "ux": "0.162996765",
      "uy": "-0.001617270"
    }
  ]
}

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": 23
}

当前第一名的答案

{
  "squares": [
    {
      "cx": "1.333021037",
      "cy": "1.706006012",
      "ux": "0.162996787",
      "uy": "-0.001615029"
    },
    {
      "cx": "1.072427817",
      "cy": "0.844866526",
      "ux": "0.162996786",
      "uy": "-0.001615048"
    },
    {
      "cx": "0.681033867",
      "cy": "1.712466227",
      "ux": "0.162996786",
      "uy": "-0.001615045"
    },
    {
      "cx": "1.004930341",
      "cy": "1.497586486",
      "ux": "0.162996786",
      "uy": "-0.001615050"
    },
    {
      "cx": "1.023208955",
      "cy": "1.171379790",
      "ux": "0.162996787",
      "uy": "-0.001615046"
    },
    {
      "cx": "0.243584900",
      "cy": "0.771885733",
      "ux": "0.162996765",
      "uy": "-0.001617195"
    },
    {
      "cx": "1.814139147",
      "cy": "1.058380476",
      "ux": "0.162985521",
      "uy": "0.002506146"
    },
    {
      "cx": "0.696082405",
      "cy": "1.060265959",
      "ux": "0.162996787",
      "uy": "-0.001615043"
    },
    {
      "cx": "1.413250369",
      "cy": "1.051074189",
      "ux": "0.162985521",
      "uy": "0.002506139"
    },
    {
      "cx": "0.886860906",
      "cy": "0.521086234",
      "ux": "0.036119829",
      "uy": "0.158952567"
    },
    {
      "cx": "1.469458176",
      "cy": "0.376336245",
      "ux": "0.159272269",
      "uy": "0.034682923"
    },
    {
      "cx": "0.788947742",
      "cy": "0.209015054",
      "ux": "0.036119827",
      "uy": "0.158952568"
    },
    {
      "cx": "0.499106943",
      "cy": "0.404757476",
      "ux": "0.036119829",
      "uy": "0.158952568"
    },
    {
      "cx": "0.597020107",
      "cy": "0.716828655",
      "ux": "0.036119828",
      "uy": "0.158952568"
    },
    {
      "cx": "1.741521640",
      "cy": "0.723234379",
      "ux": "0.011546090",
      "uy": "0.162595352"
    },
    {
      "cx": "1.175244096",
      "cy": "0.195239096",
      "ux": "0.159272267",
      "uy": "0.034682929"
    },
    {
      "cx": "1.350177647",
      "cy": "1.376152469",
      "ux": "0.162985521",
      "uy": "0.002506138"
    },
    {
      "cx": "1.401333120",
      "cy": "0.724842830",
      "ux": "0.162985521",
      "uy": "0.002506139"
    },
    {
      "cx": "1.676131141",
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      "ux": "0.162985521",
      "uy": "0.002506146"
    },
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      "cy": "1.422860035",
      "ux": "0.162996765",
      "uy": "-0.001617271"
    },
    {
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      "cy": "1.823580062",
      "ux": "0.162996787",
      "uy": "-0.001615048"
    },
    {
      "cx": "0.677803782",
      "cy": "1.386472651",
      "ux": "0.162996786",
      "uy": "-0.001615044"
    },
    {
      "cx": "0.223928928",
      "cy": "1.098106449",
      "ux": "0.162996765",
      "uy": "-0.001617270"
    }
  ]
}

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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