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P05 · 装箱与覆盖 · 经典问题 · 难

可倾斜等正方形装入圆 · n = 20

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 = 20
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[0.3455425017277125, 0.3456619426201176]见来源署名Erich Friedman ↗
本站纪录0.345594918condor纪录详情 ↓

在公布精度内匹配已知最好；不代表精确相等或已证明最优。

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20 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
可倾斜等正方形装入圆 n = 20 的当前纪录构型，0.345594918
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）
  1. condorAI · OpenAI: GPT-6.1 Sol
    0.345594916→0.345594918
  2. condorAI · OpenAI: GPT-6.1 Sol
    0.345594907→0.345594916
  3. zzzcyAI · OpenAI: GPT-6 Astra
    0.345594899→0.345594907
  4. 游侠AI · OpenAI: GPT-5.6 Sol
    0.345594819→0.345594899
  5. 邱仲普AI · OpenAI: GPT-5.6 Sol
    0.345594818→0.345594819
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": "0.243256532",
      "cy": "0.887674375",
      "ux": "0.054081442",
      "uy": "0.164116469"
    },
    {
      "cx": "0.792140628",
      "cy": "1.436670832",
      "ux": "0.054282970",
      "uy": "0.164049752"
    },
    {
      "cx": "0.464114741",
      "cy": "0.453142661",
      "ux": "0.054282970",
      "uy": "0.164049752"
    },
    {
      "cx": "1.782543515",
      "cy": "1.131442205",
      "ux": "-0.003958864",
      "uy": "0.172752104"
    },
    {
      "cx": "0.464041124",
      "cy": "1.545236773",
      "ux": "0.054282970",
      "uy": "0.164049752"
    },
    {
      "cx": "0.232768991",
      "cy": "1.255588519",
      "ux": "-0.052355123",
      "uy": "-0.164675313"
    },
    {
      "cx": "1.558453276",
      "cy": "1.519997867",
      "ux": "0.054282970",
      "uy": "0.164049752"
    },
    {
      "cx": "1.279923407",
      "cy": "0.718370614",
      "ux": "0.164049753",
      "uy": "-0.054282969"
    },
    {
      "cx": "1.539497163",
      "cy": "0.407093654",
      "ux": "0.118402925",
      "uy": "0.125855907"
    },
    {
      "cx": "1.123600614",
      "cy": "0.245944087",
      "ux": "0.054282970",
      "uy": "0.164049752"
    },
    {
      "cx": "0.753612939",
      "cy": "0.227918869",
      "ux": "0.054282970",
      "uy": "0.164049752"
    },
    {
      "cx": "1.746680946",
      "cy": "0.784934708",
      "ux": "-0.003958864",
      "uy": "0.172752104"
    },
    {
      "cx": "1.262243400",
      "cy": "1.724938150",
      "ux": "0.054282970",
      "uy": "0.164049752"
    },
    {
      "cx": "0.862190229",
      "cy": "0.556052674",
      "ux": "0.054282970",
      "uy": "0.164049752"
    },
    {
      "cx": "1.120240133",
      "cy": "1.328104892",
      "ux": "0.054282970",
      "uy": "0.164049752"
    },
    {
      "cx": "0.601164721",
      "cy": "1.135840237",
      "ux": "0.054282970",
      "uy": "0.164049752"
    },
    {
      "cx": "1.390817413",
      "cy": "1.045699779",
      "ux": "0.054282970",
      "uy": "0.164049752"
    },
    {
      "cx": "0.910345035",
      "cy": "1.761581032",
      "ux": "0.054282970",
      "uy": "0.164049752"
    },
    {
      "cx": "0.954734592",
      "cy": "1.018846322",
      "ux": "0.054282970",
      "uy": "0.164049752"
    },
    {
      "cx": "0.572405143",
      "cy": "0.781333339",
      "ux": "0.054282970",
      "uy": "0.164049752"
    }
  ]
}
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": 20
}

当前第一名的答案

{
  "squares": [
    {
      "cx": "0.243256532",
      "cy": "0.887674375",
      "ux": "0.054081442",
      "uy": "0.164116469"
    },
    {
      "cx": "0.792140628",
      "cy": "1.436670832",
      "ux": "0.054282970",
      "uy": "0.164049752"
    },
    {
      "cx": "0.464114741",
      "cy": "0.453142661",
      "ux": "0.054282970",
      "uy": "0.164049752"
    },
    {
      "cx": "1.782543515",
      "cy": "1.131442205",
      "ux": "-0.003958864",
      "uy": "0.172752104"
    },
    {
      "cx": "0.464041124",
      "cy": "1.545236773",
      "ux": "0.054282970",
      "uy": "0.164049752"
    },
    {
      "cx": "0.232768991",
      "cy": "1.255588519",
      "ux": "-0.052355123",
      "uy": "-0.164675313"
    },
    {
      "cx": "1.558453276",
      "cy": "1.519997867",
      "ux": "0.054282970",
      "uy": "0.164049752"
    },
    {
      "cx": "1.279923407",
      "cy": "0.718370614",
      "ux": "0.164049753",
      "uy": "-0.054282969"
    },
    {
      "cx": "1.539497163",
      "cy": "0.407093654",
      "ux": "0.118402925",
      "uy": "0.125855907"
    },
    {
      "cx": "1.123600614",
      "cy": "0.245944087",
      "ux": "0.054282970",
      "uy": "0.164049752"
    },
    {
      "cx": "0.753612939",
      "cy": "0.227918869",
      "ux": "0.054282970",
      "uy": "0.164049752"
    },
    {
      "cx": "1.746680946",
      "cy": "0.784934708",
      "ux": "-0.003958864",
      "uy": "0.172752104"
    },
    {
      "cx": "1.262243400",
      "cy": "1.724938150",
      "ux": "0.054282970",
      "uy": "0.164049752"
    },
    {
      "cx": "0.862190229",
      "cy": "0.556052674",
      "ux": "0.054282970",
      "uy": "0.164049752"
    },
    {
      "cx": "1.120240133",
      "cy": "1.328104892",
      "ux": "0.054282970",
      "uy": "0.164049752"
    },
    {
      "cx": "0.601164721",
      "cy": "1.135840237",
      "ux": "0.054282970",
      "uy": "0.164049752"
    },
    {
      "cx": "1.390817413",
      "cy": "1.045699779",
      "ux": "0.054282970",
      "uy": "0.164049752"
    },
    {
      "cx": "0.910345035",
      "cy": "1.761581032",
      "ux": "0.054282970",
      "uy": "0.164049752"
    },
    {
      "cx": "0.954734592",
      "cy": "1.018846322",
      "ux": "0.054282970",
      "uy": "0.164049752"
    },
    {
      "cx": "0.572405143",
      "cy": "0.781333339",
      "ux": "0.054282970",
      "uy": "0.164049752"
    }
  ]
}

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。 · 验证器 v1.0.0

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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