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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 = 37

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 = 37
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[0.2651626640362530, 0.2651633671505015]Ephraim Duncan2026-09Erich Friedman ↗
本站纪录0.265163209zzzcy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37 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 = 37 的当前纪录构型，0.265163209
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（5 次易主）
  1. zzzcyAI · OpenAI: GPT-6 Astra
    0.265163207→0.265163209
  2. ephraimduncanAI · OpenAI: GPT-6 Astra
    0.261651959→0.265163207
  3. 游侠AI · OpenAI: GPT-5.6 Sol
    0.2616177→0.261651959
  4. 邱仲普AI · OpenAI: GPT-5.6 Sol
    0.2616177001→0.2616177003
  5. 邱仲普AI · OpenAI: GPT-5.6 Sol
    0.054034318→0.2616177
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": "0.210870185",
      "cy": "0.804302946",
      "ux": "0.058708228",
      "uy": "0.118874834"
    },
    {
      "cx": "0.353331533",
      "cy": "0.493942782",
      "ux": "0.058708229",
      "uy": "0.118874833"
    },
    {
      "cx": "0.470747990",
      "cy": "0.731692451",
      "ux": "0.058708228",
      "uy": "0.118874834"
    },
    {
      "cx": "0.427032785",
      "cy": "1.616683790",
      "ux": "0.082398775",
      "uy": "0.103866857"
    },
    {
      "cx": "0.685794389",
      "cy": "0.568305481",
      "ux": "0.058708228",
      "uy": "0.118874834"
    },
    {
      "cx": "0.793857760",
      "cy": "0.188294878",
      "ux": "0.058708228",
      "uy": "0.118874834"
    },
    {
      "cx": "0.607581852",
      "cy": "0.987646977",
      "ux": "0.082398775",
      "uy": "0.103866857"
    },
    {
      "cx": "0.911274216",
      "cy": "0.426044546",
      "ux": "0.058708228",
      "uy": "0.118874834"
    },
    {
      "cx": "1.043753101",
      "cy": "0.684149846",
      "ux": "0.082398775",
      "uy": "0.103866857"
    },
    {
      "cx": "0.691891658",
      "cy": "1.755412258",
      "ux": "0.075657802",
      "uy": "0.108875062"
    },
    {
      "cx": "1.208550652",
      "cy": "0.891883560",
      "ux": "0.082398775",
      "uy": "0.103866857"
    },
    {
      "cx": "0.905022769",
      "cy": "1.597449681",
      "ux": "0.075657802",
      "uy": "0.108875062"
    },
    {
      "cx": "1.573404223",
      "cy": "1.165277188",
      "ux": "0.075657801",
      "uy": "0.108875062"
    },
    {
      "cx": "1.417823812",
      "cy": "1.406057876",
      "ux": "0.075657801",
      "uy": "0.108875062"
    },
    {
      "cx": "1.817891200",
      "cy": "1.024766777",
      "ux": "0.061191575",
      "uy": "0.117615786"
    },
    {
      "cx": "1.056338373",
      "cy": "1.815199807",
      "ux": "0.075657801",
      "uy": "0.108875062"
    },
    {
      "cx": "1.724719826",
      "cy": "1.383027313",
      "ux": "0.075657802",
      "uy": "0.108875062"
    },
    {
      "cx": "1.569153691",
      "cy": "1.623828544",
      "ux": "0.075657801",
      "uy": "0.108875062"
    },
    {
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      "ux": "0.075657802",
      "uy": "0.108875062"
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    {
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      "ux": "0.082398775",
      "uy": "0.103866857"
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      "ux": "0.058708227",
      "uy": "0.118874834"
    },
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      "cy": "1.407668482",
      "ux": "0.082398775",
      "uy": "0.103866858"
    },
    {
      "cx": "0.551663365",
      "cy": "0.296711471",
      "ux": "0.058708229",
      "uy": "0.118874833"
    },
    {
      "cx": "1.450594222",
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      "ux": "0.059824079",
      "uy": "0.118317208"
    },
    {
      "cx": "1.638183978",
      "cy": "0.520455558",
      "ux": "0.061191575",
      "uy": "0.117615787"
    },
    {
      "cx": "1.089897627",
      "cy": "0.188906813",
      "ux": "0.058708228",
      "uy": "0.118874834"
    },
    {
      "cx": "1.208977037",
      "cy": "0.427219904",
      "ux": "0.059824079",
      "uy": "0.118317208"
    },
    {
      "cx": "1.386909445",
      "cy": "0.644613692",
      "ux": "0.056571323",
      "uy": "0.119906495"
    },
    {
      "cx": "1.796645116",
      "cy": "0.736916957",
      "ux": "0.061191575",
      "uy": "0.117615787"
    },
    {
      "cx": "1.522720440",
      "cy": "0.873731863",
      "ux": "0.056571323",
      "uy": "0.119906495"
    },
    {
      "cx": "1.042194176",
      "cy": "1.330177587",
      "ux": "0.075657801",
      "uy": "0.108875062"
    },
    {
      "cx": "1.193509778",
      "cy": "1.547927713",
      "ux": "0.075657801",
      "uy": "0.108875062"
    },
    {
      "cx": "0.641794656",
      "cy": "1.457427513",
      "ux": "0.082398775",
      "uy": "0.103866857"
    },
    {
      "cx": "0.476997106",
      "cy": "1.249693799",
      "ux": "0.082398775",
      "uy": "0.103866857"
    },
    {
      "cx": "0.772379403",
      "cy": "1.195380692",
      "ux": "0.082398775",
      "uy": "0.103866857"
    },
    {
      "cx": "0.984390915",
      "cy": "1.035975472",
      "ux": "0.082398775",
      "uy": "0.103866857"
    },
    {
      "cx": "1.265869116",
      "cy": "1.187388065",
      "ux": "0.075657801",
      "uy": "0.108875063"
    }
  ]
}
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": 37
}

当前第一名的答案

{
  "squares": [
    {
      "cx": "0.210870185",
      "cy": "0.804302946",
      "ux": "0.058708228",
      "uy": "0.118874834"
    },
    {
      "cx": "0.353331533",
      "cy": "0.493942782",
      "ux": "0.058708229",
      "uy": "0.118874833"
    },
    {
      "cx": "0.470747990",
      "cy": "0.731692451",
      "ux": "0.058708228",
      "uy": "0.118874834"
    },
    {
      "cx": "0.427032785",
      "cy": "1.616683790",
      "ux": "0.082398775",
      "uy": "0.103866857"
    },
    {
      "cx": "0.685794389",
      "cy": "0.568305481",
      "ux": "0.058708228",
      "uy": "0.118874834"
    },
    {
      "cx": "0.793857760",
      "cy": "0.188294878",
      "ux": "0.058708228",
      "uy": "0.118874834"
    },
    {
      "cx": "0.607581852",
      "cy": "0.987646977",
      "ux": "0.082398775",
      "uy": "0.103866857"
    },
    {
      "cx": "0.911274216",
      "cy": "0.426044546",
      "ux": "0.058708228",
      "uy": "0.118874834"
    },
    {
      "cx": "1.043753101",
      "cy": "0.684149846",
      "ux": "0.082398775",
      "uy": "0.103866857"
    },
    {
      "cx": "0.691891658",
      "cy": "1.755412258",
      "ux": "0.075657802",
      "uy": "0.108875062"
    },
    {
      "cx": "1.208550652",
      "cy": "0.891883560",
      "ux": "0.082398775",
      "uy": "0.103866857"
    },
    {
      "cx": "0.905022769",
      "cy": "1.597449681",
      "ux": "0.075657802",
      "uy": "0.108875062"
    },
    {
      "cx": "1.573404223",
      "cy": "1.165277188",
      "ux": "0.075657801",
      "uy": "0.108875062"
    },
    {
      "cx": "1.417823812",
      "cy": "1.406057876",
      "ux": "0.075657801",
      "uy": "0.108875062"
    },
    {
      "cx": "1.817891200",
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      "uy": "0.117615786"
    },
    {
      "cx": "1.056338373",
      "cy": "1.815199807",
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      "uy": "0.108875062"
    },
    {
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      "uy": "0.108875062"
    },
    {
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      "ux": "0.075657801",
      "uy": "0.108875062"
    },
    {
      "cx": "1.344825382",
      "cy": "1.765677838",
      "ux": "0.075657802",
      "uy": "0.108875062"
    },
    {
      "cx": "0.819209568",
      "cy": "0.827757967",
      "ux": "0.082398775",
      "uy": "0.103866857"
    },
    {
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      "ux": "0.058708227",
      "uy": "0.118874834"
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    {
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      "cy": "1.407668482",
      "ux": "0.082398775",
      "uy": "0.103866858"
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    {
      "cx": "0.551663365",
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      "ux": "0.058708229",
      "uy": "0.118874833"
    },
    {
      "cx": "1.450594222",
      "cy": "0.317426428",
      "ux": "0.059824079",
      "uy": "0.118317208"
    },
    {
      "cx": "1.638183978",
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      "ux": "0.061191575",
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    {
      "cx": "1.089897627",
      "cy": "0.188906813",
      "ux": "0.058708228",
      "uy": "0.118874834"
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    {
      "cx": "1.208977037",
      "cy": "0.427219904",
      "ux": "0.059824079",
      "uy": "0.118317208"
    },
    {
      "cx": "1.386909445",
      "cy": "0.644613692",
      "ux": "0.056571323",
      "uy": "0.119906495"
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    {
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      "uy": "0.117615787"
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    {
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      "ux": "0.056571323",
      "uy": "0.119906495"
    },
    {
      "cx": "1.042194176",
      "cy": "1.330177587",
      "ux": "0.075657801",
      "uy": "0.108875062"
    },
    {
      "cx": "1.193509778",
      "cy": "1.547927713",
      "ux": "0.075657801",
      "uy": "0.108875062"
    },
    {
      "cx": "0.641794656",
      "cy": "1.457427513",
      "ux": "0.082398775",
      "uy": "0.103866857"
    },
    {
      "cx": "0.476997106",
      "cy": "1.249693799",
      "ux": "0.082398775",
      "uy": "0.103866857"
    },
    {
      "cx": "0.772379403",
      "cy": "1.195380692",
      "ux": "0.082398775",
      "uy": "0.103866857"
    },
    {
      "cx": "0.984390915",
      "cy": "1.035975472",
      "ux": "0.082398775",
      "uy": "0.103866857"
    },
    {
      "cx": "1.265869116",
      "cy": "1.187388065",
      "ux": "0.075657801",
      "uy": "0.108875063"
    }
  ]
}

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