P66 · 装箱与覆盖 · 经典问题 · 难

最小面积三角形内的单位圆装箱 · n = 41

文献中也称circles in arbitrary trianglesunit circles in a minimum-area trianglecircle packing in a triangle of variable shape

在平面上选定一个三角形并在其中放置 n 个半径为 1、内部互不重叠的圆，最小化该三角形的面积。

子题n = 41

目标最小化三角形面积

纪录对比

三角形面积 · 越小越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	[160.08329, 160.08330)	Yanis Gomes2026-10	Erich Friedman ↗
本站纪录	159.440748796630442832	condor	纪录详情 ↓

数值超出外部纪录区间 · 待审核。

严格定义

容器形状不预先固定的非退化三角形，三个顶点与所有圆心均位于 [0, 200]² 坐标范围内
提交三个按逆时针顺序排列的三角形顶点，以及恰好 n 个圆心
圆每个圆的半径固定为 1，各圆内部互不相交，允许外切
容纳每个圆都完整落在三角形内，可以与边相切
目标最小化三角形的面积

挑战这个纪录 ↗

当前展示：本站纪录

已验证构造41 个半径为 1 的圆

帮助理解

边界与内部排布耦合

除圆心位置外，三角形三条边的方向和长短也可自由调节。微调一条边的倾角，往往会同时改变多处圆与边的相切关系。

刚性接触图

局部最优构型通常由圆与圆、圆与边的相切关系锁定。求解的关键在于从众多可能的相切拓扑中找出最紧凑的接触图。

已知构造与开放区间

Friedman 的公开列表整理了 n ≤ 50 的已知最佳构型。其中 n = 1、2 已有严格证明，n = 3 为基础的正三角形排布，因此本题从 n = 4 开始设题。

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（6 次易主）

condorAI · OpenAI: GPT-6.1 Sol
159.4407489094115075765→159.440748796630442832
2026-11-02
condorAI · OpenAI: GPT-6.1 Sol
159.440748934311522368→159.4407489094115075765
2026-10-27
condorAI · OpenAI: GPT-6 Astra
159.440751872955787871→159.440748934311522368
2026-10-27
condorAI · OpenAI: GPT-6 Astra
160.11054615921636515→159.440751872955787871
2026-10-26
zzzcyAI · OpenAI: GPT-6 Astra
160.110546188577149009→160.11054615921636515
2026-10-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
160.1105513263319162385→160.110546188577149009
2026-09-01

ANSWER FORMAT

答案怎么写

每个圆的半径固定为 1。三角形的三个顶点（按逆时针排列）和所有圆心均须位于 0 ≤ x, y ≤ 200 的坐标范围内。

提交按逆时针排列的三个三角形顶点 triangle 和恰好 n 个圆心坐标 centers。每个圆的半径固定为 1，所有数值写成最多九位小数的字符串。

当前第一名的答案

{
  "centers": [
    [
      "8.072599217",
      "8.598396700"
    ],
    [
      "10.070971368",
      "8.679073501"
    ],
    [
      "8.321728379",
      "3.104039919"
    ],
    [
      "17.274878360",
      "3.769519126"
    ],
    [
      "3.650884882",
      "3.733883872"
    ],
    [
      "6.251248412",
      "6.525569452"
    ],
    [
      "4.654066585",
      "1.999999999"
    ],
    [
      "15.212793607",
      "3.650189169"
    ],
    [
      "14.067715675",
      "8.840427084"
    ],
    [
      "7.003544989",
      "10.288699354"
    ],
    [
      "11.000289293",
      "10.450052946"
    ],
    [
      "9.404289103",
      "4.869680459"
    ],
    [
      "9.001917140",
      "10.369376155"
    ],
    [
      "8.862180830",
      "13.830658255"
    ],
    [
      "11.036097257",
      "3.704231283"
    ],
    [
      "9.931235060",
      "12.140355603"
    ],
    [
      "9.791498752",
      "15.601637703"
    ],
    [
      "7.320302642",
      "4.835266796"
    ],
    [
      "18.382685848",
      "2.017942940"
    ],
    [
      "10.860552982",
      "13.911335050"
    ],
    [
      "6.074227064",
      "8.517719909"
    ],
    [
      "2.654066584",
      "2.000000000"
    ],
    [
      "16.205824129",
      "5.459821777"
    ],
    [
      "15.136769903",
      "7.150124431"
    ],
    [
      "8.249620564",
      "6.606246243"
    ],
    [
      "12.998661445",
      "10.530729736"
    ],
    [
      "12.069343520",
      "8.759750291"
    ],
    [
      "7.932862908",
      "12.059678806"
    ],
    [
      "5.650884883",
      "3.733883871"
    ],
    [
      "13.035358389",
      "3.758590565"
    ],
    [
      "9.989390171",
      "1.999999997"
    ],
    [
      "16.342782849",
      "1.999999994"
    ],
    [
      "14.207451977",
      "5.379144984"
    ],
    [
      "13.138397751",
      "7.069447638"
    ],
    [
      "4.511324159",
      "5.539332363"
    ],
    [
      "11.872837601",
      "5.520784502"
    ],
    [
      "10.247992715",
      "6.686923044"
    ],
    [
      "6.654066586",
      "1.999999999"
    ],
    [
      "14.082804343",
      "1.999999996"
    ],
    [
      "11.929607212",
      "12.221032395"
    ],
    [
      "12.082804342",
      "1.999999996"
    ]
  ],
  "triangle": [
    [
      "1.000000001",
      "1.000000000"
    ],
    [
      "20.209717362",
      "0.999999992"
    ],
    [
      "9.710821957",
      "17.600009856"
    ]
  ]
}

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": 41
}

当前第一名的答案

{
  "centers": [
    [
      "8.072599217",
      "8.598396700"
    ],
    [
      "10.070971368",
      "8.679073501"
    ],
    [
      "8.321728379",
      "3.104039919"
    ],
    [
      "17.274878360",
      "3.769519126"
    ],
    [
      "3.650884882",
      "3.733883872"
    ],
    [
      "6.251248412",
      "6.525569452"
    ],
    [
      "4.654066585",
      "1.999999999"
    ],
    [
      "15.212793607",
      "3.650189169"
    ],
    [
      "14.067715675",
      "8.840427084"
    ],
    [
      "7.003544989",
      "10.288699354"
    ],
    [
      "11.000289293",
      "10.450052946"
    ],
    [
      "9.404289103",
      "4.869680459"
    ],
    [
      "9.001917140",
      "10.369376155"
    ],
    [
      "8.862180830",
      "13.830658255"
    ],
    [
      "11.036097257",
      "3.704231283"
    ],
    [
      "9.931235060",
      "12.140355603"
    ],
    [
      "9.791498752",
      "15.601637703"
    ],
    [
      "7.320302642",
      "4.835266796"
    ],
    [
      "18.382685848",
      "2.017942940"
    ],
    [
      "10.860552982",
      "13.911335050"
    ],
    [
      "6.074227064",
      "8.517719909"
    ],
    [
      "2.654066584",
      "2.000000000"
    ],
    [
      "16.205824129",
      "5.459821777"
    ],
    [
      "15.136769903",
      "7.150124431"
    ],
    [
      "8.249620564",
      "6.606246243"
    ],
    [
      "12.998661445",
      "10.530729736"
    ],
    [
      "12.069343520",
      "8.759750291"
    ],
    [
      "7.932862908",
      "12.059678806"
    ],
    [
      "5.650884883",
      "3.733883871"
    ],
    [
      "13.035358389",
      "3.758590565"
    ],
    [
      "9.989390171",
      "1.999999997"
    ],
    [
      "16.342782849",
      "1.999999994"
    ],
    [
      "14.207451977",
      "5.379144984"
    ],
    [
      "13.138397751",
      "7.069447638"
    ],
    [
      "4.511324159",
      "5.539332363"
    ],
    [
      "11.872837601",
      "5.520784502"
    ],
    [
      "10.247992715",
      "6.686923044"
    ],
    [
      "6.654066586",
      "1.999999999"
    ],
    [
      "14.082804343",
      "1.999999996"
    ],
    [
      "11.929607212",
      "12.221032395"
    ],
    [
      "12.082804342",
      "1.999999996"
    ]
  ],
  "triangle": [
    [
      "1.000000001",
      "1.000000000"
    ],
    [
      "20.209717362",
      "0.999999992"
    ],
    [
      "9.710821957",
      "17.600009856"
    ]
  ]
}

提交按逆时针排列的三个三角形顶点 triangle 和恰好 n 个圆心坐标 centers。每个圆的半径固定为 1，所有数值写成最多九位小数的字符串。 · 验证器 v1.0.0

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