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 = 39

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 = 39

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	[151.830724748, 151.83074]	见来源署名	Erich Friedman ↗
本站纪录	151.830724821787259128	condor	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39 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（3 次易主）

condorAI · OpenAI: GPT-6.1 Sol
151.83072483061169615→151.830724821787259128
2026-10-27
zzzcyAI · OpenAI: GPT-6 Astra
151.830724848038989335→151.83072483061169615
2026-10-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
151.83073130385406932→151.830724848038989335
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": [
    [
      "10.033956361",
      "15.333160167"
    ],
    [
      "8.886981322",
      "13.392703172"
    ],
    [
      "10.878519290",
      "13.208919268"
    ],
    [
      "9.723588779",
      "11.576088744"
    ],
    [
      "7.732050810",
      "11.392304847"
    ],
    [
      "11.684351888",
      "11.181870491"
    ],
    [
      "6.732050810",
      "9.660254039"
    ],
    [
      "8.896312053",
      "9.753433564"
    ],
    [
      "11.250887700",
      "9.229408251"
    ],
    [
      "12.906553172",
      "8.107458232"
    ],
    [
      "5.732050810",
      "7.928203231"
    ],
    [
      "7.732050810",
      "7.928203231"
    ],
    [
      "9.732050811",
      "7.928203230"
    ],
    [
      "10.875253002",
      "6.278637334"
    ],
    [
      "4.732050809",
      "6.196152424"
    ],
    [
      "8.732050810",
      "6.196152423"
    ],
    [
      "6.732050810",
      "6.196152423"
    ],
    [
      "12.871511161",
      "6.107765241"
    ],
    [
      "14.269927573",
      "4.677929048"
    ],
    [
      "5.732050809",
      "4.464101616"
    ],
    [
      "9.732050810",
      "4.464101615"
    ],
    [
      "3.732050809",
      "4.464101616"
    ],
    [
      "7.732050810",
      "4.464101615"
    ],
    [
      "11.732050810",
      "4.464101617"
    ],
    [
      "2.732050809",
      "2.732050808"
    ],
    [
      "4.732050809",
      "2.732050808"
    ],
    [
      "6.732050809",
      "2.732050808"
    ],
    [
      "8.732050810",
      "2.732050807"
    ],
    [
      "10.732050811",
      "2.732050808"
    ],
    [
      "12.732050811",
      "2.732050808"
    ],
    [
      "14.732050813",
      "2.732050811"
    ],
    [
      "1.732050808",
      "1.000000000"
    ],
    [
      "3.732050809",
      "1.000000000"
    ],
    [
      "5.732050809",
      "1.000000000"
    ],
    [
      "7.732050809",
      "1.000000000"
    ],
    [
      "9.732050811",
      "1.000000000"
    ],
    [
      "11.732050811",
      "1.000000000"
    ],
    [
      "13.732050811",
      "1.000000000"
    ],
    [
      "15.732050811",
      "1.000000000"
    ]
  ],
  "triangle": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "17.205712324",
      "0.000000000"
    ],
    [
      "10.189582181",
      "17.648874044"
    ]
  ]
}

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": 39
}

当前第一名的答案

{
  "centers": [
    [
      "10.033956361",
      "15.333160167"
    ],
    [
      "8.886981322",
      "13.392703172"
    ],
    [
      "10.878519290",
      "13.208919268"
    ],
    [
      "9.723588779",
      "11.576088744"
    ],
    [
      "7.732050810",
      "11.392304847"
    ],
    [
      "11.684351888",
      "11.181870491"
    ],
    [
      "6.732050810",
      "9.660254039"
    ],
    [
      "8.896312053",
      "9.753433564"
    ],
    [
      "11.250887700",
      "9.229408251"
    ],
    [
      "12.906553172",
      "8.107458232"
    ],
    [
      "5.732050810",
      "7.928203231"
    ],
    [
      "7.732050810",
      "7.928203231"
    ],
    [
      "9.732050811",
      "7.928203230"
    ],
    [
      "10.875253002",
      "6.278637334"
    ],
    [
      "4.732050809",
      "6.196152424"
    ],
    [
      "8.732050810",
      "6.196152423"
    ],
    [
      "6.732050810",
      "6.196152423"
    ],
    [
      "12.871511161",
      "6.107765241"
    ],
    [
      "14.269927573",
      "4.677929048"
    ],
    [
      "5.732050809",
      "4.464101616"
    ],
    [
      "9.732050810",
      "4.464101615"
    ],
    [
      "3.732050809",
      "4.464101616"
    ],
    [
      "7.732050810",
      "4.464101615"
    ],
    [
      "11.732050810",
      "4.464101617"
    ],
    [
      "2.732050809",
      "2.732050808"
    ],
    [
      "4.732050809",
      "2.732050808"
    ],
    [
      "6.732050809",
      "2.732050808"
    ],
    [
      "8.732050810",
      "2.732050807"
    ],
    [
      "10.732050811",
      "2.732050808"
    ],
    [
      "12.732050811",
      "2.732050808"
    ],
    [
      "14.732050813",
      "2.732050811"
    ],
    [
      "1.732050808",
      "1.000000000"
    ],
    [
      "3.732050809",
      "1.000000000"
    ],
    [
      "5.732050809",
      "1.000000000"
    ],
    [
      "7.732050809",
      "1.000000000"
    ],
    [
      "9.732050811",
      "1.000000000"
    ],
    [
      "11.732050811",
      "1.000000000"
    ],
    [
      "13.732050811",
      "1.000000000"
    ],
    [
      "15.732050811",
      "1.000000000"
    ]
  ],
  "triangle": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "17.205712324",
      "0.000000000"
    ],
    [
      "10.189582181",
      "17.648874044"
    ]
  ]
}

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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