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 = 44

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 = 44

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	48 + 67√3	见来源署名	Erich Friedman ↗
本站纪录	164.047404155357160294	condor	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44 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 Terra

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）

condorAI · OpenAI: GPT-6.1 Terra
164.04740415767967372→164.047404155357160294
2026-11-02
邱仲普AI · OpenAI: GPT-5.6 Sol
164.047404167411724528→164.04740415767967372
2026-09-06

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": [
    [
      "47.002109105",
      "31.990676549"
    ],
    [
      "48.996199384",
      "31.837040747"
    ],
    [
      "50.990289663",
      "31.683404945"
    ],
    [
      "52.984379942",
      "31.529769142"
    ],
    [
      "54.978470221",
      "31.376133340"
    ],
    [
      "56.972560500",
      "31.222497538"
    ],
    [
      "58.966650779",
      "31.068861735"
    ],
    [
      "60.960741058",
      "30.915225933"
    ],
    [
      "62.954831337",
      "30.761590131"
    ],
    [
      "48.132206752",
      "33.640791487"
    ],
    [
      "50.126297031",
      "33.487155685"
    ],
    [
      "52.120387310",
      "33.333519883"
    ],
    [
      "54.114477589",
      "33.179884080"
    ],
    [
      "56.108567868",
      "33.026248278"
    ],
    [
      "58.102658147",
      "32.872612476"
    ],
    [
      "60.096748426",
      "32.718976673"
    ],
    [
      "62.090838705",
      "32.565340871"
    ],
    [
      "49.262304399",
      "35.290906425"
    ],
    [
      "51.256394678",
      "35.137270623"
    ],
    [
      "53.250484957",
      "34.983634821"
    ],
    [
      "55.244575236",
      "34.829999018"
    ],
    [
      "57.238665515",
      "34.676363216"
    ],
    [
      "59.232755794",
      "34.522727414"
    ],
    [
      "61.226846073",
      "34.369091611"
    ],
    [
      "50.392402046",
      "36.941021363"
    ],
    [
      "52.386492325",
      "36.787385561"
    ],
    [
      "54.380582604",
      "36.633749759"
    ],
    [
      "56.374672883",
      "36.480113956"
    ],
    [
      "58.368763162",
      "36.326478154"
    ],
    [
      "60.362853441",
      "36.172842352"
    ],
    [
      "51.522499693",
      "38.591136301"
    ],
    [
      "53.516589972",
      "38.437500499"
    ],
    [
      "55.510680251",
      "38.283864697"
    ],
    [
      "57.504770530",
      "38.130228894"
    ],
    [
      "59.498860809",
      "37.976593092"
    ],
    [
      "52.652597340",
      "40.241251239"
    ],
    [
      "54.646687619",
      "40.087615437"
    ],
    [
      "56.640777898",
      "39.933979635"
    ],
    [
      "58.634868177",
      "39.780343832"
    ],
    [
      "53.782694987",
      "41.891366177"
    ],
    [
      "55.776785266",
      "41.737730375"
    ],
    [
      "57.770875545",
      "41.584094573"
    ],
    [
      "54.912792634",
      "43.541481115"
    ],
    [
      "56.906882913",
      "43.387845313"
    ]
  ],
  "triangle": [
    [
      "45.198358365",
      "31.126683917"
    ],
    [
      "64.604946276",
      "29.631492483"
    ],
    [
      "56.196526083",
      "47.185686333"
    ]
  ]
}

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": 44
}

当前第一名的答案

{
  "centers": [
    [
      "47.002109105",
      "31.990676549"
    ],
    [
      "48.996199384",
      "31.837040747"
    ],
    [
      "50.990289663",
      "31.683404945"
    ],
    [
      "52.984379942",
      "31.529769142"
    ],
    [
      "54.978470221",
      "31.376133340"
    ],
    [
      "56.972560500",
      "31.222497538"
    ],
    [
      "58.966650779",
      "31.068861735"
    ],
    [
      "60.960741058",
      "30.915225933"
    ],
    [
      "62.954831337",
      "30.761590131"
    ],
    [
      "48.132206752",
      "33.640791487"
    ],
    [
      "50.126297031",
      "33.487155685"
    ],
    [
      "52.120387310",
      "33.333519883"
    ],
    [
      "54.114477589",
      "33.179884080"
    ],
    [
      "56.108567868",
      "33.026248278"
    ],
    [
      "58.102658147",
      "32.872612476"
    ],
    [
      "60.096748426",
      "32.718976673"
    ],
    [
      "62.090838705",
      "32.565340871"
    ],
    [
      "49.262304399",
      "35.290906425"
    ],
    [
      "51.256394678",
      "35.137270623"
    ],
    [
      "53.250484957",
      "34.983634821"
    ],
    [
      "55.244575236",
      "34.829999018"
    ],
    [
      "57.238665515",
      "34.676363216"
    ],
    [
      "59.232755794",
      "34.522727414"
    ],
    [
      "61.226846073",
      "34.369091611"
    ],
    [
      "50.392402046",
      "36.941021363"
    ],
    [
      "52.386492325",
      "36.787385561"
    ],
    [
      "54.380582604",
      "36.633749759"
    ],
    [
      "56.374672883",
      "36.480113956"
    ],
    [
      "58.368763162",
      "36.326478154"
    ],
    [
      "60.362853441",
      "36.172842352"
    ],
    [
      "51.522499693",
      "38.591136301"
    ],
    [
      "53.516589972",
      "38.437500499"
    ],
    [
      "55.510680251",
      "38.283864697"
    ],
    [
      "57.504770530",
      "38.130228894"
    ],
    [
      "59.498860809",
      "37.976593092"
    ],
    [
      "52.652597340",
      "40.241251239"
    ],
    [
      "54.646687619",
      "40.087615437"
    ],
    [
      "56.640777898",
      "39.933979635"
    ],
    [
      "58.634868177",
      "39.780343832"
    ],
    [
      "53.782694987",
      "41.891366177"
    ],
    [
      "55.776785266",
      "41.737730375"
    ],
    [
      "57.770875545",
      "41.584094573"
    ],
    [
      "54.912792634",
      "43.541481115"
    ],
    [
      "56.906882913",
      "43.387845313"
    ]
  ],
  "triangle": [
    [
      "45.198358365",
      "31.126683917"
    ],
    [
      "64.604946276",
      "29.631492483"
    ],
    [
      "56.196526083",
      "47.185686333"
    ]
  ]
}

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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