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 = 45

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 = 45

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	164.047404138929743008	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45 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→164.047404138929743008
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": [
    [
      "49.892795768",
      "34.459407656"
    ],
    [
      "50.581855254",
      "32.581856851"
    ],
    [
      "51.270914740",
      "30.704306046"
    ],
    [
      "51.959974226",
      "28.826755241"
    ],
    [
      "52.649033712",
      "26.949204436"
    ],
    [
      "53.338093198",
      "25.071653631"
    ],
    [
      "54.027152684",
      "23.194102826"
    ],
    [
      "54.716212170",
      "21.316552021"
    ],
    [
      "55.405271656",
      "19.439001216"
    ],
    [
      "51.863332205",
      "34.117375273"
    ],
    [
      "52.552391691",
      "32.239824468"
    ],
    [
      "53.241451177",
      "30.362273663"
    ],
    [
      "53.930510663",
      "28.484722858"
    ],
    [
      "54.619570149",
      "26.607172053"
    ],
    [
      "55.308629635",
      "24.729621248"
    ],
    [
      "55.997689121",
      "22.852070443"
    ],
    [
      "56.686748607",
      "20.974519638"
    ],
    [
      "53.833868642",
      "33.775342890"
    ],
    [
      "54.522928128",
      "31.897792085"
    ],
    [
      "55.211987614",
      "30.020241280"
    ],
    [
      "55.901047100",
      "28.142690475"
    ],
    [
      "56.590106586",
      "26.265139670"
    ],
    [
      "57.279166072",
      "24.387588865"
    ],
    [
      "57.968225558",
      "22.510038060"
    ],
    [
      "55.804405079",
      "33.433310507"
    ],
    [
      "56.493464565",
      "31.555759702"
    ],
    [
      "57.182524051",
      "29.678208897"
    ],
    [
      "57.871583537",
      "27.800658092"
    ],
    [
      "58.560643023",
      "25.923107287"
    ],
    [
      "59.249702509",
      "24.045556482"
    ],
    [
      "57.774941516",
      "33.091278124"
    ],
    [
      "58.464001002",
      "31.213727319"
    ],
    [
      "59.153060488",
      "29.336176514"
    ],
    [
      "59.842119974",
      "27.458625709"
    ],
    [
      "60.531179460",
      "25.581074904"
    ],
    [
      "59.745477953",
      "32.749245741"
    ],
    [
      "60.434537439",
      "30.871694936"
    ],
    [
      "61.123596925",
      "28.994144131"
    ],
    [
      "61.812656411",
      "27.116593326"
    ],
    [
      "61.716014390",
      "32.407213358"
    ],
    [
      "62.405073876",
      "30.529662553"
    ],
    [
      "63.094133362",
      "28.652111748"
    ],
    [
      "63.686550827",
      "32.065180975"
    ],
    [
      "64.375610313",
      "30.187630170"
    ],
    [
      "65.657087264",
      "31.723148592"
    ]
  ],
  "triangle": [
    [
      "48.357277346",
      "35.740884607"
    ],
    [
      "55.063239273",
      "17.468464779"
    ],
    [
      "67.534638070",
      "32.412208079"
    ]
  ]
}

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": 45
}

当前第一名的答案

{
  "centers": [
    [
      "49.892795768",
      "34.459407656"
    ],
    [
      "50.581855254",
      "32.581856851"
    ],
    [
      "51.270914740",
      "30.704306046"
    ],
    [
      "51.959974226",
      "28.826755241"
    ],
    [
      "52.649033712",
      "26.949204436"
    ],
    [
      "53.338093198",
      "25.071653631"
    ],
    [
      "54.027152684",
      "23.194102826"
    ],
    [
      "54.716212170",
      "21.316552021"
    ],
    [
      "55.405271656",
      "19.439001216"
    ],
    [
      "51.863332205",
      "34.117375273"
    ],
    [
      "52.552391691",
      "32.239824468"
    ],
    [
      "53.241451177",
      "30.362273663"
    ],
    [
      "53.930510663",
      "28.484722858"
    ],
    [
      "54.619570149",
      "26.607172053"
    ],
    [
      "55.308629635",
      "24.729621248"
    ],
    [
      "55.997689121",
      "22.852070443"
    ],
    [
      "56.686748607",
      "20.974519638"
    ],
    [
      "53.833868642",
      "33.775342890"
    ],
    [
      "54.522928128",
      "31.897792085"
    ],
    [
      "55.211987614",
      "30.020241280"
    ],
    [
      "55.901047100",
      "28.142690475"
    ],
    [
      "56.590106586",
      "26.265139670"
    ],
    [
      "57.279166072",
      "24.387588865"
    ],
    [
      "57.968225558",
      "22.510038060"
    ],
    [
      "55.804405079",
      "33.433310507"
    ],
    [
      "56.493464565",
      "31.555759702"
    ],
    [
      "57.182524051",
      "29.678208897"
    ],
    [
      "57.871583537",
      "27.800658092"
    ],
    [
      "58.560643023",
      "25.923107287"
    ],
    [
      "59.249702509",
      "24.045556482"
    ],
    [
      "57.774941516",
      "33.091278124"
    ],
    [
      "58.464001002",
      "31.213727319"
    ],
    [
      "59.153060488",
      "29.336176514"
    ],
    [
      "59.842119974",
      "27.458625709"
    ],
    [
      "60.531179460",
      "25.581074904"
    ],
    [
      "59.745477953",
      "32.749245741"
    ],
    [
      "60.434537439",
      "30.871694936"
    ],
    [
      "61.123596925",
      "28.994144131"
    ],
    [
      "61.812656411",
      "27.116593326"
    ],
    [
      "61.716014390",
      "32.407213358"
    ],
    [
      "62.405073876",
      "30.529662553"
    ],
    [
      "63.094133362",
      "28.652111748"
    ],
    [
      "63.686550827",
      "32.065180975"
    ],
    [
      "64.375610313",
      "30.187630170"
    ],
    [
      "65.657087264",
      "31.723148592"
    ]
  ],
  "triangle": [
    [
      "48.357277346",
      "35.740884607"
    ],
    [
      "55.063239273",
      "17.468464779"
    ],
    [
      "67.534638070",
      "32.412208079"
    ]
  ]
}

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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