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 = 32

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 = 32

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	[127.07490, 127.07491]	见来源署名	Erich Friedman ↗
本站纪录	127.074904016320079535	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32 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（3 次易主）

condorAI · OpenAI: GPT-6.1 Terra
127.074904155875398448→127.074904016320079535
2026-11-02
邱仲普AI · OpenAI: GPT-5.6 Sol
127.074904163242121064→127.074904155875398448
2026-09-09
邱仲普AI · OpenAI: GPT-5.6 Sol
127.07490417923377167→127.074904163242121064
2026-09-05

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": [
    [
      "21.544940164",
      "20.999999999"
    ],
    [
      "24.417384758",
      "20.999999999"
    ],
    [
      "26.417384758",
      "21.000000000"
    ],
    [
      "28.417384759",
      "21.000000001"
    ],
    [
      "30.417384760",
      "21.000000000"
    ],
    [
      "32.417384762",
      "21.000000000"
    ],
    [
      "34.417384763",
      "21.000000000"
    ],
    [
      "22.981162461",
      "22.391856858"
    ],
    [
      "25.204169322",
      "22.863046289"
    ],
    [
      "27.417384758",
      "22.732050808"
    ],
    [
      "29.417384760",
      "22.732050808"
    ],
    [
      "31.417384761",
      "22.732050807"
    ],
    [
      "33.417384762",
      "22.732050808"
    ],
    [
      "23.066499777",
      "24.390035415"
    ],
    [
      "26.417384758",
      "24.464101617"
    ],
    [
      "28.417384759",
      "24.464101615"
    ],
    [
      "30.417384761",
      "24.464101615"
    ],
    [
      "32.417384762",
      "24.464101616"
    ],
    [
      "24.717820710",
      "25.518370107"
    ],
    [
      "27.417384759",
      "26.196152424"
    ],
    [
      "29.417384760",
      "26.196152422"
    ],
    [
      "31.417384761",
      "26.196152423"
    ],
    [
      "24.463300923",
      "27.502108921"
    ],
    [
      "26.417384759",
      "27.928203233"
    ],
    [
      "28.417384760",
      "27.928203232"
    ],
    [
      "30.417384761",
      "27.928203231"
    ],
    [
      "25.436952521",
      "29.671405065"
    ],
    [
      "27.427721945",
      "29.863335010"
    ],
    [
      "29.417384761",
      "29.660254039"
    ],
    [
      "26.255910648",
      "31.496044095"
    ],
    [
      "28.246680072",
      "31.687974040"
    ],
    [
      "27.075266421",
      "33.321569065"
    ]
  ],
  "triangle": [
    [
      "19.999999996",
      "19.999999998"
    ],
    [
      "36.149435571",
      "20.000000000"
    ],
    [
      "27.063454861",
      "35.737380222"
    ]
  ]
}

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": 32
}

当前第一名的答案

{
  "centers": [
    [
      "21.544940164",
      "20.999999999"
    ],
    [
      "24.417384758",
      "20.999999999"
    ],
    [
      "26.417384758",
      "21.000000000"
    ],
    [
      "28.417384759",
      "21.000000001"
    ],
    [
      "30.417384760",
      "21.000000000"
    ],
    [
      "32.417384762",
      "21.000000000"
    ],
    [
      "34.417384763",
      "21.000000000"
    ],
    [
      "22.981162461",
      "22.391856858"
    ],
    [
      "25.204169322",
      "22.863046289"
    ],
    [
      "27.417384758",
      "22.732050808"
    ],
    [
      "29.417384760",
      "22.732050808"
    ],
    [
      "31.417384761",
      "22.732050807"
    ],
    [
      "33.417384762",
      "22.732050808"
    ],
    [
      "23.066499777",
      "24.390035415"
    ],
    [
      "26.417384758",
      "24.464101617"
    ],
    [
      "28.417384759",
      "24.464101615"
    ],
    [
      "30.417384761",
      "24.464101615"
    ],
    [
      "32.417384762",
      "24.464101616"
    ],
    [
      "24.717820710",
      "25.518370107"
    ],
    [
      "27.417384759",
      "26.196152424"
    ],
    [
      "29.417384760",
      "26.196152422"
    ],
    [
      "31.417384761",
      "26.196152423"
    ],
    [
      "24.463300923",
      "27.502108921"
    ],
    [
      "26.417384759",
      "27.928203233"
    ],
    [
      "28.417384760",
      "27.928203232"
    ],
    [
      "30.417384761",
      "27.928203231"
    ],
    [
      "25.436952521",
      "29.671405065"
    ],
    [
      "27.427721945",
      "29.863335010"
    ],
    [
      "29.417384761",
      "29.660254039"
    ],
    [
      "26.255910648",
      "31.496044095"
    ],
    [
      "28.246680072",
      "31.687974040"
    ],
    [
      "27.075266421",
      "33.321569065"
    ]
  ],
  "triangle": [
    [
      "19.999999996",
      "19.999999998"
    ],
    [
      "36.149435571",
      "20.000000000"
    ],
    [
      "27.063454861",
      "35.737380222"
    ]
  ]
}

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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