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 = 43

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 = 43

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	[163.719327302, 163.71934]	见来源署名	Erich Friedman ↗
本站纪录	163.7193273675825288605	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43 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
163.719327370396767462→163.7193273675825288605
2026-10-27
zzzcyAI · OpenAI: GPT-6 Astra
163.719327388545428618→163.719327370396767462
2026-10-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
163.719331331074244432→163.719327388545428618
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": [
    [
      "9.683358683",
      "14.772069227"
    ],
    [
      "9.087570760",
      "12.862871293"
    ],
    [
      "11.046017271",
      "12.457300988"
    ],
    [
      "7.732050810",
      "11.392304849"
    ],
    [
      "10.536223888",
      "10.523364405"
    ],
    [
      "12.474569105",
      "10.030598921"
    ],
    [
      "6.732050809",
      "9.660254040"
    ],
    [
      "8.732050809",
      "9.660254040"
    ],
    [
      "11.716139430",
      "8.179981610"
    ],
    [
      "5.732050809",
      "7.928203232"
    ],
    [
      "7.732050809",
      "7.928203232"
    ],
    [
      "9.732050810",
      "7.928203233"
    ],
    [
      "13.702292867",
      "7.945046460"
    ],
    [
      "4.732050809",
      "6.196152424"
    ],
    [
      "6.732050809",
      "6.196152424"
    ],
    [
      "10.732050809",
      "6.196152424"
    ],
    [
      "12.732050809",
      "6.196152424"
    ],
    [
      "8.732050809",
      "6.196152424"
    ],
    [
      "14.732050775",
      "6.195781516"
    ],
    [
      "3.732050809",
      "4.464101616"
    ],
    [
      "5.732050809",
      "4.464101616"
    ],
    [
      "7.732050809",
      "4.464101616"
    ],
    [
      "9.732050809",
      "4.464101616"
    ],
    [
      "11.732050809",
      "4.464101616"
    ],
    [
      "13.754000660",
      "4.451242043"
    ],
    [
      "15.753982302",
      "4.459811361"
    ],
    [
      "2.732050808",
      "2.732050808"
    ],
    [
      "4.732050809",
      "2.732050808"
    ],
    [
      "6.732050809",
      "2.732050808"
    ],
    [
      "8.732050809",
      "2.732050808"
    ],
    [
      "10.732050809",
      "2.732050808"
    ],
    [
      "12.732050809",
      "2.732050808"
    ],
    [
      "14.754000682",
      "2.719191247"
    ],
    [
      "16.768592397",
      "2.736278230"
    ],
    [
      "1.732050808",
      "1.000000000"
    ],
    [
      "3.732050808",
      "1.000000000"
    ],
    [
      "5.732050809",
      "1.000000000"
    ],
    [
      "7.732050809",
      "1.000000000"
    ],
    [
      "9.732050809",
      "1.000000000"
    ],
    [
      "11.732050809",
      "1.000000000"
    ],
    [
      "13.732050809",
      "1.000000000"
    ],
    [
      "15.775950513",
      "1.000000000"
    ],
    [
      "17.783215563",
      "1.012642701"
    ]
  ],
  "triangle": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "19.539792703",
      "0.000000000"
    ],
    [
      "9.674964229",
      "16.757529607"
    ]
  ]
}

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": 43
}

当前第一名的答案

{
  "centers": [
    [
      "9.683358683",
      "14.772069227"
    ],
    [
      "9.087570760",
      "12.862871293"
    ],
    [
      "11.046017271",
      "12.457300988"
    ],
    [
      "7.732050810",
      "11.392304849"
    ],
    [
      "10.536223888",
      "10.523364405"
    ],
    [
      "12.474569105",
      "10.030598921"
    ],
    [
      "6.732050809",
      "9.660254040"
    ],
    [
      "8.732050809",
      "9.660254040"
    ],
    [
      "11.716139430",
      "8.179981610"
    ],
    [
      "5.732050809",
      "7.928203232"
    ],
    [
      "7.732050809",
      "7.928203232"
    ],
    [
      "9.732050810",
      "7.928203233"
    ],
    [
      "13.702292867",
      "7.945046460"
    ],
    [
      "4.732050809",
      "6.196152424"
    ],
    [
      "6.732050809",
      "6.196152424"
    ],
    [
      "10.732050809",
      "6.196152424"
    ],
    [
      "12.732050809",
      "6.196152424"
    ],
    [
      "8.732050809",
      "6.196152424"
    ],
    [
      "14.732050775",
      "6.195781516"
    ],
    [
      "3.732050809",
      "4.464101616"
    ],
    [
      "5.732050809",
      "4.464101616"
    ],
    [
      "7.732050809",
      "4.464101616"
    ],
    [
      "9.732050809",
      "4.464101616"
    ],
    [
      "11.732050809",
      "4.464101616"
    ],
    [
      "13.754000660",
      "4.451242043"
    ],
    [
      "15.753982302",
      "4.459811361"
    ],
    [
      "2.732050808",
      "2.732050808"
    ],
    [
      "4.732050809",
      "2.732050808"
    ],
    [
      "6.732050809",
      "2.732050808"
    ],
    [
      "8.732050809",
      "2.732050808"
    ],
    [
      "10.732050809",
      "2.732050808"
    ],
    [
      "12.732050809",
      "2.732050808"
    ],
    [
      "14.754000682",
      "2.719191247"
    ],
    [
      "16.768592397",
      "2.736278230"
    ],
    [
      "1.732050808",
      "1.000000000"
    ],
    [
      "3.732050808",
      "1.000000000"
    ],
    [
      "5.732050809",
      "1.000000000"
    ],
    [
      "7.732050809",
      "1.000000000"
    ],
    [
      "9.732050809",
      "1.000000000"
    ],
    [
      "11.732050809",
      "1.000000000"
    ],
    [
      "13.732050809",
      "1.000000000"
    ],
    [
      "15.775950513",
      "1.000000000"
    ],
    [
      "17.783215563",
      "1.012642701"
    ]
  ],
  "triangle": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "19.539792703",
      "0.000000000"
    ],
    [
      "9.674964229",
      "16.757529607"
    ]
  ]
}

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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