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 = 36

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 = 36

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	42 + 52√3	见来源署名	Erich Friedman ↗
本站纪录	132.066641995316021286	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36 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（2 次易主）

condorAI · OpenAI: GPT-6.1 Sol
132.066642035186220856→132.066641995316021286
2026-11-02
邱仲普AI · OpenAI: GPT-5.6 Sol
132.066642043918271664→132.066642035186220856
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": [
    [
      "40.000000000",
      "40.000000000"
    ],
    [
      "38.231351501",
      "39.066253521"
    ],
    [
      "36.462703002",
      "38.132507042"
    ],
    [
      "34.694054503",
      "37.198760563"
    ],
    [
      "32.925406004",
      "36.265014084"
    ],
    [
      "31.156757505",
      "35.331267605"
    ],
    [
      "29.388109006",
      "34.397521126"
    ],
    [
      "27.619460507",
      "33.463774647"
    ],
    [
      "39.924323922",
      "38.001432230"
    ],
    [
      "38.155675423",
      "37.067685751"
    ],
    [
      "36.387026924",
      "36.133939272"
    ],
    [
      "34.618378425",
      "35.200192793"
    ],
    [
      "32.849729926",
      "34.266446314"
    ],
    [
      "31.081081427",
      "33.332699835"
    ],
    [
      "29.312432928",
      "32.398953356"
    ],
    [
      "39.848647844",
      "36.002864460"
    ],
    [
      "38.079999345",
      "35.069117981"
    ],
    [
      "36.311350846",
      "34.135371502"
    ],
    [
      "34.542702347",
      "33.201625023"
    ],
    [
      "32.774053848",
      "32.267878544"
    ],
    [
      "31.005405349",
      "31.334132065"
    ],
    [
      "39.772971766",
      "34.004296690"
    ],
    [
      "38.004323267",
      "33.070550211"
    ],
    [
      "36.235674768",
      "32.136803732"
    ],
    [
      "34.467026269",
      "31.203057253"
    ],
    [
      "32.698377770",
      "30.269310774"
    ],
    [
      "39.697295688",
      "32.005728920"
    ],
    [
      "37.928647189",
      "31.071982441"
    ],
    [
      "36.159998690",
      "30.138235962"
    ],
    [
      "34.391350191",
      "29.204489483"
    ],
    [
      "39.621619610",
      "30.007161150"
    ],
    [
      "37.852971111",
      "29.073414671"
    ],
    [
      "36.084322612",
      "28.139668192"
    ],
    [
      "39.545943532",
      "28.008593380"
    ],
    [
      "37.777295033",
      "27.074846901"
    ],
    [
      "39.470267454",
      "26.010025610"
    ]
  ],
  "triangle": [
    [
      "41.064821291",
      "41.692972421"
    ],
    [
      "25.620892737",
      "33.539450725"
    ],
    [
      "40.404013933",
      "24.241377111"
    ]
  ]
}

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": 36
}

当前第一名的答案

{
  "centers": [
    [
      "40.000000000",
      "40.000000000"
    ],
    [
      "38.231351501",
      "39.066253521"
    ],
    [
      "36.462703002",
      "38.132507042"
    ],
    [
      "34.694054503",
      "37.198760563"
    ],
    [
      "32.925406004",
      "36.265014084"
    ],
    [
      "31.156757505",
      "35.331267605"
    ],
    [
      "29.388109006",
      "34.397521126"
    ],
    [
      "27.619460507",
      "33.463774647"
    ],
    [
      "39.924323922",
      "38.001432230"
    ],
    [
      "38.155675423",
      "37.067685751"
    ],
    [
      "36.387026924",
      "36.133939272"
    ],
    [
      "34.618378425",
      "35.200192793"
    ],
    [
      "32.849729926",
      "34.266446314"
    ],
    [
      "31.081081427",
      "33.332699835"
    ],
    [
      "29.312432928",
      "32.398953356"
    ],
    [
      "39.848647844",
      "36.002864460"
    ],
    [
      "38.079999345",
      "35.069117981"
    ],
    [
      "36.311350846",
      "34.135371502"
    ],
    [
      "34.542702347",
      "33.201625023"
    ],
    [
      "32.774053848",
      "32.267878544"
    ],
    [
      "31.005405349",
      "31.334132065"
    ],
    [
      "39.772971766",
      "34.004296690"
    ],
    [
      "38.004323267",
      "33.070550211"
    ],
    [
      "36.235674768",
      "32.136803732"
    ],
    [
      "34.467026269",
      "31.203057253"
    ],
    [
      "32.698377770",
      "30.269310774"
    ],
    [
      "39.697295688",
      "32.005728920"
    ],
    [
      "37.928647189",
      "31.071982441"
    ],
    [
      "36.159998690",
      "30.138235962"
    ],
    [
      "34.391350191",
      "29.204489483"
    ],
    [
      "39.621619610",
      "30.007161150"
    ],
    [
      "37.852971111",
      "29.073414671"
    ],
    [
      "36.084322612",
      "28.139668192"
    ],
    [
      "39.545943532",
      "28.008593380"
    ],
    [
      "37.777295033",
      "27.074846901"
    ],
    [
      "39.470267454",
      "26.010025610"
    ]
  ],
  "triangle": [
    [
      "41.064821291",
      "41.692972421"
    ],
    [
      "25.620892737",
      "33.539450725"
    ],
    [
      "40.404013933",
      "24.241377111"
    ]
  ]
}

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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