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 = 35

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 = 35

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	132.0666420093831393505	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35 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
132.066642035186220856→132.0666420093831393505
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": [
    [
      "43.801936853",
      "48.442856522"
    ],
    [
      "41.810100688",
      "48.623379906"
    ],
    [
      "39.818264523",
      "48.803903290"
    ],
    [
      "37.826428358",
      "48.984426674"
    ],
    [
      "35.834592193",
      "49.164950058"
    ],
    [
      "33.842756028",
      "49.345473442"
    ],
    [
      "31.850919863",
      "49.525996826"
    ],
    [
      "42.649680934",
      "46.808137495"
    ],
    [
      "40.657844769",
      "46.988660879"
    ],
    [
      "38.666008604",
      "47.169184263"
    ],
    [
      "36.674172439",
      "47.349707647"
    ],
    [
      "34.682336274",
      "47.530231031"
    ],
    [
      "32.690500109",
      "47.710754415"
    ],
    [
      "30.698663944",
      "47.891277799"
    ],
    [
      "41.497425015",
      "45.173418468"
    ],
    [
      "39.505588850",
      "45.353941852"
    ],
    [
      "37.513752685",
      "45.534465236"
    ],
    [
      "35.521916520",
      "45.714988620"
    ],
    [
      "33.530080355",
      "45.895512004"
    ],
    [
      "31.538244190",
      "46.076035388"
    ],
    [
      "40.345169096",
      "43.538699441"
    ],
    [
      "38.353332931",
      "43.719222825"
    ],
    [
      "36.361496766",
      "43.899746209"
    ],
    [
      "34.369660601",
      "44.080269593"
    ],
    [
      "32.377824436",
      "44.260792977"
    ],
    [
      "39.192913177",
      "41.903980414"
    ],
    [
      "37.201077012",
      "42.084503798"
    ],
    [
      "35.209240847",
      "42.265027182"
    ],
    [
      "33.217404682",
      "42.445550566"
    ],
    [
      "38.040657258",
      "40.269261387"
    ],
    [
      "36.048821093",
      "40.449784771"
    ],
    [
      "34.056984928",
      "40.630308155"
    ],
    [
      "36.888401339",
      "38.634542360"
    ],
    [
      "34.896565174",
      "38.815065744"
    ],
    [
      "35.736145420",
      "36.999823333"
    ]
  ],
  "triangle": [
    [
      "45.617179264",
      "49.282436768"
    ],
    [
      "28.224364671",
      "50.858776129"
    ],
    [
      "35.555622036",
      "35.007987167"
    ]
  ]
}

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": 35
}

当前第一名的答案

{
  "centers": [
    [
      "43.801936853",
      "48.442856522"
    ],
    [
      "41.810100688",
      "48.623379906"
    ],
    [
      "39.818264523",
      "48.803903290"
    ],
    [
      "37.826428358",
      "48.984426674"
    ],
    [
      "35.834592193",
      "49.164950058"
    ],
    [
      "33.842756028",
      "49.345473442"
    ],
    [
      "31.850919863",
      "49.525996826"
    ],
    [
      "42.649680934",
      "46.808137495"
    ],
    [
      "40.657844769",
      "46.988660879"
    ],
    [
      "38.666008604",
      "47.169184263"
    ],
    [
      "36.674172439",
      "47.349707647"
    ],
    [
      "34.682336274",
      "47.530231031"
    ],
    [
      "32.690500109",
      "47.710754415"
    ],
    [
      "30.698663944",
      "47.891277799"
    ],
    [
      "41.497425015",
      "45.173418468"
    ],
    [
      "39.505588850",
      "45.353941852"
    ],
    [
      "37.513752685",
      "45.534465236"
    ],
    [
      "35.521916520",
      "45.714988620"
    ],
    [
      "33.530080355",
      "45.895512004"
    ],
    [
      "31.538244190",
      "46.076035388"
    ],
    [
      "40.345169096",
      "43.538699441"
    ],
    [
      "38.353332931",
      "43.719222825"
    ],
    [
      "36.361496766",
      "43.899746209"
    ],
    [
      "34.369660601",
      "44.080269593"
    ],
    [
      "32.377824436",
      "44.260792977"
    ],
    [
      "39.192913177",
      "41.903980414"
    ],
    [
      "37.201077012",
      "42.084503798"
    ],
    [
      "35.209240847",
      "42.265027182"
    ],
    [
      "33.217404682",
      "42.445550566"
    ],
    [
      "38.040657258",
      "40.269261387"
    ],
    [
      "36.048821093",
      "40.449784771"
    ],
    [
      "34.056984928",
      "40.630308155"
    ],
    [
      "36.888401339",
      "38.634542360"
    ],
    [
      "34.896565174",
      "38.815065744"
    ],
    [
      "35.736145420",
      "36.999823333"
    ]
  ],
  "triangle": [
    [
      "45.617179264",
      "49.282436768"
    ],
    [
      "28.224364671",
      "50.858776129"
    ],
    [
      "35.555622036",
      "35.007987167"
    ]
  ]
}

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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