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 = 50

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 = 50

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	[194.49896, 194.49897]	见来源署名	Erich Friedman ↗
本站纪录	193.318101562948860608	zzzcy	纪录详情 ↓

数值超出外部纪录区间 · 待审核。

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50 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zzzcy

解题方式AI · OpenAI: GPT-6 Astra

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）

zzzcyAI · OpenAI: GPT-6 Astra
193.3181016005745782485→193.318101562948860608
2026-10-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
194.498962963929252736→193.3181016005745782485
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": [
    [
      "11.921291368",
      "16.258961982"
    ],
    [
      "10.820145879",
      "14.589387600"
    ],
    [
      "12.816612451",
      "14.470554823"
    ],
    [
      "9.719000389",
      "12.919813220"
    ],
    [
      "11.715466961",
      "12.800980442"
    ],
    [
      "13.711933533",
      "12.682147665"
    ],
    [
      "8.617854898",
      "11.250238838"
    ],
    [
      "10.614321470",
      "11.131406062"
    ],
    [
      "12.610788043",
      "11.012573284"
    ],
    [
      "14.607254615",
      "10.893740507"
    ],
    [
      "7.516709408",
      "9.580664457"
    ],
    [
      "9.513175981",
      "9.461831681"
    ],
    [
      "13.506109126",
      "9.224166126"
    ],
    [
      "11.509642553",
      "9.342998904"
    ],
    [
      "15.502575698",
      "9.105333350"
    ],
    [
      "6.415563919",
      "7.911090076"
    ],
    [
      "8.412030491",
      "7.792257300"
    ],
    [
      "10.408497063",
      "7.673424523"
    ],
    [
      "12.404963636",
      "7.554591746"
    ],
    [
      "15.038961223",
      "7.159809868"
    ],
    [
      "16.874343823",
      "6.365221472"
    ],
    [
      "7.310885002",
      "6.122682919"
    ],
    [
      "9.307351576",
      "6.003850141"
    ],
    [
      "5.314418430",
      "6.241515695"
    ],
    [
      "11.303818149",
      "5.885017364"
    ],
    [
      "13.500788373",
      "5.881520275"
    ],
    [
      "15.755580527",
      "4.707400996"
    ],
    [
      "17.769664904",
      "4.576814314"
    ],
    [
      "4.213272940",
      "4.571941315"
    ],
    [
      "6.208012579",
      "4.419262515"
    ],
    [
      "8.206206085",
      "4.334275761"
    ],
    [
      "10.922897244",
      "3.921627624"
    ],
    [
      "13.034502248",
      "3.936635381"
    ],
    [
      "14.672052843",
      "2.788407157"
    ],
    [
      "16.668519415",
      "2.907239933"
    ],
    [
      "18.664985987",
      "2.788407157"
    ],
    [
      "3.000751550",
      "2.733497016"
    ],
    [
      "5.000751550",
      "2.733497016"
    ],
    [
      "7.138331507",
      "2.643227599"
    ],
    [
      "9.346295142",
      "2.691048162"
    ],
    [
      "11.990771822",
      "2.230579462"
    ],
    [
      "8.278420564",
      "1.000000000"
    ],
    [
      "10.414169720",
      "1.000000000"
    ],
    [
      "13.567373926",
      "1.000000000"
    ],
    [
      "19.560307070",
      "1.000000000"
    ],
    [
      "1.981551033",
      "1.012674439"
    ],
    [
      "3.998242450",
      "1.000000000"
    ],
    [
      "5.998242450",
      "1.000000000"
    ],
    [
      "15.567373926",
      "1.000000000"
    ],
    [
      "17.563840498",
      "1.118832769"
    ]
  ],
  "triangle": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "21.179245504",
      "0.000000000"
    ],
    [
      "12.040124145",
      "18.255428554"
    ]
  ]
}

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": 50
}

当前第一名的答案

{
  "centers": [
    [
      "11.921291368",
      "16.258961982"
    ],
    [
      "10.820145879",
      "14.589387600"
    ],
    [
      "12.816612451",
      "14.470554823"
    ],
    [
      "9.719000389",
      "12.919813220"
    ],
    [
      "11.715466961",
      "12.800980442"
    ],
    [
      "13.711933533",
      "12.682147665"
    ],
    [
      "8.617854898",
      "11.250238838"
    ],
    [
      "10.614321470",
      "11.131406062"
    ],
    [
      "12.610788043",
      "11.012573284"
    ],
    [
      "14.607254615",
      "10.893740507"
    ],
    [
      "7.516709408",
      "9.580664457"
    ],
    [
      "9.513175981",
      "9.461831681"
    ],
    [
      "13.506109126",
      "9.224166126"
    ],
    [
      "11.509642553",
      "9.342998904"
    ],
    [
      "15.502575698",
      "9.105333350"
    ],
    [
      "6.415563919",
      "7.911090076"
    ],
    [
      "8.412030491",
      "7.792257300"
    ],
    [
      "10.408497063",
      "7.673424523"
    ],
    [
      "12.404963636",
      "7.554591746"
    ],
    [
      "15.038961223",
      "7.159809868"
    ],
    [
      "16.874343823",
      "6.365221472"
    ],
    [
      "7.310885002",
      "6.122682919"
    ],
    [
      "9.307351576",
      "6.003850141"
    ],
    [
      "5.314418430",
      "6.241515695"
    ],
    [
      "11.303818149",
      "5.885017364"
    ],
    [
      "13.500788373",
      "5.881520275"
    ],
    [
      "15.755580527",
      "4.707400996"
    ],
    [
      "17.769664904",
      "4.576814314"
    ],
    [
      "4.213272940",
      "4.571941315"
    ],
    [
      "6.208012579",
      "4.419262515"
    ],
    [
      "8.206206085",
      "4.334275761"
    ],
    [
      "10.922897244",
      "3.921627624"
    ],
    [
      "13.034502248",
      "3.936635381"
    ],
    [
      "14.672052843",
      "2.788407157"
    ],
    [
      "16.668519415",
      "2.907239933"
    ],
    [
      "18.664985987",
      "2.788407157"
    ],
    [
      "3.000751550",
      "2.733497016"
    ],
    [
      "5.000751550",
      "2.733497016"
    ],
    [
      "7.138331507",
      "2.643227599"
    ],
    [
      "9.346295142",
      "2.691048162"
    ],
    [
      "11.990771822",
      "2.230579462"
    ],
    [
      "8.278420564",
      "1.000000000"
    ],
    [
      "10.414169720",
      "1.000000000"
    ],
    [
      "13.567373926",
      "1.000000000"
    ],
    [
      "19.560307070",
      "1.000000000"
    ],
    [
      "1.981551033",
      "1.012674439"
    ],
    [
      "3.998242450",
      "1.000000000"
    ],
    [
      "5.998242450",
      "1.000000000"
    ],
    [
      "15.567373926",
      "1.000000000"
    ],
    [
      "17.563840498",
      "1.118832769"
    ]
  ],
  "triangle": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "21.179245504",
      "0.000000000"
    ],
    [
      "12.040124145",
      "18.255428554"
    ]
  ]
}

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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