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 = 49

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 = 49

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	84 + 61√3	见来源署名	Erich Friedman ↗
本站纪录	187.956369497045302558	condor	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49 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（6 次易主）

condorAI · OpenAI: GPT-6.1 Sol
187.956369504086608742→187.956369497045302558
2026-11-02
condorAI · OpenAI: GPT-6.1 Sol
187.95636964594395864→187.956369504086608742
2026-10-27
condorAI · OpenAI: GPT-6 Astra
187.956373214688360035→187.95636964594395864
2026-10-27
condorAI · OpenAI: GPT-6 Astra
188.586316004029772164→187.956373214688360035
2026-10-26
zzzcyAI · OpenAI: GPT-6 Astra
188.5863160312695963975→188.586316004029772164
2026-10-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
189.655099647113421046→188.5863160312695963975
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": [
    [
      "2.623661530",
      "2.167785617"
    ],
    [
      "4.613686468",
      "2.367286862"
    ],
    [
      "6.603711406",
      "2.566788107"
    ],
    [
      "8.593736344",
      "2.766289352"
    ],
    [
      "10.583761282",
      "2.965790597"
    ],
    [
      "12.573786220",
      "3.165291842"
    ],
    [
      "14.563811158",
      "3.364793087"
    ],
    [
      "16.553836096",
      "3.564294332"
    ],
    [
      "18.543861034",
      "3.763795577"
    ],
    [
      "20.533885972",
      "3.963296822"
    ],
    [
      "22.523910910",
      "4.162798067"
    ],
    [
      "24.513935848",
      "4.362299312"
    ],
    [
      "26.503960786",
      "4.561800557"
    ],
    [
      "3.445900853",
      "3.990948390"
    ],
    [
      "5.435925791",
      "4.190449635"
    ],
    [
      "7.425950729",
      "4.389950880"
    ],
    [
      "9.415975667",
      "4.589452125"
    ],
    [
      "11.406000605",
      "4.788953370"
    ],
    [
      "13.396025543",
      "4.988454615"
    ],
    [
      "15.386050481",
      "5.187955860"
    ],
    [
      "17.376075419",
      "5.387457105"
    ],
    [
      "19.366100357",
      "5.586958350"
    ],
    [
      "21.356125295",
      "5.786459595"
    ],
    [
      "23.346150233",
      "5.985960840"
    ],
    [
      "4.268140176",
      "5.814111163"
    ],
    [
      "6.258165114",
      "6.013612408"
    ],
    [
      "8.248190052",
      "6.213113653"
    ],
    [
      "10.238214990",
      "6.412614898"
    ],
    [
      "12.228239928",
      "6.612116143"
    ],
    [
      "14.218264866",
      "6.811617388"
    ],
    [
      "16.208289804",
      "7.011118633"
    ],
    [
      "18.198314742",
      "7.210619878"
    ],
    [
      "20.188339680",
      "7.410121123"
    ],
    [
      "5.090379499",
      "7.637273936"
    ],
    [
      "7.080404437",
      "7.836775181"
    ],
    [
      "9.070429375",
      "8.036276426"
    ],
    [
      "11.060454313",
      "8.235777671"
    ],
    [
      "13.050479251",
      "8.435278916"
    ],
    [
      "15.040504189",
      "8.634780161"
    ],
    [
      "17.030529127",
      "8.834281406"
    ],
    [
      "5.912618822",
      "9.460436709"
    ],
    [
      "7.902643760",
      "9.659937954"
    ],
    [
      "9.892668698",
      "9.859439199"
    ],
    [
      "11.882693636",
      "10.058940444"
    ],
    [
      "13.872718574",
      "10.258441689"
    ],
    [
      "6.734858145",
      "11.283599482"
    ],
    [
      "8.724883083",
      "11.483100727"
    ],
    [
      "10.714908021",
      "11.682601972"
    ],
    [
      "7.557097468",
      "13.106762255"
    ]
  ],
  "triangle": [
    [
      "1.000000001",
      "1.000000001"
    ],
    [
      "30.317148497",
      "3.939062479"
    ],
    [
      "7.056635743",
      "14.429463303"
    ]
  ]
}

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": 49
}

当前第一名的答案

{
  "centers": [
    [
      "2.623661530",
      "2.167785617"
    ],
    [
      "4.613686468",
      "2.367286862"
    ],
    [
      "6.603711406",
      "2.566788107"
    ],
    [
      "8.593736344",
      "2.766289352"
    ],
    [
      "10.583761282",
      "2.965790597"
    ],
    [
      "12.573786220",
      "3.165291842"
    ],
    [
      "14.563811158",
      "3.364793087"
    ],
    [
      "16.553836096",
      "3.564294332"
    ],
    [
      "18.543861034",
      "3.763795577"
    ],
    [
      "20.533885972",
      "3.963296822"
    ],
    [
      "22.523910910",
      "4.162798067"
    ],
    [
      "24.513935848",
      "4.362299312"
    ],
    [
      "26.503960786",
      "4.561800557"
    ],
    [
      "3.445900853",
      "3.990948390"
    ],
    [
      "5.435925791",
      "4.190449635"
    ],
    [
      "7.425950729",
      "4.389950880"
    ],
    [
      "9.415975667",
      "4.589452125"
    ],
    [
      "11.406000605",
      "4.788953370"
    ],
    [
      "13.396025543",
      "4.988454615"
    ],
    [
      "15.386050481",
      "5.187955860"
    ],
    [
      "17.376075419",
      "5.387457105"
    ],
    [
      "19.366100357",
      "5.586958350"
    ],
    [
      "21.356125295",
      "5.786459595"
    ],
    [
      "23.346150233",
      "5.985960840"
    ],
    [
      "4.268140176",
      "5.814111163"
    ],
    [
      "6.258165114",
      "6.013612408"
    ],
    [
      "8.248190052",
      "6.213113653"
    ],
    [
      "10.238214990",
      "6.412614898"
    ],
    [
      "12.228239928",
      "6.612116143"
    ],
    [
      "14.218264866",
      "6.811617388"
    ],
    [
      "16.208289804",
      "7.011118633"
    ],
    [
      "18.198314742",
      "7.210619878"
    ],
    [
      "20.188339680",
      "7.410121123"
    ],
    [
      "5.090379499",
      "7.637273936"
    ],
    [
      "7.080404437",
      "7.836775181"
    ],
    [
      "9.070429375",
      "8.036276426"
    ],
    [
      "11.060454313",
      "8.235777671"
    ],
    [
      "13.050479251",
      "8.435278916"
    ],
    [
      "15.040504189",
      "8.634780161"
    ],
    [
      "17.030529127",
      "8.834281406"
    ],
    [
      "5.912618822",
      "9.460436709"
    ],
    [
      "7.902643760",
      "9.659937954"
    ],
    [
      "9.892668698",
      "9.859439199"
    ],
    [
      "11.882693636",
      "10.058940444"
    ],
    [
      "13.872718574",
      "10.258441689"
    ],
    [
      "6.734858145",
      "11.283599482"
    ],
    [
      "8.724883083",
      "11.483100727"
    ],
    [
      "10.714908021",
      "11.682601972"
    ],
    [
      "7.557097468",
      "13.106762255"
    ]
  ],
  "triangle": [
    [
      "1.000000001",
      "1.000000001"
    ],
    [
      "30.317148497",
      "3.939062479"
    ],
    [
      "7.056635743",
      "14.429463303"
    ]
  ]
}

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

DISCUSSION

讨论区

欢迎交流解题思路、分享构造方法或探讨卡壳之处。所有登录用户均可发帖；发言使用与排行榜一致的公开署名，并附带唯一注册编号（#UID）以防冒名。新发言经审核后公开。

暂无讨论，欢迎发表第一篇帖子。