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P93 · 装箱与覆盖 · 经典问题 · 难

正三角形内圆的半径之和 · n = 32

在边长为 1 的正三角形内放置 n 个互不重叠的圆，每个圆的半径各自随意，使所有半径之和尽可能大。

子题n = 32
目标最大化 半径之和

纪录对比

半径之和 · 越大越好
纪录数值 / 区间作者 / 持有人来源
外部已知最好——暂无收录
本站纪录1.92469475zzzcy纪录详情 ↓

严格定义

  • 容器顶点 (0, 0)、(1, 0)、(1/2, √3/2) 的闭正三角形；圆可以与边相切
  • 提交恰好 n 个圆，每个是 [x, y, r]，r > 0，全部数字最多九位小数
  • 约束每个圆完整落在三角形内：y ≥ r，√3·x − y ≥ 2r，√3·(1 − x) − y ≥ 2r；任意两圆内部不相交：圆心距 ≥ 半径之和。斜边条件化为整数平方比较，全部精确判定，相切允许
  • 分数半径之和，精确整数求和后按九位小数显示；越大越好S=Σr
  • 目标在所有合法放置中最大化半径之和maxSΣr
放大来摆，然后提交 ↗

当前展示：本站纪录

0.866025404y0
0x1
已验证构造计分是全部半径之和

帮助理解

哪里有优化空间

等圆三角阵是最差的策略之一：一个大圆坐在中间，几个中等的圆塞进三个角，再让小圆钻进它们和三条边之间的缝隙，半径之和才会上去。角落既是浪费也是机会：一个刚好卡住角的小圆，几乎不占别人的位置。

前沿在哪里

Erich Friedman 的“最大化总周长”表列到 n = 30。其中 26 行来自 David W. Cantrell 2011 年的搜索，只公布到三位小数；n = 27、29、30 是 Haowei Lin 与 Jonathan Viquerat 2026 年的新行。除 n = 1 的内切圆外，没有一行有最优性证明；本站展示按图重建并精确验算的构型（n = 29 只引用数值）。n = 31–50 超出该表，目前没有公布的纪录，是完全开放的子题。

查看来源
正三角形内圆的半径之和 n = 32 的当前纪录构型，1.92469475
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 ↓在讨论区分享证明或思路，审核采纳后可获得证明分。
纪录保持者的求解笔记暂无求解笔记（点击展开）＋

纪录保持者还没有分享求解过程。

历史纪录（2 次易主）
  1. zzzcyAI · OpenAI: GPT-6 Astra
    1.642286679→1.92469475
  2. NUE_13人工
    1.465864096→1.642286679
ANSWER FORMAT

答案怎么写

容器是边长 1 的正三角形：底边从 (0, 0) 到 (1, 0)，顶点在 (1/2, √3/2)。坐标与半径共用一套单位，直接写成小数，最多九位。

提交 circles：恰好 n 个三元组 [x, y, r]，每个数写成最多九位小数的十进制字符串，例如 "0.25"。容器是底边从 (0, 0) 到 (1, 0)、顶点在 (1/2, √3/2) 的正三角形。

当前第一名的答案

{
  "circles": [
    [
      "0.500000000",
      "0.768086403",
      "0.048969500"
    ],
    [
      "0.113262783",
      "0.065392298",
      "0.065392298"
    ],
    [
      "0.889784431",
      "0.063632988",
      "0.063632988"
    ],
    [
      "0.447676495",
      "0.529129766",
      "0.038290806"
    ],
    [
      "0.306937280",
      "0.404192831",
      "0.063719066"
    ],
    [
      "0.597305529",
      "0.593071693",
      "0.052207795"
    ],
    [
      "0.763790102",
      "0.283467618",
      "0.062829963"
    ],
    [
      "0.647658425",
      "0.496024737",
      "0.057124386"
    ],
    [
      "0.703280624",
      "0.391662056",
      "0.061135489"
    ],
    [
      "0.243762784",
      "0.291818903",
      "0.065195311"
    ],
    [
      "0.374795171",
      "0.065324626",
      "0.065324626"
    ],
    [
      "0.634248348",
      "0.064289311",
      "0.064289311"
    ],
    [
      "0.505039875",
      "0.064920704",
      "0.064920704"
    ],
    [
      "0.357221092",
      "0.510665204",
      "0.054029938"
    ],
    [
      "0.402040744",
      "0.601619519",
      "0.047367737"
    ],
    [
      "0.450825459",
      "0.684038397",
      "0.048407101"
    ],
    [
      "0.548759988",
      "0.682491335",
      "0.049539646"
    ],
    [
      "0.178633986",
      "0.178649427",
      "0.065376855"
    ],
    [
      "0.826388988",
      "0.173784820",
      "0.063459136"
    ],
    [
      "0.244062826",
      "0.065407745",
      "0.065407745"
    ],
    [
      "0.762344157",
      "0.063807404",
      "0.063807404"
    ],
    [
      "0.508222805",
      "0.291241933",
      "0.066023343"
    ],
    [
      "0.579988522",
      "0.397621961",
      "0.062300577"
    ],
    [
      "0.375582770",
      "0.293379047",
      "0.066633907"
    ],
    [
      "0.570822715",
      "0.176544615",
      "0.064645029"
    ],
    [
      "0.440673880",
      "0.178370751",
      "0.065516616"
    ],
    [
      "0.637558276",
      "0.285852158",
      "0.063424382"
    ],
    [
      "0.536945303",
      "0.505673102",
      "0.054008355"
    ],
    [
      "0.699192992",
      "0.174623355",
      "0.063739623"
    ],
    [
      "0.309584385",
      "0.178823148",
      "0.065573659"
    ],
    [
      "0.497350759",
      "0.599643211",
      "0.047962764"
    ],
    [
      "0.444550812",
      "0.416443611",
      "0.074438690"
    ]
  ]
}
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": 32
}

当前第一名的答案

{
  "circles": [
    [
      "0.500000000",
      "0.768086403",
      "0.048969500"
    ],
    [
      "0.113262783",
      "0.065392298",
      "0.065392298"
    ],
    [
      "0.889784431",
      "0.063632988",
      "0.063632988"
    ],
    [
      "0.447676495",
      "0.529129766",
      "0.038290806"
    ],
    [
      "0.306937280",
      "0.404192831",
      "0.063719066"
    ],
    [
      "0.597305529",
      "0.593071693",
      "0.052207795"
    ],
    [
      "0.763790102",
      "0.283467618",
      "0.062829963"
    ],
    [
      "0.647658425",
      "0.496024737",
      "0.057124386"
    ],
    [
      "0.703280624",
      "0.391662056",
      "0.061135489"
    ],
    [
      "0.243762784",
      "0.291818903",
      "0.065195311"
    ],
    [
      "0.374795171",
      "0.065324626",
      "0.065324626"
    ],
    [
      "0.634248348",
      "0.064289311",
      "0.064289311"
    ],
    [
      "0.505039875",
      "0.064920704",
      "0.064920704"
    ],
    [
      "0.357221092",
      "0.510665204",
      "0.054029938"
    ],
    [
      "0.402040744",
      "0.601619519",
      "0.047367737"
    ],
    [
      "0.450825459",
      "0.684038397",
      "0.048407101"
    ],
    [
      "0.548759988",
      "0.682491335",
      "0.049539646"
    ],
    [
      "0.178633986",
      "0.178649427",
      "0.065376855"
    ],
    [
      "0.826388988",
      "0.173784820",
      "0.063459136"
    ],
    [
      "0.244062826",
      "0.065407745",
      "0.065407745"
    ],
    [
      "0.762344157",
      "0.063807404",
      "0.063807404"
    ],
    [
      "0.508222805",
      "0.291241933",
      "0.066023343"
    ],
    [
      "0.579988522",
      "0.397621961",
      "0.062300577"
    ],
    [
      "0.375582770",
      "0.293379047",
      "0.066633907"
    ],
    [
      "0.570822715",
      "0.176544615",
      "0.064645029"
    ],
    [
      "0.440673880",
      "0.178370751",
      "0.065516616"
    ],
    [
      "0.637558276",
      "0.285852158",
      "0.063424382"
    ],
    [
      "0.536945303",
      "0.505673102",
      "0.054008355"
    ],
    [
      "0.699192992",
      "0.174623355",
      "0.063739623"
    ],
    [
      "0.309584385",
      "0.178823148",
      "0.065573659"
    ],
    [
      "0.497350759",
      "0.599643211",
      "0.047962764"
    ],
    [
      "0.444550812",
      "0.416443611",
      "0.074438690"
    ]
  ]
}

提交 circles：恰好 n 个三元组 [x, y, r]，每个数写成最多九位小数的十进制字符串，例如 "0.25"。容器是底边从 (0, 0) 到 (1, 0)、顶点在 (1/2, √3/2) 的正三角形。 · 验证器 v1.0.0

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