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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 = 41

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 = 41
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2.190472445zzzcy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 = 41 的当前纪录构型，2.190472445
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
    1.746806551→2.190472445
  2. NUE_13人工
    1.685153546→1.746806551
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": [
    [
      "0.098665780",
      "0.056964715",
      "0.056964714"
    ],
    [
      "0.212712671",
      "0.057082236",
      "0.057082235"
    ],
    [
      "0.327185271",
      "0.057390781",
      "0.057390780"
    ],
    [
      "0.442287802",
      "0.057712198",
      "0.057712198"
    ],
    [
      "0.557712198",
      "0.057712198",
      "0.057712198"
    ],
    [
      "0.672814729",
      "0.057390781",
      "0.057390780"
    ],
    [
      "0.787287329",
      "0.057082236",
      "0.057082235"
    ],
    [
      "0.901334220",
      "0.056964715",
      "0.056964714"
    ],
    [
      "0.155470319",
      "0.155587532",
      "0.056847479"
    ],
    [
      "0.269374258",
      "0.156165234",
      "0.057057923"
    ],
    [
      "0.384135809",
      "0.157415410",
      "0.057710436"
    ],
    [
      "0.500000000",
      "0.158185149",
      "0.058156311"
    ],
    [
      "0.615864191",
      "0.157415410",
      "0.057710436"
    ],
    [
      "0.730625742",
      "0.156165234",
      "0.057057923"
    ],
    [
      "0.844529681",
      "0.155587532",
      "0.056847479"
    ],
    [
      "0.211715750",
      "0.253888310",
      "0.056407062"
    ],
    [
      "0.325028393",
      "0.255630557",
      "0.056918974"
    ],
    [
      "0.440942495",
      "0.259433803",
      "0.059057505"
    ],
    [
      "0.559057505",
      "0.259433803",
      "0.059057505"
    ],
    [
      "0.674971607",
      "0.255630557",
      "0.056918974"
    ],
    [
      "0.788284250",
      "0.253888310",
      "0.056407062"
    ],
    [
      "0.266385431",
      "0.351117392",
      "0.055137853"
    ],
    [
      "0.377654108",
      "0.355795553",
      "0.056229123"
    ],
    [
      "0.500000000",
      "0.370845296",
      "0.067038924"
    ],
    [
      "0.622345892",
      "0.355795553",
      "0.056229123"
    ],
    [
      "0.733614569",
      "0.351117392",
      "0.055137853"
    ],
    [
      "0.317098906",
      "0.445397965",
      "0.051916725"
    ],
    [
      "0.417964233",
      "0.453274449",
      "0.049255668"
    ],
    [
      "0.500000000",
      "0.473003274",
      "0.035119053"
    ],
    [
      "0.582035767",
      "0.453274449",
      "0.049255668"
    ],
    [
      "0.682901094",
      "0.445397965",
      "0.051916725"
    ],
    [
      "0.363535540",
      "0.533793819",
      "0.047934102"
    ],
    [
      "0.455663046",
      "0.538938785",
      "0.044336954"
    ],
    [
      "0.544336954",
      "0.538938785",
      "0.044336954"
    ],
    [
      "0.636464460",
      "0.533793819",
      "0.047934102"
    ],
    [
      "0.408743380",
      "0.616072209",
      "0.045946046"
    ],
    [
      "0.500000000",
      "0.616858755",
      "0.045313963"
    ],
    [
      "0.591256620",
      "0.616072209",
      "0.045946046"
    ],
    [
      "0.454322453",
      "0.695554477",
      "0.045677547"
    ],
    [
      "0.545677547",
      "0.695554477",
      "0.045677547"
    ],
    [
      "0.500000000",
      "0.774670310",
      "0.045677546"
    ]
  ]
}
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": 41
}

当前第一名的答案

{
  "circles": [
    [
      "0.098665780",
      "0.056964715",
      "0.056964714"
    ],
    [
      "0.212712671",
      "0.057082236",
      "0.057082235"
    ],
    [
      "0.327185271",
      "0.057390781",
      "0.057390780"
    ],
    [
      "0.442287802",
      "0.057712198",
      "0.057712198"
    ],
    [
      "0.557712198",
      "0.057712198",
      "0.057712198"
    ],
    [
      "0.672814729",
      "0.057390781",
      "0.057390780"
    ],
    [
      "0.787287329",
      "0.057082236",
      "0.057082235"
    ],
    [
      "0.901334220",
      "0.056964715",
      "0.056964714"
    ],
    [
      "0.155470319",
      "0.155587532",
      "0.056847479"
    ],
    [
      "0.269374258",
      "0.156165234",
      "0.057057923"
    ],
    [
      "0.384135809",
      "0.157415410",
      "0.057710436"
    ],
    [
      "0.500000000",
      "0.158185149",
      "0.058156311"
    ],
    [
      "0.615864191",
      "0.157415410",
      "0.057710436"
    ],
    [
      "0.730625742",
      "0.156165234",
      "0.057057923"
    ],
    [
      "0.844529681",
      "0.155587532",
      "0.056847479"
    ],
    [
      "0.211715750",
      "0.253888310",
      "0.056407062"
    ],
    [
      "0.325028393",
      "0.255630557",
      "0.056918974"
    ],
    [
      "0.440942495",
      "0.259433803",
      "0.059057505"
    ],
    [
      "0.559057505",
      "0.259433803",
      "0.059057505"
    ],
    [
      "0.674971607",
      "0.255630557",
      "0.056918974"
    ],
    [
      "0.788284250",
      "0.253888310",
      "0.056407062"
    ],
    [
      "0.266385431",
      "0.351117392",
      "0.055137853"
    ],
    [
      "0.377654108",
      "0.355795553",
      "0.056229123"
    ],
    [
      "0.500000000",
      "0.370845296",
      "0.067038924"
    ],
    [
      "0.622345892",
      "0.355795553",
      "0.056229123"
    ],
    [
      "0.733614569",
      "0.351117392",
      "0.055137853"
    ],
    [
      "0.317098906",
      "0.445397965",
      "0.051916725"
    ],
    [
      "0.417964233",
      "0.453274449",
      "0.049255668"
    ],
    [
      "0.500000000",
      "0.473003274",
      "0.035119053"
    ],
    [
      "0.582035767",
      "0.453274449",
      "0.049255668"
    ],
    [
      "0.682901094",
      "0.445397965",
      "0.051916725"
    ],
    [
      "0.363535540",
      "0.533793819",
      "0.047934102"
    ],
    [
      "0.455663046",
      "0.538938785",
      "0.044336954"
    ],
    [
      "0.544336954",
      "0.538938785",
      "0.044336954"
    ],
    [
      "0.636464460",
      "0.533793819",
      "0.047934102"
    ],
    [
      "0.408743380",
      "0.616072209",
      "0.045946046"
    ],
    [
      "0.500000000",
      "0.616858755",
      "0.045313963"
    ],
    [
      "0.591256620",
      "0.616072209",
      "0.045946046"
    ],
    [
      "0.454322453",
      "0.695554477",
      "0.045677547"
    ],
    [
      "0.545677547",
      "0.695554477",
      "0.045677547"
    ],
    [
      "0.500000000",
      "0.774670310",
      "0.045677546"
    ]
  ]
}

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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