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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 = 42

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 = 42
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2.218631935zzzcy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 = 42 的当前纪录构型，2.218631935
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
    1.770008327→2.218631935
  2. NUE_13人工
    1.726254852→1.770008327
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": [
    [
      "0.273053977",
      "0.047971558",
      "0.047971557"
    ],
    [
      "0.129729004",
      "0.129817277",
      "0.047439974"
    ],
    [
      "0.500000000",
      "0.751283305",
      "0.057371049"
    ],
    [
      "0.903448487",
      "0.055744042",
      "0.055744042"
    ],
    [
      "0.397886714",
      "0.287308607",
      "0.035834059"
    ],
    [
      "0.469926334",
      "0.050934453",
      "0.050934452"
    ],
    [
      "0.385033402",
      "0.555023886",
      "0.055936764"
    ],
    [
      "0.224118886",
      "0.293713088",
      "0.047236104"
    ],
    [
      "0.273773518",
      "0.376164461",
      "0.049012590"
    ],
    [
      "0.327987557",
      "0.462397637",
      "0.052846737"
    ],
    [
      "0.790443116",
      "0.249589289",
      "0.056686940"
    ],
    [
      "0.614652099",
      "0.550580906",
      "0.058430618"
    ],
    [
      "0.732062839",
      "0.348222751",
      "0.057929012"
    ],
    [
      "0.672947802",
      "0.448874168",
      "0.058798427"
    ],
    [
      "0.681919702",
      "0.054758260",
      "0.054758260"
    ],
    [
      "0.574004988",
      "0.053168168",
      "0.053168168"
    ],
    [
      "0.176880004",
      "0.211908120",
      "0.047228516"
    ],
    [
      "0.442520276",
      "0.652320369",
      "0.057073616"
    ],
    [
      "0.847389684",
      "0.152379721",
      "0.055974550"
    ],
    [
      "0.557261315",
      "0.651505260",
      "0.057670317"
    ],
    [
      "0.792190025",
      "0.055514656",
      "0.055514656"
    ],
    [
      "0.082321366",
      "0.047528263",
      "0.047528262"
    ],
    [
      "0.563484984",
      "0.241593341",
      "0.056650432"
    ],
    [
      "0.416384638",
      "0.135563030",
      "0.049208991"
    ],
    [
      "0.363048052",
      "0.213819314",
      "0.045494954"
    ],
    [
      "0.457900828",
      "0.225788731",
      "0.050110045"
    ],
    [
      "0.270837852",
      "0.212184457",
      "0.046729736"
    ],
    [
      "0.437666055",
      "0.455010649",
      "0.057080241"
    ],
    [
      "0.554367782",
      "0.448895468",
      "0.059781593"
    ],
    [
      "0.614624467",
      "0.345915515",
      "0.059532022"
    ],
    [
      "0.624965153",
      "0.148489842",
      "0.054920498"
    ],
    [
      "0.676877593",
      "0.247526809",
      "0.056897309"
    ],
    [
      "0.517781045",
      "0.142551689",
      "0.052427974"
    ],
    [
      "0.372765785",
      "0.369591047",
      "0.050197684"
    ],
    [
      "0.316766715",
      "0.292079650",
      "0.045426122"
    ],
    [
      "0.735633851",
      "0.151374922",
      "0.055785799"
    ],
    [
      "0.498584505",
      "0.552230355",
      "0.057648695"
    ],
    [
      "0.224600963",
      "0.130155842",
      "0.047432588"
    ],
    [
      "0.177466333",
      "0.047616745",
      "0.047616745"
    ],
    [
      "0.370015414",
      "0.048995284",
      "0.048995283"
    ],
    [
      "0.487234005",
      "0.340185138",
      "0.067987258"
    ],
    [
      "0.319644175",
      "0.131439904",
      "0.047619296"
    ]
  ]
}
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": 42
}

当前第一名的答案

{
  "circles": [
    [
      "0.273053977",
      "0.047971558",
      "0.047971557"
    ],
    [
      "0.129729004",
      "0.129817277",
      "0.047439974"
    ],
    [
      "0.500000000",
      "0.751283305",
      "0.057371049"
    ],
    [
      "0.903448487",
      "0.055744042",
      "0.055744042"
    ],
    [
      "0.397886714",
      "0.287308607",
      "0.035834059"
    ],
    [
      "0.469926334",
      "0.050934453",
      "0.050934452"
    ],
    [
      "0.385033402",
      "0.555023886",
      "0.055936764"
    ],
    [
      "0.224118886",
      "0.293713088",
      "0.047236104"
    ],
    [
      "0.273773518",
      "0.376164461",
      "0.049012590"
    ],
    [
      "0.327987557",
      "0.462397637",
      "0.052846737"
    ],
    [
      "0.790443116",
      "0.249589289",
      "0.056686940"
    ],
    [
      "0.614652099",
      "0.550580906",
      "0.058430618"
    ],
    [
      "0.732062839",
      "0.348222751",
      "0.057929012"
    ],
    [
      "0.672947802",
      "0.448874168",
      "0.058798427"
    ],
    [
      "0.681919702",
      "0.054758260",
      "0.054758260"
    ],
    [
      "0.574004988",
      "0.053168168",
      "0.053168168"
    ],
    [
      "0.176880004",
      "0.211908120",
      "0.047228516"
    ],
    [
      "0.442520276",
      "0.652320369",
      "0.057073616"
    ],
    [
      "0.847389684",
      "0.152379721",
      "0.055974550"
    ],
    [
      "0.557261315",
      "0.651505260",
      "0.057670317"
    ],
    [
      "0.792190025",
      "0.055514656",
      "0.055514656"
    ],
    [
      "0.082321366",
      "0.047528263",
      "0.047528262"
    ],
    [
      "0.563484984",
      "0.241593341",
      "0.056650432"
    ],
    [
      "0.416384638",
      "0.135563030",
      "0.049208991"
    ],
    [
      "0.363048052",
      "0.213819314",
      "0.045494954"
    ],
    [
      "0.457900828",
      "0.225788731",
      "0.050110045"
    ],
    [
      "0.270837852",
      "0.212184457",
      "0.046729736"
    ],
    [
      "0.437666055",
      "0.455010649",
      "0.057080241"
    ],
    [
      "0.554367782",
      "0.448895468",
      "0.059781593"
    ],
    [
      "0.614624467",
      "0.345915515",
      "0.059532022"
    ],
    [
      "0.624965153",
      "0.148489842",
      "0.054920498"
    ],
    [
      "0.676877593",
      "0.247526809",
      "0.056897309"
    ],
    [
      "0.517781045",
      "0.142551689",
      "0.052427974"
    ],
    [
      "0.372765785",
      "0.369591047",
      "0.050197684"
    ],
    [
      "0.316766715",
      "0.292079650",
      "0.045426122"
    ],
    [
      "0.735633851",
      "0.151374922",
      "0.055785799"
    ],
    [
      "0.498584505",
      "0.552230355",
      "0.057648695"
    ],
    [
      "0.224600963",
      "0.130155842",
      "0.047432588"
    ],
    [
      "0.177466333",
      "0.047616745",
      "0.047616745"
    ],
    [
      "0.370015414",
      "0.048995284",
      "0.048995283"
    ],
    [
      "0.487234005",
      "0.340185138",
      "0.067987258"
    ],
    [
      "0.319644175",
      "0.131439904",
      "0.047619296"
    ]
  ]
}

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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