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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 = 47

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 = 47
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2.356195344zzzcy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 = 47 的当前纪录构型，2.356195344
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（3 次易主）
  1. zzzcyAI · OpenAI: GPT-6 Astra
    2.356195336→2.356195344
  2. zzzcyAI · OpenAI: GPT-6 Astra
    1.847434223→2.356195336
  3. NUE_13人工
    1.751762098→1.847434223
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",
      "0.768133155",
      "0.048946124"
    ],
    [
      "0.084777172",
      "0.048946123",
      "0.048946123"
    ],
    [
      "0.908449790",
      "0.052856538",
      "0.052856538"
    ],
    [
      "0.338841970",
      "0.381719766",
      "0.032664597"
    ],
    [
      "0.401941016",
      "0.600666026",
      "0.047758117"
    ],
    [
      "0.180778469",
      "0.217601259",
      "0.047758117"
    ],
    [
      "0.225461172",
      "0.299476971",
      "0.045516617"
    ],
    [
      "0.353375921",
      "0.521031815",
      "0.045516617"
    ],
    [
      "0.597775518",
      "0.596691184",
      "0.049991027"
    ],
    [
      "0.282138034",
      "0.049991027",
      "0.049991027"
    ],
    [
      "0.802763280",
      "0.235949838",
      "0.052837091"
    ],
    [
      "0.697043086",
      "0.052837091",
      "0.052837091"
    ],
    [
      "0.698118211",
      "0.418501630",
      "0.052186482"
    ],
    [
      "0.486626062",
      "0.052186483",
      "0.052186483"
    ],
    [
      "0.647334853",
      "0.508511580",
      "0.051161186"
    ],
    [
      "0.383283479",
      "0.051161186",
      "0.051161186"
    ],
    [
      "0.750117065",
      "0.327415151",
      "0.052697393"
    ],
    [
      "0.591508694",
      "0.052697393",
      "0.052697393"
    ],
    [
      "0.306350611",
      "0.448111803",
      "0.041251510"
    ],
    [
      "0.265099100",
      "0.376662090",
      "0.041251510"
    ],
    [
      "0.133360138",
      "0.133625724",
      "0.048680405"
    ],
    [
      "0.450956797",
      "0.683719273",
      "0.048680405"
    ],
    [
      "0.855593252",
      "0.144406748",
      "0.052856538"
    ],
    [
      "0.802736714",
      "0.052856538",
      "0.052856538"
    ],
    [
      "0.548848053",
      "0.682990972",
      "0.049213560"
    ],
    [
      "0.182936493",
      "0.049213560",
      "0.049213560"
    ],
    [
      "0.489849613",
      "0.425089387",
      "0.051916673"
    ],
    [
      "0.376786598",
      "0.229258500",
      "0.051916673"
    ],
    [
      "0.537190142",
      "0.329955651",
      "0.054345047"
    ],
    [
      "0.482845094",
      "0.235827268",
      "0.054345047"
    ],
    [
      "0.590601896",
      "0.236366104",
      "0.053413102"
    ],
    [
      "0.644014673",
      "0.144380119",
      "0.052955813"
    ],
    [
      "0.696970486",
      "0.236102277",
      "0.052955812"
    ],
    [
      "0.447116876",
      "0.515815365",
      "0.048369365"
    ],
    [
      "0.276849228",
      "0.220903148",
      "0.048369365"
    ],
    [
      "0.330531238",
      "0.137971465",
      "0.050420422"
    ],
    [
      "0.545778826",
      "0.510791221",
      "0.050420423"
    ],
    [
      "0.538108671",
      "0.143857264",
      "0.052951479"
    ],
    [
      "0.644470290",
      "0.328080991",
      "0.052951479"
    ],
    [
      "0.433031508",
      "0.141703284",
      "0.052147757"
    ],
    [
      "0.593797110",
      "0.420157473",
      "0.052147758"
    ],
    [
      "0.316612706",
      "0.306370552",
      "0.045895216"
    ],
    [
      "0.392981672",
      "0.438645480",
      "0.045895216"
    ],
    [
      "0.749853597",
      "0.144422093",
      "0.052883118"
    ],
    [
      "0.231095643",
      "0.134881592",
      "0.049063168"
    ],
    [
      "0.498736937",
      "0.598449909",
      "0.049063168"
    ],
    [
      "0.420872420",
      "0.334359463",
      "0.062056008"
    ]
  ]
}
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": 47
}

当前第一名的答案

{
  "circles": [
    [
      "0.500000000",
      "0.768133155",
      "0.048946124"
    ],
    [
      "0.084777172",
      "0.048946123",
      "0.048946123"
    ],
    [
      "0.908449790",
      "0.052856538",
      "0.052856538"
    ],
    [
      "0.338841970",
      "0.381719766",
      "0.032664597"
    ],
    [
      "0.401941016",
      "0.600666026",
      "0.047758117"
    ],
    [
      "0.180778469",
      "0.217601259",
      "0.047758117"
    ],
    [
      "0.225461172",
      "0.299476971",
      "0.045516617"
    ],
    [
      "0.353375921",
      "0.521031815",
      "0.045516617"
    ],
    [
      "0.597775518",
      "0.596691184",
      "0.049991027"
    ],
    [
      "0.282138034",
      "0.049991027",
      "0.049991027"
    ],
    [
      "0.802763280",
      "0.235949838",
      "0.052837091"
    ],
    [
      "0.697043086",
      "0.052837091",
      "0.052837091"
    ],
    [
      "0.698118211",
      "0.418501630",
      "0.052186482"
    ],
    [
      "0.486626062",
      "0.052186483",
      "0.052186483"
    ],
    [
      "0.647334853",
      "0.508511580",
      "0.051161186"
    ],
    [
      "0.383283479",
      "0.051161186",
      "0.051161186"
    ],
    [
      "0.750117065",
      "0.327415151",
      "0.052697393"
    ],
    [
      "0.591508694",
      "0.052697393",
      "0.052697393"
    ],
    [
      "0.306350611",
      "0.448111803",
      "0.041251510"
    ],
    [
      "0.265099100",
      "0.376662090",
      "0.041251510"
    ],
    [
      "0.133360138",
      "0.133625724",
      "0.048680405"
    ],
    [
      "0.450956797",
      "0.683719273",
      "0.048680405"
    ],
    [
      "0.855593252",
      "0.144406748",
      "0.052856538"
    ],
    [
      "0.802736714",
      "0.052856538",
      "0.052856538"
    ],
    [
      "0.548848053",
      "0.682990972",
      "0.049213560"
    ],
    [
      "0.182936493",
      "0.049213560",
      "0.049213560"
    ],
    [
      "0.489849613",
      "0.425089387",
      "0.051916673"
    ],
    [
      "0.376786598",
      "0.229258500",
      "0.051916673"
    ],
    [
      "0.537190142",
      "0.329955651",
      "0.054345047"
    ],
    [
      "0.482845094",
      "0.235827268",
      "0.054345047"
    ],
    [
      "0.590601896",
      "0.236366104",
      "0.053413102"
    ],
    [
      "0.644014673",
      "0.144380119",
      "0.052955813"
    ],
    [
      "0.696970486",
      "0.236102277",
      "0.052955812"
    ],
    [
      "0.447116876",
      "0.515815365",
      "0.048369365"
    ],
    [
      "0.276849228",
      "0.220903148",
      "0.048369365"
    ],
    [
      "0.330531238",
      "0.137971465",
      "0.050420422"
    ],
    [
      "0.545778826",
      "0.510791221",
      "0.050420423"
    ],
    [
      "0.538108671",
      "0.143857264",
      "0.052951479"
    ],
    [
      "0.644470290",
      "0.328080991",
      "0.052951479"
    ],
    [
      "0.433031508",
      "0.141703284",
      "0.052147757"
    ],
    [
      "0.593797110",
      "0.420157473",
      "0.052147758"
    ],
    [
      "0.316612706",
      "0.306370552",
      "0.045895216"
    ],
    [
      "0.392981672",
      "0.438645480",
      "0.045895216"
    ],
    [
      "0.749853597",
      "0.144422093",
      "0.052883118"
    ],
    [
      "0.231095643",
      "0.134881592",
      "0.049063168"
    ],
    [
      "0.498736937",
      "0.598449909",
      "0.049063168"
    ],
    [
      "0.420872420",
      "0.334359463",
      "0.062056008"
    ]
  ]
}

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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