P93 · 装箱与覆盖 · 经典问题 · 难

正三角形内圆的半径之和 · n = 33

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 = 33

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	1.956503782	zzzcy	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 $S = Σ r$
目标在所有合法放置中最大化半径之和 $\max_{S} Σ r$

放大来摆，然后提交 ↗

当前展示：本站纪录

0.866025404y0

123456789101112131415161718192021222324252627282930313233

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

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）

zzzcyAI · OpenAI: GPT-6 Astra
1.647229014→1.956503782
2026-10-04
NUE_13人工
1.511672349→1.647229014
2026-10-04

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",
      "0.762812917",
      "0.051606243"
    ],
    [
      "0.110818831",
      "0.063981281",
      "0.063981281"
    ],
    [
      "0.889181169",
      "0.063981281",
      "0.063981281"
    ],
    [
      "0.500000000",
      "0.421985235",
      "0.039680745"
    ],
    [
      "0.396898692",
      "0.583629932",
      "0.051909384"
    ],
    [
      "0.603101308",
      "0.583629932",
      "0.051909384"
    ],
    [
      "0.236060878",
      "0.284353878",
      "0.062257778"
    ],
    [
      "0.763939122",
      "0.284353878",
      "0.062257778"
    ],
    [
      "0.345822732",
      "0.490674752",
      "0.054153895"
    ],
    [
      "0.654177267",
      "0.490674751",
      "0.054153895"
    ],
    [
      "0.293357766",
      "0.390818743",
      "0.058645906"
    ],
    [
      "0.706642235",
      "0.390818743",
      "0.058645905"
    ],
    [
      "0.368783931",
      "0.065295910",
      "0.065295910"
    ],
    [
      "0.631216069",
      "0.065295910",
      "0.065295910"
    ],
    [
      "0.500000000",
      "0.065921651",
      "0.065921651"
    ],
    [
      "0.174305042",
      "0.174667081",
      "0.063619053"
    ],
    [
      "0.825694958",
      "0.174667082",
      "0.063619052"
    ],
    [
      "0.448393757",
      "0.673428282",
      "0.051606243"
    ],
    [
      "0.551606244",
      "0.673428281",
      "0.051606243"
    ],
    [
      "0.760854452",
      "0.064345953",
      "0.064345953"
    ],
    [
      "0.239145547",
      "0.064345952",
      "0.064345952"
    ],
    [
      "0.638258697",
      "0.289572154",
      "0.063530932"
    ],
    [
      "0.361741303",
      "0.289572154",
      "0.063530932"
    ],
    [
      "0.433212226",
      "0.180600266",
      "0.066787774"
    ],
    [
      "0.566787774",
      "0.180600266",
      "0.066787774"
    ],
    [
      "0.592690516",
      "0.399699125",
      "0.055651318"
    ],
    [
      "0.407309485",
      "0.399699125",
      "0.055651318"
    ],
    [
      "0.550092666",
      "0.496483491",
      "0.050092666"
    ],
    [
      "0.449907334",
      "0.496483491",
      "0.050092666"
    ],
    [
      "0.500000000",
      "0.584517960",
      "0.051195748"
    ],
    [
      "0.697799458",
      "0.176466629",
      "0.064289105"
    ],
    [
      "0.302200542",
      "0.176466629",
      "0.064289105"
    ],
    [
      "0.500000000",
      "0.306539487",
      "0.075765002"
    ]
  ]
}

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": 33
}

当前第一名的答案

{
  "circles": [
    [
      "0.500000000",
      "0.762812917",
      "0.051606243"
    ],
    [
      "0.110818831",
      "0.063981281",
      "0.063981281"
    ],
    [
      "0.889181169",
      "0.063981281",
      "0.063981281"
    ],
    [
      "0.500000000",
      "0.421985235",
      "0.039680745"
    ],
    [
      "0.396898692",
      "0.583629932",
      "0.051909384"
    ],
    [
      "0.603101308",
      "0.583629932",
      "0.051909384"
    ],
    [
      "0.236060878",
      "0.284353878",
      "0.062257778"
    ],
    [
      "0.763939122",
      "0.284353878",
      "0.062257778"
    ],
    [
      "0.345822732",
      "0.490674752",
      "0.054153895"
    ],
    [
      "0.654177267",
      "0.490674751",
      "0.054153895"
    ],
    [
      "0.293357766",
      "0.390818743",
      "0.058645906"
    ],
    [
      "0.706642235",
      "0.390818743",
      "0.058645905"
    ],
    [
      "0.368783931",
      "0.065295910",
      "0.065295910"
    ],
    [
      "0.631216069",
      "0.065295910",
      "0.065295910"
    ],
    [
      "0.500000000",
      "0.065921651",
      "0.065921651"
    ],
    [
      "0.174305042",
      "0.174667081",
      "0.063619053"
    ],
    [
      "0.825694958",
      "0.174667082",
      "0.063619052"
    ],
    [
      "0.448393757",
      "0.673428282",
      "0.051606243"
    ],
    [
      "0.551606244",
      "0.673428281",
      "0.051606243"
    ],
    [
      "0.760854452",
      "0.064345953",
      "0.064345953"
    ],
    [
      "0.239145547",
      "0.064345952",
      "0.064345952"
    ],
    [
      "0.638258697",
      "0.289572154",
      "0.063530932"
    ],
    [
      "0.361741303",
      "0.289572154",
      "0.063530932"
    ],
    [
      "0.433212226",
      "0.180600266",
      "0.066787774"
    ],
    [
      "0.566787774",
      "0.180600266",
      "0.066787774"
    ],
    [
      "0.592690516",
      "0.399699125",
      "0.055651318"
    ],
    [
      "0.407309485",
      "0.399699125",
      "0.055651318"
    ],
    [
      "0.550092666",
      "0.496483491",
      "0.050092666"
    ],
    [
      "0.449907334",
      "0.496483491",
      "0.050092666"
    ],
    [
      "0.500000000",
      "0.584517960",
      "0.051195748"
    ],
    [
      "0.697799458",
      "0.176466629",
      "0.064289105"
    ],
    [
      "0.302200542",
      "0.176466629",
      "0.064289105"
    ],
    [
      "0.500000000",
      "0.306539487",
      "0.075765002"
    ]
  ]
}

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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