P93 · 装箱与覆盖 · 经典问题 · 难

正三角形内圆的半径之和 · n = 35

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 = 35

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	2.020699891	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

1234567891011121314151617181920212223242526272829303132333435

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
1.626192982→2.020699891
2026-10-04
NUE_13人工
1.603288855→1.626192982
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",
      "0.755404205",
      "0.055310599"
    ],
    [
      "0.105563318",
      "0.060947010",
      "0.060947010"
    ],
    [
      "0.894436682",
      "0.060947010",
      "0.060947010"
    ],
    [
      "0.500000000",
      "0.202178073",
      "0.041656782"
    ],
    [
      "0.355256791",
      "0.065257999",
      "0.065257999"
    ],
    [
      "0.644743209",
      "0.065257999",
      "0.065257999"
    ],
    [
      "0.389391410",
      "0.563755624",
      "0.055345041"
    ],
    [
      "0.610608590",
      "0.563755623",
      "0.055345041"
    ],
    [
      "0.223035536",
      "0.269691138",
      "0.058308871"
    ],
    [
      "0.776964464",
      "0.269691138",
      "0.058308871"
    ],
    [
      "0.334137499",
      "0.467555858",
      "0.055593633"
    ],
    [
      "0.665862501",
      "0.467555858",
      "0.055593633"
    ],
    [
      "0.278881555",
      "0.370014842",
      "0.056511089"
    ],
    [
      "0.721118446",
      "0.370014842",
      "0.056511089"
    ],
    [
      "0.165403003",
      "0.166211640",
      "0.060137382"
    ],
    [
      "0.834596997",
      "0.166211640",
      "0.060137382"
    ],
    [
      "0.444689400",
      "0.659603436",
      "0.055310599"
    ],
    [
      "0.555310600",
      "0.659603436",
      "0.055310599"
    ],
    [
      "0.771722864",
      "0.061769564",
      "0.061769564"
    ],
    [
      "0.228277136",
      "0.061769564",
      "0.061769564"
    ],
    [
      "0.500000000",
      "0.373645650",
      "0.054329830"
    ],
    [
      "0.447261376",
      "0.280466835",
      "0.052738624"
    ],
    [
      "0.552738624",
      "0.280466835",
      "0.052738624"
    ],
    [
      "0.661983953",
      "0.273437126",
      "0.056732644"
    ],
    [
      "0.338016047",
      "0.273437126",
      "0.056732644"
    ],
    [
      "0.555135404",
      "0.468211631",
      "0.055135404"
    ],
    [
      "0.444864596",
      "0.468211631",
      "0.055135404"
    ],
    [
      "0.609475127",
      "0.372237547",
      "0.055154352"
    ],
    [
      "0.390524873",
      "0.372237547",
      "0.055154352"
    ],
    [
      "0.403071127",
      "0.178895599",
      "0.058029124"
    ],
    [
      "0.596928873",
      "0.178895599",
      "0.058029124"
    ],
    [
      "0.500000000",
      "0.563856947",
      "0.055263595"
    ],
    [
      "0.714529806",
      "0.169235654",
      "0.059967884"
    ],
    [
      "0.285470194",
      "0.169235654",
      "0.059967884"
    ],
    [
      "0.500000000",
      "0.080260645",
      "0.080260645"
    ]
  ]
}

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": 35
}

当前第一名的答案

{
  "circles": [
    [
      "0.500000000",
      "0.755404205",
      "0.055310599"
    ],
    [
      "0.105563318",
      "0.060947010",
      "0.060947010"
    ],
    [
      "0.894436682",
      "0.060947010",
      "0.060947010"
    ],
    [
      "0.500000000",
      "0.202178073",
      "0.041656782"
    ],
    [
      "0.355256791",
      "0.065257999",
      "0.065257999"
    ],
    [
      "0.644743209",
      "0.065257999",
      "0.065257999"
    ],
    [
      "0.389391410",
      "0.563755624",
      "0.055345041"
    ],
    [
      "0.610608590",
      "0.563755623",
      "0.055345041"
    ],
    [
      "0.223035536",
      "0.269691138",
      "0.058308871"
    ],
    [
      "0.776964464",
      "0.269691138",
      "0.058308871"
    ],
    [
      "0.334137499",
      "0.467555858",
      "0.055593633"
    ],
    [
      "0.665862501",
      "0.467555858",
      "0.055593633"
    ],
    [
      "0.278881555",
      "0.370014842",
      "0.056511089"
    ],
    [
      "0.721118446",
      "0.370014842",
      "0.056511089"
    ],
    [
      "0.165403003",
      "0.166211640",
      "0.060137382"
    ],
    [
      "0.834596997",
      "0.166211640",
      "0.060137382"
    ],
    [
      "0.444689400",
      "0.659603436",
      "0.055310599"
    ],
    [
      "0.555310600",
      "0.659603436",
      "0.055310599"
    ],
    [
      "0.771722864",
      "0.061769564",
      "0.061769564"
    ],
    [
      "0.228277136",
      "0.061769564",
      "0.061769564"
    ],
    [
      "0.500000000",
      "0.373645650",
      "0.054329830"
    ],
    [
      "0.447261376",
      "0.280466835",
      "0.052738624"
    ],
    [
      "0.552738624",
      "0.280466835",
      "0.052738624"
    ],
    [
      "0.661983953",
      "0.273437126",
      "0.056732644"
    ],
    [
      "0.338016047",
      "0.273437126",
      "0.056732644"
    ],
    [
      "0.555135404",
      "0.468211631",
      "0.055135404"
    ],
    [
      "0.444864596",
      "0.468211631",
      "0.055135404"
    ],
    [
      "0.609475127",
      "0.372237547",
      "0.055154352"
    ],
    [
      "0.390524873",
      "0.372237547",
      "0.055154352"
    ],
    [
      "0.403071127",
      "0.178895599",
      "0.058029124"
    ],
    [
      "0.596928873",
      "0.178895599",
      "0.058029124"
    ],
    [
      "0.500000000",
      "0.563856947",
      "0.055263595"
    ],
    [
      "0.714529806",
      "0.169235654",
      "0.059967884"
    ],
    [
      "0.285470194",
      "0.169235654",
      "0.059967884"
    ],
    [
      "0.500000000",
      "0.080260645",
      "0.080260645"
    ]
  ]
}

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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