P93 · 装箱与覆盖 · 经典问题 · 难

正三角形内圆的半径之和 · n = 28

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 = 28

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	待核实以原始来源为准	见来源署名	Erich Friedman ↗
本站纪录	1.810645101	zzzcy	纪录详情 ↓

来源比较尚待核实。

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

12345678910111213141516171819202122232425262728

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
1.810645095→1.810645101
2026-10-04
zzzcyAI · OpenAI: GPT-6 Astra
1.810645088→1.810645095
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": [
    [
      "0.112004619",
      "0.064665897",
      "0.064665896"
    ],
    [
      "0.241336412",
      "0.064665897",
      "0.064665897"
    ],
    [
      "0.370668206",
      "0.064665897",
      "0.064665897"
    ],
    [
      "0.500000000",
      "0.064665897",
      "0.064665897"
    ],
    [
      "0.629331794",
      "0.064665897",
      "0.064665897"
    ],
    [
      "0.758663588",
      "0.064665897",
      "0.064665897"
    ],
    [
      "0.887995381",
      "0.064665897",
      "0.064665896"
    ],
    [
      "0.176670516",
      "0.176670516",
      "0.064665896"
    ],
    [
      "0.306002309",
      "0.176670517",
      "0.064665897"
    ],
    [
      "0.435334103",
      "0.176670517",
      "0.064665897"
    ],
    [
      "0.564665897",
      "0.176670516",
      "0.064665896"
    ],
    [
      "0.693997691",
      "0.176670516",
      "0.064665896"
    ],
    [
      "0.823329485",
      "0.176670516",
      "0.064665896"
    ],
    [
      "0.241336412",
      "0.288675135",
      "0.064665896"
    ],
    [
      "0.370668206",
      "0.288675135",
      "0.064665896"
    ],
    [
      "0.500000000",
      "0.288675135",
      "0.064665896"
    ],
    [
      "0.629331793",
      "0.288675135",
      "0.064665897"
    ],
    [
      "0.758663587",
      "0.288675135",
      "0.064665897"
    ],
    [
      "0.306002310",
      "0.400679754",
      "0.064665897"
    ],
    [
      "0.435334104",
      "0.400679754",
      "0.064665897"
    ],
    [
      "0.564665897",
      "0.400679754",
      "0.064665896"
    ],
    [
      "0.693997690",
      "0.400679753",
      "0.064665896"
    ],
    [
      "0.370668207",
      "0.512684374",
      "0.064665896"
    ],
    [
      "0.500000000",
      "0.512684373",
      "0.064665896"
    ],
    [
      "0.629331793",
      "0.512684373",
      "0.064665897"
    ],
    [
      "0.435334104",
      "0.624688992",
      "0.064665897"
    ],
    [
      "0.564665897",
      "0.624688992",
      "0.064665896"
    ],
    [
      "0.500000000",
      "0.736693611",
      "0.064665896"
    ]
  ]
}

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": 28
}

当前第一名的答案

{
  "circles": [
    [
      "0.112004619",
      "0.064665897",
      "0.064665896"
    ],
    [
      "0.241336412",
      "0.064665897",
      "0.064665897"
    ],
    [
      "0.370668206",
      "0.064665897",
      "0.064665897"
    ],
    [
      "0.500000000",
      "0.064665897",
      "0.064665897"
    ],
    [
      "0.629331794",
      "0.064665897",
      "0.064665897"
    ],
    [
      "0.758663588",
      "0.064665897",
      "0.064665897"
    ],
    [
      "0.887995381",
      "0.064665897",
      "0.064665896"
    ],
    [
      "0.176670516",
      "0.176670516",
      "0.064665896"
    ],
    [
      "0.306002309",
      "0.176670517",
      "0.064665897"
    ],
    [
      "0.435334103",
      "0.176670517",
      "0.064665897"
    ],
    [
      "0.564665897",
      "0.176670516",
      "0.064665896"
    ],
    [
      "0.693997691",
      "0.176670516",
      "0.064665896"
    ],
    [
      "0.823329485",
      "0.176670516",
      "0.064665896"
    ],
    [
      "0.241336412",
      "0.288675135",
      "0.064665896"
    ],
    [
      "0.370668206",
      "0.288675135",
      "0.064665896"
    ],
    [
      "0.500000000",
      "0.288675135",
      "0.064665896"
    ],
    [
      "0.629331793",
      "0.288675135",
      "0.064665897"
    ],
    [
      "0.758663587",
      "0.288675135",
      "0.064665897"
    ],
    [
      "0.306002310",
      "0.400679754",
      "0.064665897"
    ],
    [
      "0.435334104",
      "0.400679754",
      "0.064665897"
    ],
    [
      "0.564665897",
      "0.400679754",
      "0.064665896"
    ],
    [
      "0.693997690",
      "0.400679753",
      "0.064665896"
    ],
    [
      "0.370668207",
      "0.512684374",
      "0.064665896"
    ],
    [
      "0.500000000",
      "0.512684373",
      "0.064665896"
    ],
    [
      "0.629331793",
      "0.512684373",
      "0.064665897"
    ],
    [
      "0.435334104",
      "0.624688992",
      "0.064665897"
    ],
    [
      "0.564665897",
      "0.624688992",
      "0.064665896"
    ],
    [
      "0.500000000",
      "0.736693611",
      "0.064665896"
    ]
  ]
}

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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