P93 · 装箱与覆盖 · 经典问题 · 难

正三角形内圆的半径之和 · n = 38

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 = 38

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	2.110458785	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

1234567891011121314151617181920212223242526272829303132333435363738

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
1.627809413→2.110458785
2026-10-04
NUE_13人工
1.561849628→1.627809413
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": [
    [
      "0.102702499",
      "0.059295315",
      "0.059295315"
    ],
    [
      "0.221235984",
      "0.059238184",
      "0.059238183"
    ],
    [
      "0.339482753",
      "0.059008807",
      "0.059008807"
    ],
    [
      "0.456843475",
      "0.058353744",
      "0.058353743"
    ],
    [
      "0.572330539",
      "0.057139700",
      "0.057139700"
    ],
    [
      "0.685252135",
      "0.055789962",
      "0.055789962"
    ],
    [
      "0.795939695",
      "0.054901166",
      "0.054901166"
    ],
    [
      "0.905436626",
      "0.054596190",
      "0.054596189"
    ],
    [
      "0.162075940",
      "0.162018738",
      "0.059352512"
    ],
    [
      "0.280788379",
      "0.161800708",
      "0.059360126"
    ],
    [
      "0.399392257",
      "0.160965731",
      "0.059246691"
    ],
    [
      "0.516863813",
      "0.158328896",
      "0.058254455"
    ],
    [
      "0.631304734",
      "0.154007432",
      "0.056268029"
    ],
    [
      "0.742243672",
      "0.150478752",
      "0.054727013"
    ],
    [
      "0.851254517",
      "0.149048303",
      "0.054293215"
    ],
    [
      "0.221613637",
      "0.264869869",
      "0.059488104"
    ],
    [
      "0.340931641",
      "0.264703951",
      "0.059830013"
    ],
    [
      "0.461470821",
      "0.263614841",
      "0.060714085"
    ],
    [
      "0.579850982",
      "0.255939709",
      "0.057914621"
    ],
    [
      "0.691290572",
      "0.246446560",
      "0.053928581"
    ],
    [
      "0.798397390",
      "0.242698363",
      "0.053243800"
    ],
    [
      "0.281074660",
      "0.367899031",
      "0.059468279"
    ],
    [
      "0.401261766",
      "0.369076794",
      "0.060724597"
    ],
    [
      "0.531055207",
      "0.373246057",
      "0.069135789"
    ],
    [
      "0.647114094",
      "0.341754409",
      "0.051119726"
    ],
    [
      "0.748638960",
      "0.333962705",
      "0.050703693"
    ],
    [
      "0.338931395",
      "0.470464991",
      "0.058290702"
    ],
    [
      "0.454882818",
      "0.474709458",
      "0.057738379"
    ],
    [
      "0.622557634",
      "0.425681041",
      "0.036325681"
    ],
    [
      "0.704571014",
      "0.419918173",
      "0.045889920"
    ],
    [
      "0.393864985",
      "0.570508694",
      "0.055842735"
    ],
    [
      "0.503108075",
      "0.574939783",
      "0.053490184"
    ],
    [
      "0.562611396",
      "0.489109864",
      "0.050948409"
    ],
    [
      "0.447020483",
      "0.666498279",
      "0.053881954"
    ],
    [
      "0.553459112",
      "0.668288061",
      "0.052571721"
    ],
    [
      "0.500000000",
      "0.759581267",
      "0.053222068"
    ],
    [
      "0.658810038",
      "0.499367261",
      "0.045795544"
    ],
    [
      "0.606826689",
      "0.580265962",
      "0.050365094"
    ]
  ]
}

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": 38
}

当前第一名的答案

{
  "circles": [
    [
      "0.102702499",
      "0.059295315",
      "0.059295315"
    ],
    [
      "0.221235984",
      "0.059238184",
      "0.059238183"
    ],
    [
      "0.339482753",
      "0.059008807",
      "0.059008807"
    ],
    [
      "0.456843475",
      "0.058353744",
      "0.058353743"
    ],
    [
      "0.572330539",
      "0.057139700",
      "0.057139700"
    ],
    [
      "0.685252135",
      "0.055789962",
      "0.055789962"
    ],
    [
      "0.795939695",
      "0.054901166",
      "0.054901166"
    ],
    [
      "0.905436626",
      "0.054596190",
      "0.054596189"
    ],
    [
      "0.162075940",
      "0.162018738",
      "0.059352512"
    ],
    [
      "0.280788379",
      "0.161800708",
      "0.059360126"
    ],
    [
      "0.399392257",
      "0.160965731",
      "0.059246691"
    ],
    [
      "0.516863813",
      "0.158328896",
      "0.058254455"
    ],
    [
      "0.631304734",
      "0.154007432",
      "0.056268029"
    ],
    [
      "0.742243672",
      "0.150478752",
      "0.054727013"
    ],
    [
      "0.851254517",
      "0.149048303",
      "0.054293215"
    ],
    [
      "0.221613637",
      "0.264869869",
      "0.059488104"
    ],
    [
      "0.340931641",
      "0.264703951",
      "0.059830013"
    ],
    [
      "0.461470821",
      "0.263614841",
      "0.060714085"
    ],
    [
      "0.579850982",
      "0.255939709",
      "0.057914621"
    ],
    [
      "0.691290572",
      "0.246446560",
      "0.053928581"
    ],
    [
      "0.798397390",
      "0.242698363",
      "0.053243800"
    ],
    [
      "0.281074660",
      "0.367899031",
      "0.059468279"
    ],
    [
      "0.401261766",
      "0.369076794",
      "0.060724597"
    ],
    [
      "0.531055207",
      "0.373246057",
      "0.069135789"
    ],
    [
      "0.647114094",
      "0.341754409",
      "0.051119726"
    ],
    [
      "0.748638960",
      "0.333962705",
      "0.050703693"
    ],
    [
      "0.338931395",
      "0.470464991",
      "0.058290702"
    ],
    [
      "0.454882818",
      "0.474709458",
      "0.057738379"
    ],
    [
      "0.622557634",
      "0.425681041",
      "0.036325681"
    ],
    [
      "0.704571014",
      "0.419918173",
      "0.045889920"
    ],
    [
      "0.393864985",
      "0.570508694",
      "0.055842735"
    ],
    [
      "0.503108075",
      "0.574939783",
      "0.053490184"
    ],
    [
      "0.562611396",
      "0.489109864",
      "0.050948409"
    ],
    [
      "0.447020483",
      "0.666498279",
      "0.053881954"
    ],
    [
      "0.553459112",
      "0.668288061",
      "0.052571721"
    ],
    [
      "0.500000000",
      "0.759581267",
      "0.053222068"
    ],
    [
      "0.658810038",
      "0.499367261",
      "0.045795544"
    ],
    [
      "0.606826689",
      "0.580265962",
      "0.050365094"
    ]
  ]
}

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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