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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 = 39

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 = 39
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2.13654198zzzcy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 = 39 的当前纪录构型，2.13654198
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）
  1. zzzcyAI · OpenAI: GPT-6 Astra
    1.652364799→2.13654198
  2. NUE_13人工
    1.602950934→1.652364799
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": [
    [
      "0.091430161",
      "0.052787228",
      "0.052787227"
    ],
    [
      "0.196670929",
      "0.052454068",
      "0.052454068"
    ],
    [
      "0.300550362",
      "0.051430407",
      "0.051430407"
    ],
    [
      "0.401648803",
      "0.049683132",
      "0.049683131"
    ],
    [
      "0.500000000",
      "0.048673250",
      "0.048673250"
    ],
    [
      "0.598351197",
      "0.049683132",
      "0.049683131"
    ],
    [
      "0.699449638",
      "0.051430407",
      "0.051430407"
    ],
    [
      "0.803329071",
      "0.052454068",
      "0.052454068"
    ],
    [
      "0.908569839",
      "0.052787228",
      "0.052787227"
    ],
    [
      "0.144675650",
      "0.144339790",
      "0.053122893"
    ],
    [
      "0.250537873",
      "0.142821616",
      "0.052750214"
    ],
    [
      "0.354382488",
      "0.138777888",
      "0.051173101"
    ],
    [
      "0.452667857",
      "0.132199916",
      "0.047332143"
    ],
    [
      "0.547332143",
      "0.132199916",
      "0.047332143"
    ],
    [
      "0.645617512",
      "0.138777888",
      "0.051173102"
    ],
    [
      "0.749462127",
      "0.142821616",
      "0.052750214"
    ],
    [
      "0.855324350",
      "0.144339790",
      "0.053122893"
    ],
    [
      "0.199320277",
      "0.236754841",
      "0.054239002"
    ],
    [
      "0.307669969",
      "0.233196736",
      "0.054169095"
    ],
    [
      "0.413365101",
      "0.223402784",
      "0.051978832"
    ],
    [
      "0.500000000",
      "0.202241729",
      "0.037202981"
    ],
    [
      "0.586634899",
      "0.223402784",
      "0.051978832"
    ],
    [
      "0.692330031",
      "0.233196736",
      "0.054169094"
    ],
    [
      "0.800679723",
      "0.236754841",
      "0.054239002"
    ],
    [
      "0.256737453",
      "0.331533118",
      "0.056574596"
    ],
    [
      "0.371889518",
      "0.326026306",
      "0.058709067"
    ],
    [
      "0.500000000",
      "0.309861877",
      "0.070417166"
    ],
    [
      "0.628110482",
      "0.326026306",
      "0.058709067"
    ],
    [
      "0.743262547",
      "0.331533118",
      "0.056574596"
    ],
    [
      "0.317009995",
      "0.430490703",
      "0.059293357"
    ],
    [
      "0.438123461",
      "0.426793132",
      "0.061876538"
    ],
    [
      "0.561876539",
      "0.426793132",
      "0.061876538"
    ],
    [
      "0.621812527",
      "0.533688830",
      "0.060675544"
    ],
    [
      "0.682990005",
      "0.430490703",
      "0.059293357"
    ],
    [
      "0.378187473",
      "0.533688830",
      "0.060675544"
    ],
    [
      "0.439129396",
      "0.638853216",
      "0.060870603"
    ],
    [
      "0.500000000",
      "0.533113192",
      "0.061138343"
    ],
    [
      "0.560870604",
      "0.638853216",
      "0.060870603"
    ],
    [
      "0.500000000",
      "0.744284195",
      "0.060870604"
    ]
  ]
}
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": 39
}

当前第一名的答案

{
  "circles": [
    [
      "0.091430161",
      "0.052787228",
      "0.052787227"
    ],
    [
      "0.196670929",
      "0.052454068",
      "0.052454068"
    ],
    [
      "0.300550362",
      "0.051430407",
      "0.051430407"
    ],
    [
      "0.401648803",
      "0.049683132",
      "0.049683131"
    ],
    [
      "0.500000000",
      "0.048673250",
      "0.048673250"
    ],
    [
      "0.598351197",
      "0.049683132",
      "0.049683131"
    ],
    [
      "0.699449638",
      "0.051430407",
      "0.051430407"
    ],
    [
      "0.803329071",
      "0.052454068",
      "0.052454068"
    ],
    [
      "0.908569839",
      "0.052787228",
      "0.052787227"
    ],
    [
      "0.144675650",
      "0.144339790",
      "0.053122893"
    ],
    [
      "0.250537873",
      "0.142821616",
      "0.052750214"
    ],
    [
      "0.354382488",
      "0.138777888",
      "0.051173101"
    ],
    [
      "0.452667857",
      "0.132199916",
      "0.047332143"
    ],
    [
      "0.547332143",
      "0.132199916",
      "0.047332143"
    ],
    [
      "0.645617512",
      "0.138777888",
      "0.051173102"
    ],
    [
      "0.749462127",
      "0.142821616",
      "0.052750214"
    ],
    [
      "0.855324350",
      "0.144339790",
      "0.053122893"
    ],
    [
      "0.199320277",
      "0.236754841",
      "0.054239002"
    ],
    [
      "0.307669969",
      "0.233196736",
      "0.054169095"
    ],
    [
      "0.413365101",
      "0.223402784",
      "0.051978832"
    ],
    [
      "0.500000000",
      "0.202241729",
      "0.037202981"
    ],
    [
      "0.586634899",
      "0.223402784",
      "0.051978832"
    ],
    [
      "0.692330031",
      "0.233196736",
      "0.054169094"
    ],
    [
      "0.800679723",
      "0.236754841",
      "0.054239002"
    ],
    [
      "0.256737453",
      "0.331533118",
      "0.056574596"
    ],
    [
      "0.371889518",
      "0.326026306",
      "0.058709067"
    ],
    [
      "0.500000000",
      "0.309861877",
      "0.070417166"
    ],
    [
      "0.628110482",
      "0.326026306",
      "0.058709067"
    ],
    [
      "0.743262547",
      "0.331533118",
      "0.056574596"
    ],
    [
      "0.317009995",
      "0.430490703",
      "0.059293357"
    ],
    [
      "0.438123461",
      "0.426793132",
      "0.061876538"
    ],
    [
      "0.561876539",
      "0.426793132",
      "0.061876538"
    ],
    [
      "0.621812527",
      "0.533688830",
      "0.060675544"
    ],
    [
      "0.682990005",
      "0.430490703",
      "0.059293357"
    ],
    [
      "0.378187473",
      "0.533688830",
      "0.060675544"
    ],
    [
      "0.439129396",
      "0.638853216",
      "0.060870603"
    ],
    [
      "0.500000000",
      "0.533113192",
      "0.061138343"
    ],
    [
      "0.560870604",
      "0.638853216",
      "0.060870603"
    ],
    [
      "0.500000000",
      "0.744284195",
      "0.060870604"
    ]
  ]
}

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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