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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 = 34

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 = 34
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1.98837524zzzcy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 = 34 的当前纪录构型，1.98837524
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
    1.768637314→1.98837524
  2. NUE_13人工
    1.557480602→1.768637314
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": [
    [
      "0.906932541",
      "0.053732522",
      "0.053732522"
    ],
    [
      "0.500000000",
      "0.737613536",
      "0.064205933"
    ],
    [
      "0.107201793",
      "0.061892984",
      "0.061892984"
    ],
    [
      "0.549936649",
      "0.324722302",
      "0.040564618"
    ],
    [
      "0.372051246",
      "0.508032949",
      "0.068189356"
    ],
    [
      "0.303730485",
      "0.390417227",
      "0.067829702"
    ],
    [
      "0.629000977",
      "0.522971420",
      "0.059808868"
    ],
    [
      "0.351695621",
      "0.059845173",
      "0.059845172"
    ],
    [
      "0.691326005",
      "0.054217124",
      "0.054217124"
    ],
    [
      "0.800006388",
      "0.239582149",
      "0.053408474"
    ],
    [
      "0.690357851",
      "0.425380030",
      "0.055467952"
    ],
    [
      "0.746559984",
      "0.332098501",
      "0.053436241"
    ],
    [
      "0.468982386",
      "0.057465725",
      "0.057465725"
    ],
    [
      "0.581771405",
      "0.055343262",
      "0.055343262"
    ],
    [
      "0.234579725",
      "0.278003131",
      "0.064150435"
    ],
    [
      "0.230476965",
      "0.061383241",
      "0.061383241"
    ],
    [
      "0.853331302",
      "0.146764772",
      "0.053636432"
    ],
    [
      "0.799371247",
      "0.053828815",
      "0.053828815"
    ],
    [
      "0.564568135",
      "0.627767963",
      "0.063211075"
    ],
    [
      "0.169797420",
      "0.169282271",
      "0.062407743"
    ],
    [
      "0.436166993",
      "0.625024654",
      "0.065219369"
    ],
    [
      "0.482166319",
      "0.254986359",
      "0.056676933"
    ],
    [
      "0.589466968",
      "0.241629698",
      "0.051451831"
    ],
    [
      "0.362681642",
      "0.272421060",
      "0.064073043"
    ],
    [
      "0.693751060",
      "0.239862972",
      "0.052847225"
    ],
    [
      "0.638628828",
      "0.148536460",
      "0.053825134"
    ],
    [
      "0.414800441",
      "0.160746710",
      "0.059164652"
    ],
    [
      "0.529303914",
      "0.153089676",
      "0.055594554"
    ],
    [
      "0.578292373",
      "0.417914398",
      "0.056845923"
    ],
    [
      "0.641739073",
      "0.330210010",
      "0.051401679"
    ],
    [
      "0.504770290",
      "0.514698579",
      "0.064696968"
    ],
    [
      "0.746070136",
      "0.147131502",
      "0.053625359"
    ],
    [
      "0.293988677",
      "0.166937792",
      "0.061805641"
    ],
    [
      "0.448560428",
      "0.384495429",
      "0.077121255"
    ]
  ]
}
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": 34
}

当前第一名的答案

{
  "circles": [
    [
      "0.906932541",
      "0.053732522",
      "0.053732522"
    ],
    [
      "0.500000000",
      "0.737613536",
      "0.064205933"
    ],
    [
      "0.107201793",
      "0.061892984",
      "0.061892984"
    ],
    [
      "0.549936649",
      "0.324722302",
      "0.040564618"
    ],
    [
      "0.372051246",
      "0.508032949",
      "0.068189356"
    ],
    [
      "0.303730485",
      "0.390417227",
      "0.067829702"
    ],
    [
      "0.629000977",
      "0.522971420",
      "0.059808868"
    ],
    [
      "0.351695621",
      "0.059845173",
      "0.059845172"
    ],
    [
      "0.691326005",
      "0.054217124",
      "0.054217124"
    ],
    [
      "0.800006388",
      "0.239582149",
      "0.053408474"
    ],
    [
      "0.690357851",
      "0.425380030",
      "0.055467952"
    ],
    [
      "0.746559984",
      "0.332098501",
      "0.053436241"
    ],
    [
      "0.468982386",
      "0.057465725",
      "0.057465725"
    ],
    [
      "0.581771405",
      "0.055343262",
      "0.055343262"
    ],
    [
      "0.234579725",
      "0.278003131",
      "0.064150435"
    ],
    [
      "0.230476965",
      "0.061383241",
      "0.061383241"
    ],
    [
      "0.853331302",
      "0.146764772",
      "0.053636432"
    ],
    [
      "0.799371247",
      "0.053828815",
      "0.053828815"
    ],
    [
      "0.564568135",
      "0.627767963",
      "0.063211075"
    ],
    [
      "0.169797420",
      "0.169282271",
      "0.062407743"
    ],
    [
      "0.436166993",
      "0.625024654",
      "0.065219369"
    ],
    [
      "0.482166319",
      "0.254986359",
      "0.056676933"
    ],
    [
      "0.589466968",
      "0.241629698",
      "0.051451831"
    ],
    [
      "0.362681642",
      "0.272421060",
      "0.064073043"
    ],
    [
      "0.693751060",
      "0.239862972",
      "0.052847225"
    ],
    [
      "0.638628828",
      "0.148536460",
      "0.053825134"
    ],
    [
      "0.414800441",
      "0.160746710",
      "0.059164652"
    ],
    [
      "0.529303914",
      "0.153089676",
      "0.055594554"
    ],
    [
      "0.578292373",
      "0.417914398",
      "0.056845923"
    ],
    [
      "0.641739073",
      "0.330210010",
      "0.051401679"
    ],
    [
      "0.504770290",
      "0.514698579",
      "0.064696968"
    ],
    [
      "0.746070136",
      "0.147131502",
      "0.053625359"
    ],
    [
      "0.293988677",
      "0.166937792",
      "0.061805641"
    ],
    [
      "0.448560428",
      "0.384495429",
      "0.077121255"
    ]
  ]
}

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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