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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 = 50

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 = 50
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2.427760297zzzcy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 = 50 的当前纪录构型，2.427760297
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（3 次易主）
  1. zzzcyAI · OpenAI: GPT-6 Astra
    2.427760287→2.427760297
  2. zzzcyAI · OpenAI: GPT-6 Astra
    2.084952742→2.427760287
  3. NUE_13人工
    1.8635767→2.084952742
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": [
    [
      "0.088746094",
      "0.051237581",
      "0.051237581"
    ],
    [
      "0.191262482",
      "0.051278815",
      "0.051278815"
    ],
    [
      "0.293932314",
      "0.051391078",
      "0.051391078"
    ],
    [
      "0.396859008",
      "0.051535717",
      "0.051535717"
    ],
    [
      "0.500000000",
      "0.051605299",
      "0.051605298"
    ],
    [
      "0.603140992",
      "0.051535717",
      "0.051535717"
    ],
    [
      "0.706067685",
      "0.051391077",
      "0.051391077"
    ],
    [
      "0.808737517",
      "0.051278815",
      "0.051278815"
    ],
    [
      "0.911253906",
      "0.051237581",
      "0.051237581"
    ],
    [
      "0.139927398",
      "0.139968589",
      "0.051196386"
    ],
    [
      "0.242388082",
      "0.140168090",
      "0.051264490"
    ],
    [
      "0.345125729",
      "0.140612615",
      "0.051474117"
    ],
    [
      "0.448299474",
      "0.141043261",
      "0.051700526"
    ],
    [
      "0.551700526",
      "0.141043261",
      "0.051700526"
    ],
    [
      "0.654874271",
      "0.140612615",
      "0.051474117"
    ],
    [
      "0.757611919",
      "0.140168088",
      "0.051264491"
    ],
    [
      "0.860072602",
      "0.139968589",
      "0.051196386"
    ],
    [
      "0.190917401",
      "0.228587949",
      "0.051045344"
    ],
    [
      "0.396112082",
      "0.230369330",
      "0.051753185"
    ],
    [
      "0.500000000",
      "0.231095151",
      "0.052137267"
    ],
    [
      "0.603887918",
      "0.230369330",
      "0.051753185"
    ],
    [
      "0.706836514",
      "0.229169895",
      "0.051202397"
    ],
    [
      "0.809082599",
      "0.228587949",
      "0.051045344"
    ],
    [
      "0.343056100",
      "0.318439363",
      "0.051063486"
    ],
    [
      "0.447031366",
      "0.321878269",
      "0.052968634"
    ],
    [
      "0.552968634",
      "0.321878269",
      "0.052968634"
    ],
    [
      "0.656943900",
      "0.318439363",
      "0.051063486"
    ],
    [
      "0.758616512",
      "0.316845221",
      "0.050621622"
    ],
    [
      "0.290428302",
      "0.404097087",
      "0.049469744"
    ],
    [
      "0.390251003",
      "0.408305540",
      "0.050441630"
    ],
    [
      "0.609748997",
      "0.408305540",
      "0.050441630"
    ],
    [
      "0.709571698",
      "0.404097085",
      "0.049469744"
    ],
    [
      "0.335922404",
      "0.488683993",
      "0.046575339"
    ],
    [
      "0.426408124",
      "0.495753057",
      "0.044186091"
    ],
    [
      "0.460226149",
      "0.572601044",
      "0.039773851"
    ],
    [
      "0.664077596",
      "0.488683993",
      "0.046575339"
    ],
    [
      "0.377579829",
      "0.567984879",
      "0.043001284"
    ],
    [
      "0.500000000",
      "0.513452074",
      "0.031504195"
    ],
    [
      "0.539773851",
      "0.572601044",
      "0.039773851"
    ],
    [
      "0.622420171",
      "0.567984879",
      "0.043001284"
    ],
    [
      "0.418135048",
      "0.641796036",
      "0.041217554"
    ],
    [
      "0.500000000",
      "0.642501740",
      "0.040650439"
    ],
    [
      "0.581864952",
      "0.641796036",
      "0.041217554"
    ],
    [
      "0.459023346",
      "0.713098453",
      "0.040976652"
    ],
    [
      "0.540976654",
      "0.713098453",
      "0.040976652"
    ],
    [
      "0.500000000",
      "0.784072097",
      "0.040976653"
    ],
    [
      "0.293163486",
      "0.229169895",
      "0.051202397"
    ],
    [
      "0.500000000",
      "0.421812478",
      "0.060135399"
    ],
    [
      "0.573591876",
      "0.495753057",
      "0.044186091"
    ],
    [
      "0.241383488",
      "0.316845221",
      "0.050621622"
    ]
  ]
}
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": 50
}

当前第一名的答案

{
  "circles": [
    [
      "0.088746094",
      "0.051237581",
      "0.051237581"
    ],
    [
      "0.191262482",
      "0.051278815",
      "0.051278815"
    ],
    [
      "0.293932314",
      "0.051391078",
      "0.051391078"
    ],
    [
      "0.396859008",
      "0.051535717",
      "0.051535717"
    ],
    [
      "0.500000000",
      "0.051605299",
      "0.051605298"
    ],
    [
      "0.603140992",
      "0.051535717",
      "0.051535717"
    ],
    [
      "0.706067685",
      "0.051391077",
      "0.051391077"
    ],
    [
      "0.808737517",
      "0.051278815",
      "0.051278815"
    ],
    [
      "0.911253906",
      "0.051237581",
      "0.051237581"
    ],
    [
      "0.139927398",
      "0.139968589",
      "0.051196386"
    ],
    [
      "0.242388082",
      "0.140168090",
      "0.051264490"
    ],
    [
      "0.345125729",
      "0.140612615",
      "0.051474117"
    ],
    [
      "0.448299474",
      "0.141043261",
      "0.051700526"
    ],
    [
      "0.551700526",
      "0.141043261",
      "0.051700526"
    ],
    [
      "0.654874271",
      "0.140612615",
      "0.051474117"
    ],
    [
      "0.757611919",
      "0.140168088",
      "0.051264491"
    ],
    [
      "0.860072602",
      "0.139968589",
      "0.051196386"
    ],
    [
      "0.190917401",
      "0.228587949",
      "0.051045344"
    ],
    [
      "0.396112082",
      "0.230369330",
      "0.051753185"
    ],
    [
      "0.500000000",
      "0.231095151",
      "0.052137267"
    ],
    [
      "0.603887918",
      "0.230369330",
      "0.051753185"
    ],
    [
      "0.706836514",
      "0.229169895",
      "0.051202397"
    ],
    [
      "0.809082599",
      "0.228587949",
      "0.051045344"
    ],
    [
      "0.343056100",
      "0.318439363",
      "0.051063486"
    ],
    [
      "0.447031366",
      "0.321878269",
      "0.052968634"
    ],
    [
      "0.552968634",
      "0.321878269",
      "0.052968634"
    ],
    [
      "0.656943900",
      "0.318439363",
      "0.051063486"
    ],
    [
      "0.758616512",
      "0.316845221",
      "0.050621622"
    ],
    [
      "0.290428302",
      "0.404097087",
      "0.049469744"
    ],
    [
      "0.390251003",
      "0.408305540",
      "0.050441630"
    ],
    [
      "0.609748997",
      "0.408305540",
      "0.050441630"
    ],
    [
      "0.709571698",
      "0.404097085",
      "0.049469744"
    ],
    [
      "0.335922404",
      "0.488683993",
      "0.046575339"
    ],
    [
      "0.426408124",
      "0.495753057",
      "0.044186091"
    ],
    [
      "0.460226149",
      "0.572601044",
      "0.039773851"
    ],
    [
      "0.664077596",
      "0.488683993",
      "0.046575339"
    ],
    [
      "0.377579829",
      "0.567984879",
      "0.043001284"
    ],
    [
      "0.500000000",
      "0.513452074",
      "0.031504195"
    ],
    [
      "0.539773851",
      "0.572601044",
      "0.039773851"
    ],
    [
      "0.622420171",
      "0.567984879",
      "0.043001284"
    ],
    [
      "0.418135048",
      "0.641796036",
      "0.041217554"
    ],
    [
      "0.500000000",
      "0.642501740",
      "0.040650439"
    ],
    [
      "0.581864952",
      "0.641796036",
      "0.041217554"
    ],
    [
      "0.459023346",
      "0.713098453",
      "0.040976652"
    ],
    [
      "0.540976654",
      "0.713098453",
      "0.040976652"
    ],
    [
      "0.500000000",
      "0.784072097",
      "0.040976653"
    ],
    [
      "0.293163486",
      "0.229169895",
      "0.051202397"
    ],
    [
      "0.500000000",
      "0.421812478",
      "0.060135399"
    ],
    [
      "0.573591876",
      "0.495753057",
      "0.044186091"
    ],
    [
      "0.241383488",
      "0.316845221",
      "0.050621622"
    ]
  ]
}

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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