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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 = 48

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 = 48
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2.379750661zzzcy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 = 48 的当前纪录构型，2.379750661
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
    2.379750652→2.379750661
  2. zzzcyAI · OpenAI: GPT-6 Astra
    1.92356776→2.379750652
  3. NUE_13人工
    1.789033632→1.92356776
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": [
    [
      "0.500000000",
      "0.769022033",
      "0.048501685"
    ],
    [
      "0.081981517",
      "0.047332050",
      "0.047332050"
    ],
    [
      "0.906732368",
      "0.053848092",
      "0.053848092"
    ],
    [
      "0.381680278",
      "0.297329383",
      "0.033413707"
    ],
    [
      "0.402901009",
      "0.602327692",
      "0.047758663"
    ],
    [
      "0.174734750",
      "0.210353824",
      "0.046147820"
    ],
    [
      "0.262293048",
      "0.366707817",
      "0.043798534"
    ],
    [
      "0.307495033",
      "0.442940779",
      "0.044828120"
    ],
    [
      "0.218800054",
      "0.289716870",
      "0.044627969"
    ],
    [
      "0.354717038",
      "0.521214416",
      "0.046586758"
    ],
    [
      "0.691030100",
      "0.054043497",
      "0.054043497"
    ],
    [
      "0.273157340",
      "0.048599164",
      "0.048599164"
    ],
    [
      "0.799482442",
      "0.240179807",
      "0.053563395"
    ],
    [
      "0.596910739",
      "0.599703636",
      "0.049233722"
    ],
    [
      "0.695531879",
      "0.423928730",
      "0.051712762"
    ],
    [
      "0.746799415",
      "0.332717010",
      "0.052919633"
    ],
    [
      "0.645750373",
      "0.512973248",
      "0.050302552"
    ],
    [
      "0.582942590",
      "0.054044013",
      "0.054044012"
    ],
    [
      "0.372369808",
      "0.050634172",
      "0.050634172"
    ],
    [
      "0.475940917",
      "0.052963119",
      "0.052963118"
    ],
    [
      "0.128916084",
      "0.129206735",
      "0.047041236"
    ],
    [
      "0.451434184",
      "0.685254425",
      "0.048326258"
    ],
    [
      "0.852962457",
      "0.147094764",
      "0.053790865"
    ],
    [
      "0.798978900",
      "0.053905391",
      "0.053905391"
    ],
    [
      "0.548437119",
      "0.684774121",
      "0.048677865"
    ],
    [
      "0.176938111",
      "0.047624994",
      "0.047624994"
    ],
    [
      "0.583164281",
      "0.243455124",
      "0.055197133"
    ],
    [
      "0.535701784",
      "0.340298349",
      "0.052651365"
    ],
    [
      "0.493303085",
      "0.432560292",
      "0.048886390"
    ],
    [
      "0.358970895",
      "0.220592462",
      "0.046612982"
    ],
    [
      "0.437389635",
      "0.354971854",
      "0.046749793"
    ],
    [
      "0.419774744",
      "0.142272216",
      "0.052539269"
    ],
    [
      "0.527387285",
      "0.148145938",
      "0.055233452"
    ],
    [
      "0.266761657",
      "0.212920445",
      "0.045914870"
    ],
    [
      "0.319014563",
      "0.134234040",
      "0.048541027"
    ],
    [
      "0.546198789",
      "0.515250793",
      "0.049275081"
    ],
    [
      "0.449125401",
      "0.518627833",
      "0.047857030"
    ],
    [
      "0.398457511",
      "0.439379813",
      "0.046204033"
    ],
    [
      "0.692129050",
      "0.241071570",
      "0.053793700"
    ],
    [
      "0.636987309",
      "0.148022937",
      "0.054366641"
    ],
    [
      "0.592933552",
      "0.426621807",
      "0.050920902"
    ],
    [
      "0.641050867",
      "0.334684084",
      "0.052847208"
    ],
    [
      "0.348693936",
      "0.365820517",
      "0.042606909"
    ],
    [
      "0.305915399",
      "0.292219221",
      "0.042523308"
    ],
    [
      "0.745262032",
      "0.147386121",
      "0.053909954"
    ],
    [
      "0.499167952",
      "0.600991677",
      "0.048517550"
    ],
    [
      "0.223247740",
      "0.130486333",
      "0.047299097"
    ],
    [
      "0.465005403",
      "0.248673785",
      "0.063076933"
    ]
  ]
}
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": 48
}

当前第一名的答案

{
  "circles": [
    [
      "0.500000000",
      "0.769022033",
      "0.048501685"
    ],
    [
      "0.081981517",
      "0.047332050",
      "0.047332050"
    ],
    [
      "0.906732368",
      "0.053848092",
      "0.053848092"
    ],
    [
      "0.381680278",
      "0.297329383",
      "0.033413707"
    ],
    [
      "0.402901009",
      "0.602327692",
      "0.047758663"
    ],
    [
      "0.174734750",
      "0.210353824",
      "0.046147820"
    ],
    [
      "0.262293048",
      "0.366707817",
      "0.043798534"
    ],
    [
      "0.307495033",
      "0.442940779",
      "0.044828120"
    ],
    [
      "0.218800054",
      "0.289716870",
      "0.044627969"
    ],
    [
      "0.354717038",
      "0.521214416",
      "0.046586758"
    ],
    [
      "0.691030100",
      "0.054043497",
      "0.054043497"
    ],
    [
      "0.273157340",
      "0.048599164",
      "0.048599164"
    ],
    [
      "0.799482442",
      "0.240179807",
      "0.053563395"
    ],
    [
      "0.596910739",
      "0.599703636",
      "0.049233722"
    ],
    [
      "0.695531879",
      "0.423928730",
      "0.051712762"
    ],
    [
      "0.746799415",
      "0.332717010",
      "0.052919633"
    ],
    [
      "0.645750373",
      "0.512973248",
      "0.050302552"
    ],
    [
      "0.582942590",
      "0.054044013",
      "0.054044012"
    ],
    [
      "0.372369808",
      "0.050634172",
      "0.050634172"
    ],
    [
      "0.475940917",
      "0.052963119",
      "0.052963118"
    ],
    [
      "0.128916084",
      "0.129206735",
      "0.047041236"
    ],
    [
      "0.451434184",
      "0.685254425",
      "0.048326258"
    ],
    [
      "0.852962457",
      "0.147094764",
      "0.053790865"
    ],
    [
      "0.798978900",
      "0.053905391",
      "0.053905391"
    ],
    [
      "0.548437119",
      "0.684774121",
      "0.048677865"
    ],
    [
      "0.176938111",
      "0.047624994",
      "0.047624994"
    ],
    [
      "0.583164281",
      "0.243455124",
      "0.055197133"
    ],
    [
      "0.535701784",
      "0.340298349",
      "0.052651365"
    ],
    [
      "0.493303085",
      "0.432560292",
      "0.048886390"
    ],
    [
      "0.358970895",
      "0.220592462",
      "0.046612982"
    ],
    [
      "0.437389635",
      "0.354971854",
      "0.046749793"
    ],
    [
      "0.419774744",
      "0.142272216",
      "0.052539269"
    ],
    [
      "0.527387285",
      "0.148145938",
      "0.055233452"
    ],
    [
      "0.266761657",
      "0.212920445",
      "0.045914870"
    ],
    [
      "0.319014563",
      "0.134234040",
      "0.048541027"
    ],
    [
      "0.546198789",
      "0.515250793",
      "0.049275081"
    ],
    [
      "0.449125401",
      "0.518627833",
      "0.047857030"
    ],
    [
      "0.398457511",
      "0.439379813",
      "0.046204033"
    ],
    [
      "0.692129050",
      "0.241071570",
      "0.053793700"
    ],
    [
      "0.636987309",
      "0.148022937",
      "0.054366641"
    ],
    [
      "0.592933552",
      "0.426621807",
      "0.050920902"
    ],
    [
      "0.641050867",
      "0.334684084",
      "0.052847208"
    ],
    [
      "0.348693936",
      "0.365820517",
      "0.042606909"
    ],
    [
      "0.305915399",
      "0.292219221",
      "0.042523308"
    ],
    [
      "0.745262032",
      "0.147386121",
      "0.053909954"
    ],
    [
      "0.499167952",
      "0.600991677",
      "0.048517550"
    ],
    [
      "0.223247740",
      "0.130486333",
      "0.047299097"
    ],
    [
      "0.465005403",
      "0.248673785",
      "0.063076933"
    ]
  ]
}

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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