P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 40

在半径为 1 的圆内放置 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 = 40

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	[5.836305, 5.836315]	见来源署名	Erich Friedman ↗
本站纪录	暂无选手纪录	—	—

严格定义

容器圆心 (1, 1)、半径 1 的闭圆盘；圆可以与容器边界相切
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圆盘内：(x − 1)² + (y − 1)² ≤ (1 − r)²；任意两圆内部不相交：圆心距 ≥ 半径之和。全部按整数精确判定，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：起步布局

2y0

12345678910111213141516171819202122232425262728293031323334353637383940

0x2

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

和正方形里一样，等圆是最差的策略之一：一个大圆压在圆心附近，或者几个大圆围成一圈，再让小圆钻进它们和圆周之间的缝隙。圆盘没有角，所以“靠墙的大圆”换成了“沿圆周的一圈”，大小的取舍换了一种几何。

前沿在哪里

Erich Friedman 的“最大化总周长”表列到 n = 50：n ≤ 4 和 6–10 是 Elser 与 Cantrell 的闭式构型，n = 5 是 Haowei Lin 2026 年 7 月的改进，n = 33–50 是 Jonathan Viquerat 2026 年 9 月新加的行。表还在被改写，没有一行有最优性证明；本站展示的是按图重建并精确验算的构型。

查看来源

构型与历史

起步布局

查看构型、求解笔记与纪录历史。

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。

ANSWER FORMAT

答案怎么写

容器是圆心 (1, 1)、半径 1 的圆盘，所以坐标都在 [0, 2] 之内，不会出现负数。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"。容器是圆心 (1, 1)、半径 1 的闭圆盘。

起步布局答案

{
  "circles": [
    [
      "0.997457065",
      "1.029381877",
      "0.190296339"
    ],
    [
      "0.929172279",
      "1.818345383",
      "0.178595268"
    ],
    [
      "1.267303369",
      "1.796049772",
      "0.160270083"
    ],
    [
      "0.599899989",
      "1.738286047",
      "0.160270099"
    ],
    [
      "1.183509061",
      "0.732828907",
      "0.159787949"
    ],
    [
      "1.229952340",
      "0.191438494",
      "0.159375358"
    ],
    [
      "0.912388820",
      "0.163953316",
      "0.159375365"
    ],
    [
      "0.865123430",
      "0.705272600",
      "0.159787956"
    ],
    [
      "1.655038237",
      "0.469974366",
      "0.157383678"
    ],
    [
      "0.445767939",
      "0.365311855",
      "0.157383706"
    ],
    [
      "1.342885377",
      "1.005405335",
      "0.155963090"
    ],
    [
      "0.661284891",
      "0.946412773",
      "0.155963105"
    ],
    [
      "1.047753591",
      "0.448255691",
      "0.155507884"
    ],
    [
      "1.805020331",
      "0.744441115",
      "0.155388801"
    ],
    [
      "0.250859098",
      "0.609928304",
      "0.155388835"
    ],
    [
      "1.496493862",
      "0.737073269",
      "0.153225629"
    ],
    [
      "0.556063588",
      "0.655678936",
      "0.153225649"
    ],
    [
      "1.813375171",
      "1.238022176",
      "0.152513289"
    ],
    [
      "0.157824788",
      "1.094734240",
      "0.152513324"
    ],
    [
      "1.676526019",
      "1.510550781",
      "0.152444954"
    ],
    [
      "0.245815002",
      "1.386722756",
      "0.152444985"
    ],
    [
      "1.513018542",
      "1.257898191",
      "0.148500266"
    ],
    [
      "0.450300086",
      "1.165919883",
      "0.148500289"
    ],
    [
      "0.745628488",
      "0.421881484",
      "0.147766210"
    ],
    [
      "1.349917381",
      "0.474182768",
      "0.147766198"
    ],
    [
      "0.734992985",
      "1.238525085",
      "0.145305023"
    ],
    [
      "1.220084660",
      "1.280509778",
      "0.145305013"
    ],
    [
      "0.810337225",
      "1.518135612",
      "0.144278818"
    ],
    [
      "1.097820113",
      "1.543017250",
      "0.144278812"
    ],
    [
      "0.367703192",
      "0.885446838",
      "0.143881953"
    ],
    [
      "1.642576093",
      "0.995787168",
      "0.143881926"
    ],
    [
      "1.382533208",
      "1.517017367",
      "0.141618965"
    ],
    [
      "0.534325024",
      "1.443604960",
      "0.141618983"
    ],
    [
      "1.491241927",
      "0.258233352",
      "0.110316690"
    ],
    [
      "0.643508467",
      "0.184861881",
      "0.110316704"
    ],
    [
      "1.524725546",
      "1.721760931",
      "0.107657162"
    ],
    [
      "0.359069531",
      "1.620873451",
      "0.107657179"
    ],
    [
      "0.971820254",
      "1.325589903",
      "0.107019049"
    ],
    [
      "1.893188139",
      "0.991328853",
      "0.106769771"
    ],
    [
      "0.121583801",
      "0.837996379",
      "0.106769798"
    ]
  ]
}

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": 40
}

起步布局答案

{
  "circles": [
    [
      "0.997457065",
      "1.029381877",
      "0.190296339"
    ],
    [
      "0.929172279",
      "1.818345383",
      "0.178595268"
    ],
    [
      "1.267303369",
      "1.796049772",
      "0.160270083"
    ],
    [
      "0.599899989",
      "1.738286047",
      "0.160270099"
    ],
    [
      "1.183509061",
      "0.732828907",
      "0.159787949"
    ],
    [
      "1.229952340",
      "0.191438494",
      "0.159375358"
    ],
    [
      "0.912388820",
      "0.163953316",
      "0.159375365"
    ],
    [
      "0.865123430",
      "0.705272600",
      "0.159787956"
    ],
    [
      "1.655038237",
      "0.469974366",
      "0.157383678"
    ],
    [
      "0.445767939",
      "0.365311855",
      "0.157383706"
    ],
    [
      "1.342885377",
      "1.005405335",
      "0.155963090"
    ],
    [
      "0.661284891",
      "0.946412773",
      "0.155963105"
    ],
    [
      "1.047753591",
      "0.448255691",
      "0.155507884"
    ],
    [
      "1.805020331",
      "0.744441115",
      "0.155388801"
    ],
    [
      "0.250859098",
      "0.609928304",
      "0.155388835"
    ],
    [
      "1.496493862",
      "0.737073269",
      "0.153225629"
    ],
    [
      "0.556063588",
      "0.655678936",
      "0.153225649"
    ],
    [
      "1.813375171",
      "1.238022176",
      "0.152513289"
    ],
    [
      "0.157824788",
      "1.094734240",
      "0.152513324"
    ],
    [
      "1.676526019",
      "1.510550781",
      "0.152444954"
    ],
    [
      "0.245815002",
      "1.386722756",
      "0.152444985"
    ],
    [
      "1.513018542",
      "1.257898191",
      "0.148500266"
    ],
    [
      "0.450300086",
      "1.165919883",
      "0.148500289"
    ],
    [
      "0.745628488",
      "0.421881484",
      "0.147766210"
    ],
    [
      "1.349917381",
      "0.474182768",
      "0.147766198"
    ],
    [
      "0.734992985",
      "1.238525085",
      "0.145305023"
    ],
    [
      "1.220084660",
      "1.280509778",
      "0.145305013"
    ],
    [
      "0.810337225",
      "1.518135612",
      "0.144278818"
    ],
    [
      "1.097820113",
      "1.543017250",
      "0.144278812"
    ],
    [
      "0.367703192",
      "0.885446838",
      "0.143881953"
    ],
    [
      "1.642576093",
      "0.995787168",
      "0.143881926"
    ],
    [
      "1.382533208",
      "1.517017367",
      "0.141618965"
    ],
    [
      "0.534325024",
      "1.443604960",
      "0.141618983"
    ],
    [
      "1.491241927",
      "0.258233352",
      "0.110316690"
    ],
    [
      "0.643508467",
      "0.184861881",
      "0.110316704"
    ],
    [
      "1.524725546",
      "1.721760931",
      "0.107657162"
    ],
    [
      "0.359069531",
      "1.620873451",
      "0.107657179"
    ],
    [
      "0.971820254",
      "1.325589903",
      "0.107019049"
    ],
    [
      "1.893188139",
      "0.991328853",
      "0.106769771"
    ],
    [
      "0.121583801",
      "0.837996379",
      "0.106769798"
    ]
  ]
}

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"。容器是圆心 (1, 1)、半径 1 的闭圆盘。 · 验证器 v1.0.0

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