P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 41

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 = 41

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	[5.910325, 5.910335]	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

1234567891011121314151617181920212223242526272829303132333435363738394041

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": [
    [
      "1.002660160",
      "1.015712102",
      "0.194305993"
    ],
    [
      "0.762148406",
      "1.542970231",
      "0.181232303"
    ],
    [
      "0.952599235",
      "1.832953051",
      "0.165699323"
    ],
    [
      "0.894001375",
      "0.679510299",
      "0.159018763"
    ],
    [
      "1.207562001",
      "0.728664314",
      "0.158371191"
    ],
    [
      "0.669186413",
      "0.902725824",
      "0.157788651"
    ],
    [
      "1.272899297",
      "1.798148883",
      "0.156486119"
    ],
    [
      "1.352652678",
      "1.007311526",
      "0.155787324"
    ],
    [
      "0.982948393",
      "0.154272037",
      "0.154100156"
    ],
    [
      "0.714414339",
      "1.211185216",
      "0.153968885"
    ],
    [
      "0.513286600",
      "0.307738570",
      "0.153763731"
    ],
    [
      "1.286700538",
      "0.203683323",
      "0.153644608"
    ],
    [
      "0.297914023",
      "0.526119511",
      "0.152953698"
    ],
    [
      "0.592848365",
      "0.602729405",
      "0.151768046"
    ],
    [
      "1.092801038",
      "0.440134835",
      "0.152143443"
    ],
    [
      "1.681313200",
      "0.495260283",
      "0.152090890"
    ],
    [
      "1.674855399",
      "1.515899031",
      "0.150540395"
    ],
    [
      "0.151102030",
      "0.991391468",
      "0.151058382"
    ],
    [
      "1.816318762",
      "0.766584526",
      "0.150965780"
    ],
    [
      "1.515613739",
      "0.748210818",
      "0.150300055"
    ],
    [
      "0.432162794",
      "1.098810645",
      "0.149830355"
    ],
    [
      "1.092894638",
      "1.550584758",
      "0.149601569"
    ],
    [
      "1.812843563",
      "1.249076951",
      "0.149850609"
    ],
    [
      "0.200046157",
      "1.288166781",
      "0.149725782"
    ],
    [
      "1.219719998",
      "1.280916382",
      "0.148401291"
    ],
    [
      "1.515481966",
      "1.263616222",
      "0.147866218"
    ],
    [
      "0.800062438",
      "0.390629935",
      "0.144751526"
    ],
    [
      "0.475681977",
      "1.389264256",
      "0.143865438"
    ],
    [
      "1.385525053",
      "0.484335899",
      "0.143898922"
    ],
    [
      "0.382804858",
      "0.809907281",
      "0.143258981"
    ],
    [
      "1.382669660",
      "1.520915207",
      "0.141688399"
    ],
    [
      "1.650048447",
      "1.007320605",
      "0.141608445"
    ],
    [
      "0.467714175",
      "1.674100519",
      "0.141082245"
    ],
    [
      "0.959516206",
      "1.322024287",
      "0.115029673"
    ],
    [
      "0.673479692",
      "1.824111632",
      "0.113560214"
    ],
    [
      "0.721906213",
      "0.151361434",
      "0.106958248"
    ],
    [
      "1.524651892",
      "1.723866018",
      "0.105996856"
    ],
    [
      "1.534012280",
      "0.283175889",
      "0.106128688"
    ],
    [
      "0.145648330",
      "0.734920934",
      "0.105470130"
    ],
    [
      "1.895802938",
      "1.009004104",
      "0.104151811"
    ],
    [
      "0.273527753",
      "1.528523936",
      "0.101612847"
    ]
  ]
}

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": 41
}

起步布局答案

{
  "circles": [
    [
      "1.002660160",
      "1.015712102",
      "0.194305993"
    ],
    [
      "0.762148406",
      "1.542970231",
      "0.181232303"
    ],
    [
      "0.952599235",
      "1.832953051",
      "0.165699323"
    ],
    [
      "0.894001375",
      "0.679510299",
      "0.159018763"
    ],
    [
      "1.207562001",
      "0.728664314",
      "0.158371191"
    ],
    [
      "0.669186413",
      "0.902725824",
      "0.157788651"
    ],
    [
      "1.272899297",
      "1.798148883",
      "0.156486119"
    ],
    [
      "1.352652678",
      "1.007311526",
      "0.155787324"
    ],
    [
      "0.982948393",
      "0.154272037",
      "0.154100156"
    ],
    [
      "0.714414339",
      "1.211185216",
      "0.153968885"
    ],
    [
      "0.513286600",
      "0.307738570",
      "0.153763731"
    ],
    [
      "1.286700538",
      "0.203683323",
      "0.153644608"
    ],
    [
      "0.297914023",
      "0.526119511",
      "0.152953698"
    ],
    [
      "0.592848365",
      "0.602729405",
      "0.151768046"
    ],
    [
      "1.092801038",
      "0.440134835",
      "0.152143443"
    ],
    [
      "1.681313200",
      "0.495260283",
      "0.152090890"
    ],
    [
      "1.674855399",
      "1.515899031",
      "0.150540395"
    ],
    [
      "0.151102030",
      "0.991391468",
      "0.151058382"
    ],
    [
      "1.816318762",
      "0.766584526",
      "0.150965780"
    ],
    [
      "1.515613739",
      "0.748210818",
      "0.150300055"
    ],
    [
      "0.432162794",
      "1.098810645",
      "0.149830355"
    ],
    [
      "1.092894638",
      "1.550584758",
      "0.149601569"
    ],
    [
      "1.812843563",
      "1.249076951",
      "0.149850609"
    ],
    [
      "0.200046157",
      "1.288166781",
      "0.149725782"
    ],
    [
      "1.219719998",
      "1.280916382",
      "0.148401291"
    ],
    [
      "1.515481966",
      "1.263616222",
      "0.147866218"
    ],
    [
      "0.800062438",
      "0.390629935",
      "0.144751526"
    ],
    [
      "0.475681977",
      "1.389264256",
      "0.143865438"
    ],
    [
      "1.385525053",
      "0.484335899",
      "0.143898922"
    ],
    [
      "0.382804858",
      "0.809907281",
      "0.143258981"
    ],
    [
      "1.382669660",
      "1.520915207",
      "0.141688399"
    ],
    [
      "1.650048447",
      "1.007320605",
      "0.141608445"
    ],
    [
      "0.467714175",
      "1.674100519",
      "0.141082245"
    ],
    [
      "0.959516206",
      "1.322024287",
      "0.115029673"
    ],
    [
      "0.673479692",
      "1.824111632",
      "0.113560214"
    ],
    [
      "0.721906213",
      "0.151361434",
      "0.106958248"
    ],
    [
      "1.524651892",
      "1.723866018",
      "0.105996856"
    ],
    [
      "1.534012280",
      "0.283175889",
      "0.106128688"
    ],
    [
      "0.145648330",
      "0.734920934",
      "0.105470130"
    ],
    [
      "1.895802938",
      "1.009004104",
      "0.104151811"
    ],
    [
      "0.273527753",
      "1.528523936",
      "0.101612847"
    ]
  ]
}

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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