P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 38

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 = 38

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	[5.692485, 5.692495]	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

1234567891011121314151617181920212223242526272829303132333435363738

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": [
    [
      "1.477826183",
      "1.326065238",
      "0.194222612"
    ],
    [
      "0.726464333",
      "1.790079087",
      "0.163909858"
    ],
    [
      "1.635978148",
      "0.457249120",
      "0.163909859"
    ],
    [
      "1.050641420",
      "1.835088938",
      "0.163376973"
    ],
    [
      "1.307176743",
      "1.011597262",
      "0.163564109"
    ],
    [
      "1.796076336",
      "0.742705321",
      "0.163376975"
    ],
    [
      "1.122787358",
      "1.281807230",
      "0.163564109"
    ],
    [
      "0.309647248",
      "1.474613703",
      "0.162238047"
    ],
    [
      "1.190264529",
      "0.184129608",
      "0.162238048"
    ],
    [
      "0.180181009",
      "1.178454070",
      "0.160983292"
    ],
    [
      "0.867288244",
      "0.171545643",
      "0.160983294"
    ],
    [
      "0.228559497",
      "0.666966420",
      "0.159743006"
    ],
    [
      "0.408641267",
      "0.403068962",
      "0.159743006"
    ],
    [
      "0.927239699",
      "1.537762073",
      "0.158541104"
    ],
    [
      "1.474224112",
      "0.736194020",
      "0.158541105"
    ],
    [
      "0.986196936",
      "0.990580890",
      "0.158102992"
    ],
    [
      "1.159187752",
      "0.727676571",
      "0.156610374"
    ],
    [
      "0.804454781",
      "1.247513356",
      "0.156610373"
    ],
    [
      "0.494158204",
      "1.216234296",
      "0.155258741"
    ],
    [
      "1.016935787",
      "0.450139683",
      "0.155258742"
    ],
    [
      "0.848793919",
      "0.710034441",
      "0.154284425"
    ],
    [
      "0.674865725",
      "0.964914253",
      "0.154284425"
    ],
    [
      "1.237354733",
      "1.578198486",
      "0.154199103"
    ],
    [
      "1.624936988",
      "1.010223340",
      "0.154199104"
    ],
    [
      "0.541507890",
      "0.687128208",
      "0.153854178"
    ],
    [
      "1.321967901",
      "0.467527936",
      "0.150268574"
    ],
    [
      "0.621556117",
      "1.493933256",
      "0.150268573"
    ],
    [
      "1.528226440",
      "1.666296323",
      "0.149721244"
    ],
    [
      "1.813015391",
      "1.248957698",
      "0.149721244"
    ],
    [
      "0.714030344",
      "0.439644003",
      "0.147828484"
    ],
    [
      "0.373953806",
      "0.938002794",
      "0.147828483"
    ],
    [
      "0.462320770",
      "1.704520780",
      "0.113744684"
    ],
    [
      "1.460008842",
      "0.242477495",
      "0.113744686"
    ],
    [
      "1.732130786",
      "1.499600915",
      "0.113649864"
    ],
    [
      "0.597231229",
      "0.208070880",
      "0.111532547"
    ],
    [
      "0.115736950",
      "0.913667654",
      "0.111532546"
    ],
    [
      "1.324458319",
      "1.828184202",
      "0.110526968"
    ],
    [
      "1.889473010",
      "1.000194024",
      "0.110526968"
    ]
  ]
}

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": 38
}

起步布局答案

{
  "circles": [
    [
      "1.477826183",
      "1.326065238",
      "0.194222612"
    ],
    [
      "0.726464333",
      "1.790079087",
      "0.163909858"
    ],
    [
      "1.635978148",
      "0.457249120",
      "0.163909859"
    ],
    [
      "1.050641420",
      "1.835088938",
      "0.163376973"
    ],
    [
      "1.307176743",
      "1.011597262",
      "0.163564109"
    ],
    [
      "1.796076336",
      "0.742705321",
      "0.163376975"
    ],
    [
      "1.122787358",
      "1.281807230",
      "0.163564109"
    ],
    [
      "0.309647248",
      "1.474613703",
      "0.162238047"
    ],
    [
      "1.190264529",
      "0.184129608",
      "0.162238048"
    ],
    [
      "0.180181009",
      "1.178454070",
      "0.160983292"
    ],
    [
      "0.867288244",
      "0.171545643",
      "0.160983294"
    ],
    [
      "0.228559497",
      "0.666966420",
      "0.159743006"
    ],
    [
      "0.408641267",
      "0.403068962",
      "0.159743006"
    ],
    [
      "0.927239699",
      "1.537762073",
      "0.158541104"
    ],
    [
      "1.474224112",
      "0.736194020",
      "0.158541105"
    ],
    [
      "0.986196936",
      "0.990580890",
      "0.158102992"
    ],
    [
      "1.159187752",
      "0.727676571",
      "0.156610374"
    ],
    [
      "0.804454781",
      "1.247513356",
      "0.156610373"
    ],
    [
      "0.494158204",
      "1.216234296",
      "0.155258741"
    ],
    [
      "1.016935787",
      "0.450139683",
      "0.155258742"
    ],
    [
      "0.848793919",
      "0.710034441",
      "0.154284425"
    ],
    [
      "0.674865725",
      "0.964914253",
      "0.154284425"
    ],
    [
      "1.237354733",
      "1.578198486",
      "0.154199103"
    ],
    [
      "1.624936988",
      "1.010223340",
      "0.154199104"
    ],
    [
      "0.541507890",
      "0.687128208",
      "0.153854178"
    ],
    [
      "1.321967901",
      "0.467527936",
      "0.150268574"
    ],
    [
      "0.621556117",
      "1.493933256",
      "0.150268573"
    ],
    [
      "1.528226440",
      "1.666296323",
      "0.149721244"
    ],
    [
      "1.813015391",
      "1.248957698",
      "0.149721244"
    ],
    [
      "0.714030344",
      "0.439644003",
      "0.147828484"
    ],
    [
      "0.373953806",
      "0.938002794",
      "0.147828483"
    ],
    [
      "0.462320770",
      "1.704520780",
      "0.113744684"
    ],
    [
      "1.460008842",
      "0.242477495",
      "0.113744686"
    ],
    [
      "1.732130786",
      "1.499600915",
      "0.113649864"
    ],
    [
      "0.597231229",
      "0.208070880",
      "0.111532547"
    ],
    [
      "0.115736950",
      "0.913667654",
      "0.111532546"
    ],
    [
      "1.324458319",
      "1.828184202",
      "0.110526968"
    ],
    [
      "1.889473010",
      "1.000194024",
      "0.110526968"
    ]
  ]
}

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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