P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 26

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 = 26

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	[4.664, 4.665]	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

1234567891011121314151617181920212223242526

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": [
    [
      "1.354537306",
      "1.294714840",
      "0.229509709"
    ],
    [
      "0.646640125",
      "1.296125499",
      "0.229509716"
    ],
    [
      "0.781320310",
      "0.248345779",
      "0.217181454"
    ],
    [
      "1.215682352",
      "0.247480229",
      "0.217181449"
    ],
    [
      "0.673389634",
      "1.727423620",
      "0.202617121"
    ],
    [
      "1.329506950",
      "1.726116131",
      "0.202617114"
    ],
    [
      "1.772931752",
      "1.227771559",
      "0.194206368"
    ],
    [
      "0.999193125",
      "0.595076580",
      "0.192319231"
    ],
    [
      "0.227982179",
      "1.230850245",
      "0.194206383"
    ],
    [
      "1.001027849",
      "1.515783639",
      "0.187431953"
    ],
    [
      "0.455741968",
      "0.932899024",
      "0.180826217"
    ],
    [
      "0.818130037",
      "0.922364994",
      "0.181714922"
    ],
    [
      "1.181559157",
      "0.921640779",
      "0.181714919"
    ],
    [
      "1.371506946",
      "0.613296612",
      "0.180440141"
    ],
    [
      "0.626954869",
      "0.614780289",
      "0.180440148"
    ],
    [
      "1.543986317",
      "0.930730449",
      "0.180826207"
    ],
    [
      "1.730232200",
      "0.623520345",
      "0.178430771"
    ],
    [
      "0.268273190",
      "0.626433608",
      "0.178430785"
    ],
    [
      "1.641014391",
      "1.550881862",
      "0.154795719"
    ],
    [
      "0.361186254",
      "1.553432251",
      "0.154795729"
    ],
    [
      "1.001697066",
      "1.851606618",
      "0.148391691"
    ],
    [
      "1.560292663",
      "0.351081302",
      "0.142664974"
    ],
    [
      "0.437125611",
      "0.353319434",
      "0.142664982"
    ],
    [
      "1.000374578",
      "1.187959410",
      "0.140392926"
    ],
    [
      "1.859683365",
      "0.909602996",
      "0.135577009"
    ],
    [
      "0.139963203",
      "0.913029927",
      "0.135577021"
    ]
  ]
}

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": 26
}

起步布局答案

{
  "circles": [
    [
      "1.354537306",
      "1.294714840",
      "0.229509709"
    ],
    [
      "0.646640125",
      "1.296125499",
      "0.229509716"
    ],
    [
      "0.781320310",
      "0.248345779",
      "0.217181454"
    ],
    [
      "1.215682352",
      "0.247480229",
      "0.217181449"
    ],
    [
      "0.673389634",
      "1.727423620",
      "0.202617121"
    ],
    [
      "1.329506950",
      "1.726116131",
      "0.202617114"
    ],
    [
      "1.772931752",
      "1.227771559",
      "0.194206368"
    ],
    [
      "0.999193125",
      "0.595076580",
      "0.192319231"
    ],
    [
      "0.227982179",
      "1.230850245",
      "0.194206383"
    ],
    [
      "1.001027849",
      "1.515783639",
      "0.187431953"
    ],
    [
      "0.455741968",
      "0.932899024",
      "0.180826217"
    ],
    [
      "0.818130037",
      "0.922364994",
      "0.181714922"
    ],
    [
      "1.181559157",
      "0.921640779",
      "0.181714919"
    ],
    [
      "1.371506946",
      "0.613296612",
      "0.180440141"
    ],
    [
      "0.626954869",
      "0.614780289",
      "0.180440148"
    ],
    [
      "1.543986317",
      "0.930730449",
      "0.180826207"
    ],
    [
      "1.730232200",
      "0.623520345",
      "0.178430771"
    ],
    [
      "0.268273190",
      "0.626433608",
      "0.178430785"
    ],
    [
      "1.641014391",
      "1.550881862",
      "0.154795719"
    ],
    [
      "0.361186254",
      "1.553432251",
      "0.154795729"
    ],
    [
      "1.001697066",
      "1.851606618",
      "0.148391691"
    ],
    [
      "1.560292663",
      "0.351081302",
      "0.142664974"
    ],
    [
      "0.437125611",
      "0.353319434",
      "0.142664982"
    ],
    [
      "1.000374578",
      "1.187959410",
      "0.140392926"
    ],
    [
      "1.859683365",
      "0.909602996",
      "0.135577009"
    ],
    [
      "0.139963203",
      "0.913029927",
      "0.135577021"
    ]
  ]
}

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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