P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 36

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 = 36

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	[5.530325, 5.530335]	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

123456789101112131415161718192021222324252627282930313233343536

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": [
    [
      "1.596489141",
      "0.437267415",
      "0.179958990"
    ],
    [
      "1.298072620",
      "0.236049770",
      "0.179959000"
    ],
    [
      "0.794045332",
      "1.800918557",
      "0.173024873"
    ],
    [
      "0.334704725",
      "1.491192459",
      "0.173024888"
    ],
    [
      "0.191115336",
      "1.177643845",
      "0.171838339"
    ],
    [
      "1.138556849",
      "1.816488694",
      "0.171838308"
    ],
    [
      "0.955507137",
      "0.170090560",
      "0.168898745"
    ],
    [
      "1.617697739",
      "1.556093063",
      "0.168862230"
    ],
    [
      "1.786062916",
      "0.730121450",
      "0.168898718"
    ],
    [
      "0.252893603",
      "0.635827034",
      "0.168862275"
    ],
    [
      "0.456952545",
      "0.368711870",
      "0.167278412"
    ],
    [
      "1.788831589",
      "1.266777131",
      "0.167278369"
    ],
    [
      "0.980336663",
      "1.523563227",
      "0.161086543"
    ],
    [
      "0.521992853",
      "1.214509218",
      "0.161086557"
    ],
    [
      "1.463859583",
      "1.268040707",
      "0.157696092"
    ],
    [
      "0.577610644",
      "0.670456912",
      "0.157696119"
    ],
    [
      "0.839041332",
      "1.238710525",
      "0.156884261"
    ],
    [
      "0.663951034",
      "1.498378906",
      "0.156299839"
    ],
    [
      "0.705309765",
      "0.956418217",
      "0.155482637"
    ],
    [
      "1.150880525",
      "1.256859593",
      "0.155482623"
    ],
    [
      "1.295588382",
      "1.530705351",
      "0.154246069"
    ],
    [
      "0.397186120",
      "0.924926826",
      "0.154246096"
    ],
    [
      "1.011784897",
      "0.982522504",
      "0.152102213"
    ],
    [
      "1.314654194",
      "0.997633881",
      "0.151143833"
    ],
    [
      "0.884232250",
      "0.707407083",
      "0.151143846"
    ],
    [
      "0.768761769",
      "0.428958473",
      "0.150297790"
    ],
    [
      "1.616084050",
      "1.000294692",
      "0.150297765"
    ],
    [
      "1.468494835",
      "0.740116266",
      "0.148826606"
    ],
    [
      "1.065295051",
      "0.468244917",
      "0.148826617"
    ],
    [
      "1.176734147",
      "0.737893898",
      "0.142942545"
    ],
    [
      "0.508752117",
      "1.728547319",
      "0.121304558"
    ],
    [
      "1.425966348",
      "1.771008427",
      "0.119147387"
    ],
    [
      "0.125530982",
      "0.894145338",
      "0.119147417"
    ],
    [
      "0.669997651",
      "0.180779843",
      "0.116810770"
    ],
    [
      "1.883188463",
      "0.998815559",
      "0.116810742"
    ],
    [
      "1.319037777",
      "0.526849984",
      "0.111595971"
    ]
  ]
}

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": 36
}

起步布局答案

{
  "circles": [
    [
      "1.596489141",
      "0.437267415",
      "0.179958990"
    ],
    [
      "1.298072620",
      "0.236049770",
      "0.179959000"
    ],
    [
      "0.794045332",
      "1.800918557",
      "0.173024873"
    ],
    [
      "0.334704725",
      "1.491192459",
      "0.173024888"
    ],
    [
      "0.191115336",
      "1.177643845",
      "0.171838339"
    ],
    [
      "1.138556849",
      "1.816488694",
      "0.171838308"
    ],
    [
      "0.955507137",
      "0.170090560",
      "0.168898745"
    ],
    [
      "1.617697739",
      "1.556093063",
      "0.168862230"
    ],
    [
      "1.786062916",
      "0.730121450",
      "0.168898718"
    ],
    [
      "0.252893603",
      "0.635827034",
      "0.168862275"
    ],
    [
      "0.456952545",
      "0.368711870",
      "0.167278412"
    ],
    [
      "1.788831589",
      "1.266777131",
      "0.167278369"
    ],
    [
      "0.980336663",
      "1.523563227",
      "0.161086543"
    ],
    [
      "0.521992853",
      "1.214509218",
      "0.161086557"
    ],
    [
      "1.463859583",
      "1.268040707",
      "0.157696092"
    ],
    [
      "0.577610644",
      "0.670456912",
      "0.157696119"
    ],
    [
      "0.839041332",
      "1.238710525",
      "0.156884261"
    ],
    [
      "0.663951034",
      "1.498378906",
      "0.156299839"
    ],
    [
      "0.705309765",
      "0.956418217",
      "0.155482637"
    ],
    [
      "1.150880525",
      "1.256859593",
      "0.155482623"
    ],
    [
      "1.295588382",
      "1.530705351",
      "0.154246069"
    ],
    [
      "0.397186120",
      "0.924926826",
      "0.154246096"
    ],
    [
      "1.011784897",
      "0.982522504",
      "0.152102213"
    ],
    [
      "1.314654194",
      "0.997633881",
      "0.151143833"
    ],
    [
      "0.884232250",
      "0.707407083",
      "0.151143846"
    ],
    [
      "0.768761769",
      "0.428958473",
      "0.150297790"
    ],
    [
      "1.616084050",
      "1.000294692",
      "0.150297765"
    ],
    [
      "1.468494835",
      "0.740116266",
      "0.148826606"
    ],
    [
      "1.065295051",
      "0.468244917",
      "0.148826617"
    ],
    [
      "1.176734147",
      "0.737893898",
      "0.142942545"
    ],
    [
      "0.508752117",
      "1.728547319",
      "0.121304558"
    ],
    [
      "1.425966348",
      "1.771008427",
      "0.119147387"
    ],
    [
      "0.125530982",
      "0.894145338",
      "0.119147417"
    ],
    [
      "0.669997651",
      "0.180779843",
      "0.116810770"
    ],
    [
      "1.883188463",
      "0.998815559",
      "0.116810742"
    ],
    [
      "1.319037777",
      "0.526849984",
      "0.111595971"
    ]
  ]
}

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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