P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 35

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 = 35

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	[5.446245, 5.446255]	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

1234567891011121314151617181920212223242526272829303132333435

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": [
    [
      "1.183349882",
      "0.485513991",
      "0.194205246"
    ],
    [
      "1.558094557",
      "0.410228350",
      "0.188027011"
    ],
    [
      "1.769029911",
      "0.709111399",
      "0.177793710"
    ],
    [
      "0.317019665",
      "1.459925218",
      "0.176596487"
    ],
    [
      "0.186922739",
      "1.131889584",
      "0.176295262"
    ],
    [
      "0.771396305",
      "1.792369470",
      "0.175312770"
    ],
    [
      "0.288711020",
      "0.580459649",
      "0.174199709"
    ],
    [
      "1.119595728",
      "1.817505532",
      "0.173792742"
    ],
    [
      "0.521938899",
      "0.324264458",
      "0.172255510"
    ],
    [
      "1.609001937",
      "1.561549010",
      "0.171615638"
    ],
    [
      "1.784258855",
      "1.266241957",
      "0.171780987"
    ],
    [
      "0.521171305",
      "1.188832795",
      "0.162769087"
    ],
    [
      "0.967624612",
      "1.518716438",
      "0.161423818"
    ],
    [
      "1.034539198",
      "0.162316859",
      "0.161605103"
    ],
    [
      "0.617888185",
      "0.643333651",
      "0.160928245"
    ],
    [
      "1.433857464",
      "0.733464203",
      "0.158262278"
    ],
    [
      "1.454497710",
      "1.270380753",
      "0.158006129"
    ],
    [
      "0.650601423",
      "1.482019858",
      "0.157716184"
    ],
    [
      "0.839028295",
      "0.414373504",
      "0.157388729"
    ],
    [
      "0.413774073",
      "0.887391003",
      "0.157232973"
    ],
    [
      "0.721244195",
      "0.941912810",
      "0.155033744"
    ],
    [
      "0.836708633",
      "1.230139488",
      "0.155460465"
    ],
    [
      "1.283705148",
      "1.532639628",
      "0.154963223"
    ],
    [
      "1.602646325",
      "0.995979426",
      "0.153833734"
    ],
    [
      "1.143798626",
      "1.258374753",
      "0.152924834"
    ],
    [
      "0.922265359",
      "0.711247555",
      "0.150933522"
    ],
    [
      "1.304164587",
      "1.007449439",
      "0.144868306"
    ],
    [
      "1.016034565",
      "0.990617299",
      "0.143752951"
    ],
    [
      "1.324897109",
      "0.193384645",
      "0.130410175"
    ],
    [
      "0.484500238",
      "1.708910663",
      "0.123475994"
    ],
    [
      "0.138257468",
      "0.836663895",
      "0.122914556"
    ],
    [
      "1.878143448",
      "0.988102843",
      "0.121775963"
    ],
    [
      "1.411375741",
      "1.777045212",
      "0.120779173"
    ],
    [
      "0.755403754",
      "0.152292602",
      "0.117710276"
    ],
    [
      "1.171344085",
      "0.789686550",
      "0.110204156"
    ]
  ]
}

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": 35
}

起步布局答案

{
  "circles": [
    [
      "1.183349882",
      "0.485513991",
      "0.194205246"
    ],
    [
      "1.558094557",
      "0.410228350",
      "0.188027011"
    ],
    [
      "1.769029911",
      "0.709111399",
      "0.177793710"
    ],
    [
      "0.317019665",
      "1.459925218",
      "0.176596487"
    ],
    [
      "0.186922739",
      "1.131889584",
      "0.176295262"
    ],
    [
      "0.771396305",
      "1.792369470",
      "0.175312770"
    ],
    [
      "0.288711020",
      "0.580459649",
      "0.174199709"
    ],
    [
      "1.119595728",
      "1.817505532",
      "0.173792742"
    ],
    [
      "0.521938899",
      "0.324264458",
      "0.172255510"
    ],
    [
      "1.609001937",
      "1.561549010",
      "0.171615638"
    ],
    [
      "1.784258855",
      "1.266241957",
      "0.171780987"
    ],
    [
      "0.521171305",
      "1.188832795",
      "0.162769087"
    ],
    [
      "0.967624612",
      "1.518716438",
      "0.161423818"
    ],
    [
      "1.034539198",
      "0.162316859",
      "0.161605103"
    ],
    [
      "0.617888185",
      "0.643333651",
      "0.160928245"
    ],
    [
      "1.433857464",
      "0.733464203",
      "0.158262278"
    ],
    [
      "1.454497710",
      "1.270380753",
      "0.158006129"
    ],
    [
      "0.650601423",
      "1.482019858",
      "0.157716184"
    ],
    [
      "0.839028295",
      "0.414373504",
      "0.157388729"
    ],
    [
      "0.413774073",
      "0.887391003",
      "0.157232973"
    ],
    [
      "0.721244195",
      "0.941912810",
      "0.155033744"
    ],
    [
      "0.836708633",
      "1.230139488",
      "0.155460465"
    ],
    [
      "1.283705148",
      "1.532639628",
      "0.154963223"
    ],
    [
      "1.602646325",
      "0.995979426",
      "0.153833734"
    ],
    [
      "1.143798626",
      "1.258374753",
      "0.152924834"
    ],
    [
      "0.922265359",
      "0.711247555",
      "0.150933522"
    ],
    [
      "1.304164587",
      "1.007449439",
      "0.144868306"
    ],
    [
      "1.016034565",
      "0.990617299",
      "0.143752951"
    ],
    [
      "1.324897109",
      "0.193384645",
      "0.130410175"
    ],
    [
      "0.484500238",
      "1.708910663",
      "0.123475994"
    ],
    [
      "0.138257468",
      "0.836663895",
      "0.122914556"
    ],
    [
      "1.878143448",
      "0.988102843",
      "0.121775963"
    ],
    [
      "1.411375741",
      "1.777045212",
      "0.120779173"
    ],
    [
      "0.755403754",
      "0.152292602",
      "0.117710276"
    ],
    [
      "1.171344085",
      "0.789686550",
      "0.110204156"
    ]
  ]
}

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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