P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 24

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 = 24

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	[4.466, 4.467]	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

123456789101112131415161718192021222324

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": [
    [
      "1.414179996",
      "0.898055357",
      "0.236934194"
    ],
    [
      "0.585345863",
      "0.900001599",
      "0.236934207"
    ],
    [
      "1.001080343",
      "1.460059785",
      "0.221680789"
    ],
    [
      "1.764497615",
      "1.182152187",
      "0.214101772"
    ],
    [
      "1.594873743",
      "0.485473683",
      "0.213481022"
    ],
    [
      "0.236366290",
      "1.185740528",
      "0.214101796"
    ],
    [
      "0.402716495",
      "0.488273034",
      "0.213481040"
    ],
    [
      "0.658392031",
      "1.716431662",
      "0.206293296"
    ],
    [
      "1.344968896",
      "1.714819423",
      "0.206293286"
    ],
    [
      "0.998109810",
      "0.195014326",
      "0.195012106"
    ],
    [
      "1.191420014",
      "0.530890890",
      "0.192520990"
    ],
    [
      "0.806379090",
      "0.531795021",
      "0.192520995"
    ],
    [
      "0.614199339",
      "1.324288257",
      "0.188332404"
    ],
    [
      "1.387319436",
      "1.322472818",
      "0.188332393"
    ],
    [
      "0.999649246",
      "0.850611832",
      "0.180302705"
    ],
    [
      "1.807889529",
      "0.802258074",
      "0.168262444"
    ],
    [
      "0.191190745",
      "0.806054344",
      "0.168262464"
    ],
    [
      "1.001974568",
      "1.840868603",
      "0.159129078"
    ],
    [
      "0.346108443",
      "1.536977392",
      "0.153880098"
    ],
    [
      "1.656406227",
      "1.533900532",
      "0.153880083"
    ],
    [
      "0.653701529",
      "0.224455713",
      "0.150652266"
    ],
    [
      "1.342652548",
      "0.222837973",
      "0.150652259"
    ],
    [
      "1.131010045",
      "1.132509130",
      "0.130698493"
    ],
    [
      "0.869613777",
      "1.133122937",
      "0.130698495"
    ]
  ]
}

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": 24
}

起步布局答案

{
  "circles": [
    [
      "1.414179996",
      "0.898055357",
      "0.236934194"
    ],
    [
      "0.585345863",
      "0.900001599",
      "0.236934207"
    ],
    [
      "1.001080343",
      "1.460059785",
      "0.221680789"
    ],
    [
      "1.764497615",
      "1.182152187",
      "0.214101772"
    ],
    [
      "1.594873743",
      "0.485473683",
      "0.213481022"
    ],
    [
      "0.236366290",
      "1.185740528",
      "0.214101796"
    ],
    [
      "0.402716495",
      "0.488273034",
      "0.213481040"
    ],
    [
      "0.658392031",
      "1.716431662",
      "0.206293296"
    ],
    [
      "1.344968896",
      "1.714819423",
      "0.206293286"
    ],
    [
      "0.998109810",
      "0.195014326",
      "0.195012106"
    ],
    [
      "1.191420014",
      "0.530890890",
      "0.192520990"
    ],
    [
      "0.806379090",
      "0.531795021",
      "0.192520995"
    ],
    [
      "0.614199339",
      "1.324288257",
      "0.188332404"
    ],
    [
      "1.387319436",
      "1.322472818",
      "0.188332393"
    ],
    [
      "0.999649246",
      "0.850611832",
      "0.180302705"
    ],
    [
      "1.807889529",
      "0.802258074",
      "0.168262444"
    ],
    [
      "0.191190745",
      "0.806054344",
      "0.168262464"
    ],
    [
      "1.001974568",
      "1.840868603",
      "0.159129078"
    ],
    [
      "0.346108443",
      "1.536977392",
      "0.153880098"
    ],
    [
      "1.656406227",
      "1.533900532",
      "0.153880083"
    ],
    [
      "0.653701529",
      "0.224455713",
      "0.150652266"
    ],
    [
      "1.342652548",
      "0.222837973",
      "0.150652259"
    ],
    [
      "1.131010045",
      "1.132509130",
      "0.130698493"
    ],
    [
      "0.869613777",
      "1.133122937",
      "0.130698495"
    ]
  ]
}

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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