P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 25

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 = 25

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	[4.564, 4.565]	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

12345678910111213141516171819202122232425

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": [
    [
      "1.296740062",
      "1.336162884",
      "0.230873910"
    ],
    [
      "1.231078797",
      "0.262038136",
      "0.226705021"
    ],
    [
      "0.780887983",
      "0.255096561",
      "0.223539305"
    ],
    [
      "0.533370381",
      "1.624998322",
      "0.220021727"
    ],
    [
      "0.266320032",
      "1.283602475",
      "0.213414556"
    ],
    [
      "1.204080958",
      "1.765198981",
      "0.208053967"
    ],
    [
      "1.730000955",
      "1.320694331",
      "0.202663027"
    ],
    [
      "0.997593621",
      "0.611666954",
      "0.193718143"
    ],
    [
      "1.748162569",
      "0.685619946",
      "0.188469317"
    ],
    [
      "1.527483917",
      "0.988570578",
      "0.186334835"
    ],
    [
      "1.375514352",
      "0.648652231",
      "0.186008059"
    ],
    [
      "0.916594179",
      "1.499996994",
      "0.183073559"
    ],
    [
      "0.261586473",
      "0.648384483",
      "0.182144261"
    ],
    [
      "0.623661497",
      "0.627335211",
      "0.180542096"
    ],
    [
      "1.163080396",
      "0.947204289",
      "0.180409074"
    ],
    [
      "0.658104782",
      "1.244598015",
      "0.180306966"
    ],
    [
      "0.462126986",
      "0.946077286",
      "0.176795151"
    ],
    [
      "0.810825974",
      "0.926618215",
      "0.172446369"
    ],
    [
      "1.548886308",
      "1.635346970",
      "0.160391786"
    ],
    [
      "0.846921977",
      "1.830916335",
      "0.155100694"
    ],
    [
      "1.587415472",
      "0.386801439",
      "0.150841939"
    ],
    [
      "0.428249308",
      "0.365678504",
      "0.146031256"
    ],
    [
      "1.856827241",
      "0.998953686",
      "0.143172120"
    ],
    [
      "0.143403179",
      "0.950167477",
      "0.141954899"
    ],
    [
      "0.965261390",
      "1.188836175",
      "0.131870170"
    ]
  ]
}

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": 25
}

起步布局答案

{
  "circles": [
    [
      "1.296740062",
      "1.336162884",
      "0.230873910"
    ],
    [
      "1.231078797",
      "0.262038136",
      "0.226705021"
    ],
    [
      "0.780887983",
      "0.255096561",
      "0.223539305"
    ],
    [
      "0.533370381",
      "1.624998322",
      "0.220021727"
    ],
    [
      "0.266320032",
      "1.283602475",
      "0.213414556"
    ],
    [
      "1.204080958",
      "1.765198981",
      "0.208053967"
    ],
    [
      "1.730000955",
      "1.320694331",
      "0.202663027"
    ],
    [
      "0.997593621",
      "0.611666954",
      "0.193718143"
    ],
    [
      "1.748162569",
      "0.685619946",
      "0.188469317"
    ],
    [
      "1.527483917",
      "0.988570578",
      "0.186334835"
    ],
    [
      "1.375514352",
      "0.648652231",
      "0.186008059"
    ],
    [
      "0.916594179",
      "1.499996994",
      "0.183073559"
    ],
    [
      "0.261586473",
      "0.648384483",
      "0.182144261"
    ],
    [
      "0.623661497",
      "0.627335211",
      "0.180542096"
    ],
    [
      "1.163080396",
      "0.947204289",
      "0.180409074"
    ],
    [
      "0.658104782",
      "1.244598015",
      "0.180306966"
    ],
    [
      "0.462126986",
      "0.946077286",
      "0.176795151"
    ],
    [
      "0.810825974",
      "0.926618215",
      "0.172446369"
    ],
    [
      "1.548886308",
      "1.635346970",
      "0.160391786"
    ],
    [
      "0.846921977",
      "1.830916335",
      "0.155100694"
    ],
    [
      "1.587415472",
      "0.386801439",
      "0.150841939"
    ],
    [
      "0.428249308",
      "0.365678504",
      "0.146031256"
    ],
    [
      "1.856827241",
      "0.998953686",
      "0.143172120"
    ],
    [
      "0.143403179",
      "0.950167477",
      "0.141954899"
    ],
    [
      "0.965261390",
      "1.188836175",
      "0.131870170"
    ]
  ]
}

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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