P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 31

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 = 31

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	[5.104, 5.105]	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

12345678910111213141516171819202122232425262728293031

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": [
    [
      "1.161213569",
      "0.882860225",
      "0.219904860"
    ],
    [
      "1.047070501",
      "1.531060705",
      "0.206124950"
    ],
    [
      "0.307159060",
      "1.413399510",
      "0.193199080"
    ],
    [
      "1.128677180",
      "0.202490405",
      "0.192176151"
    ],
    [
      "1.814836209",
      "0.999551062",
      "0.185163667"
    ],
    [
      "0.953808810",
      "0.535906712",
      "0.184314725"
    ],
    [
      "0.750699625",
      "0.225652100",
      "0.186510388"
    ],
    [
      "0.186475592",
      "1.055403719",
      "0.184591185"
    ],
    [
      "1.734493149",
      "1.358513370",
      "0.182679975"
    ],
    [
      "0.763854619",
      "0.845815919",
      "0.179177105"
    ],
    [
      "0.542206705",
      "1.125999575",
      "0.178077256"
    ],
    [
      "0.894547977",
      "1.178367914",
      "0.178134499"
    ],
    [
      "0.673438896",
      "1.454149334",
      "0.175340559"
    ],
    [
      "0.768552241",
      "1.791600122",
      "0.175258332"
    ],
    [
      "1.311114005",
      "0.519265537",
      "0.173377783"
    ],
    [
      "1.476924314",
      "1.114853685",
      "0.171878548"
    ],
    [
      "1.536947292",
      "0.778131619",
      "0.170151431"
    ],
    [
      "1.342033578",
      "1.759554666",
      "0.166987239"
    ],
    [
      "1.642214981",
      "0.462522516",
      "0.162550223"
    ],
    [
      "0.608260415",
      "0.542964453",
      "0.161305737"
    ],
    [
      "0.426664026",
      "0.808163139",
      "0.160109245"
    ],
    [
      "1.401155646",
      "1.437962040",
      "0.159994770"
    ],
    [
      "0.296236032",
      "0.525485101",
      "0.151207851"
    ],
    [
      "0.477066202",
      "1.691703365",
      "0.132870654"
    ],
    [
      "1.064759248",
      "1.867152867",
      "0.130432374"
    ],
    [
      "1.198196000",
      "1.230814899",
      "0.130009636"
    ],
    [
      "1.444989620",
      "0.249662761",
      "0.127634403"
    ],
    [
      "1.818303748",
      "0.689602543",
      "0.124804247"
    ],
    [
      "1.604281896",
      "1.635302179",
      "0.123207283"
    ],
    [
      "0.456044648",
      "0.307295509",
      "0.119246380"
    ],
    [
      "0.153399171",
      "0.754187962",
      "0.118435185"
    ]
  ]
}

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": 31
}

起步布局答案

{
  "circles": [
    [
      "1.161213569",
      "0.882860225",
      "0.219904860"
    ],
    [
      "1.047070501",
      "1.531060705",
      "0.206124950"
    ],
    [
      "0.307159060",
      "1.413399510",
      "0.193199080"
    ],
    [
      "1.128677180",
      "0.202490405",
      "0.192176151"
    ],
    [
      "1.814836209",
      "0.999551062",
      "0.185163667"
    ],
    [
      "0.953808810",
      "0.535906712",
      "0.184314725"
    ],
    [
      "0.750699625",
      "0.225652100",
      "0.186510388"
    ],
    [
      "0.186475592",
      "1.055403719",
      "0.184591185"
    ],
    [
      "1.734493149",
      "1.358513370",
      "0.182679975"
    ],
    [
      "0.763854619",
      "0.845815919",
      "0.179177105"
    ],
    [
      "0.542206705",
      "1.125999575",
      "0.178077256"
    ],
    [
      "0.894547977",
      "1.178367914",
      "0.178134499"
    ],
    [
      "0.673438896",
      "1.454149334",
      "0.175340559"
    ],
    [
      "0.768552241",
      "1.791600122",
      "0.175258332"
    ],
    [
      "1.311114005",
      "0.519265537",
      "0.173377783"
    ],
    [
      "1.476924314",
      "1.114853685",
      "0.171878548"
    ],
    [
      "1.536947292",
      "0.778131619",
      "0.170151431"
    ],
    [
      "1.342033578",
      "1.759554666",
      "0.166987239"
    ],
    [
      "1.642214981",
      "0.462522516",
      "0.162550223"
    ],
    [
      "0.608260415",
      "0.542964453",
      "0.161305737"
    ],
    [
      "0.426664026",
      "0.808163139",
      "0.160109245"
    ],
    [
      "1.401155646",
      "1.437962040",
      "0.159994770"
    ],
    [
      "0.296236032",
      "0.525485101",
      "0.151207851"
    ],
    [
      "0.477066202",
      "1.691703365",
      "0.132870654"
    ],
    [
      "1.064759248",
      "1.867152867",
      "0.130432374"
    ],
    [
      "1.198196000",
      "1.230814899",
      "0.130009636"
    ],
    [
      "1.444989620",
      "0.249662761",
      "0.127634403"
    ],
    [
      "1.818303748",
      "0.689602543",
      "0.124804247"
    ],
    [
      "1.604281896",
      "1.635302179",
      "0.123207283"
    ],
    [
      "0.456044648",
      "0.307295509",
      "0.119246380"
    ],
    [
      "0.153399171",
      "0.754187962",
      "0.118435185"
    ]
  ]
}

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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