P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 28

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 = 28

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	[4.849, 4.850]	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

12345678910111213141516171819202122232425262728

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": [
    [
      "0.999972038",
      "0.532483327",
      "0.234828277"
    ],
    [
      "1.756908782",
      "1.252536654",
      "0.202074147"
    ],
    [
      "0.243121525",
      "1.252627391",
      "0.202074173"
    ],
    [
      "1.539252607",
      "1.590759661",
      "0.200131039"
    ],
    [
      "0.460818280",
      "1.590824335",
      "0.200131059"
    ],
    [
      "0.809891975",
      "0.912530711",
      "0.190102827"
    ],
    [
      "1.190097624",
      "0.912507932",
      "0.190102821"
    ],
    [
      "0.679201257",
      "0.255279965",
      "0.189124075"
    ],
    [
      "1.320709620",
      "0.255241569",
      "0.189124064"
    ],
    [
      "0.631470324",
      "1.244038179",
      "0.186369596"
    ],
    [
      "1.368558996",
      "1.243993998",
      "0.186369585"
    ],
    [
      "1.000013993",
      "1.232798291",
      "0.182345429"
    ],
    [
      "0.594371984",
      "0.613218221",
      "0.178728885"
    ],
    [
      "1.405581746",
      "0.613169643",
      "0.178728872"
    ],
    [
      "0.445284145",
      "0.934154359",
      "0.175145651"
    ],
    [
      "1.554708019",
      "0.934087890",
      "0.175145634"
    ],
    [
      "1.171179192",
      "1.542086391",
      "0.171146654"
    ],
    [
      "0.828885879",
      "1.542106916",
      "0.171146659"
    ],
    [
      "1.752619854",
      "0.650891114",
      "0.170353292"
    ],
    [
      "0.247338361",
      "0.650981262",
      "0.170353314"
    ],
    [
      "1.000050005",
      "1.833339580",
      "0.166660418"
    ],
    [
      "0.999949074",
      "0.148827526",
      "0.148827524"
    ],
    [
      "1.863670217",
      "0.934064416",
      "0.133816563"
    ],
    [
      "0.136321909",
      "0.934167906",
      "0.133816584"
    ],
    [
      "0.702901945",
      "1.816684545",
      "0.130954028"
    ],
    [
      "1.297196057",
      "1.816648894",
      "0.130954021"
    ],
    [
      "1.613426937",
      "0.383995154",
      "0.130658537"
    ],
    [
      "0.386499334",
      "0.384068603",
      "0.130658551"
    ]
  ]
}

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": 28
}

起步布局答案

{
  "circles": [
    [
      "0.999972038",
      "0.532483327",
      "0.234828277"
    ],
    [
      "1.756908782",
      "1.252536654",
      "0.202074147"
    ],
    [
      "0.243121525",
      "1.252627391",
      "0.202074173"
    ],
    [
      "1.539252607",
      "1.590759661",
      "0.200131039"
    ],
    [
      "0.460818280",
      "1.590824335",
      "0.200131059"
    ],
    [
      "0.809891975",
      "0.912530711",
      "0.190102827"
    ],
    [
      "1.190097624",
      "0.912507932",
      "0.190102821"
    ],
    [
      "0.679201257",
      "0.255279965",
      "0.189124075"
    ],
    [
      "1.320709620",
      "0.255241569",
      "0.189124064"
    ],
    [
      "0.631470324",
      "1.244038179",
      "0.186369596"
    ],
    [
      "1.368558996",
      "1.243993998",
      "0.186369585"
    ],
    [
      "1.000013993",
      "1.232798291",
      "0.182345429"
    ],
    [
      "0.594371984",
      "0.613218221",
      "0.178728885"
    ],
    [
      "1.405581746",
      "0.613169643",
      "0.178728872"
    ],
    [
      "0.445284145",
      "0.934154359",
      "0.175145651"
    ],
    [
      "1.554708019",
      "0.934087890",
      "0.175145634"
    ],
    [
      "1.171179192",
      "1.542086391",
      "0.171146654"
    ],
    [
      "0.828885879",
      "1.542106916",
      "0.171146659"
    ],
    [
      "1.752619854",
      "0.650891114",
      "0.170353292"
    ],
    [
      "0.247338361",
      "0.650981262",
      "0.170353314"
    ],
    [
      "1.000050005",
      "1.833339580",
      "0.166660418"
    ],
    [
      "0.999949074",
      "0.148827526",
      "0.148827524"
    ],
    [
      "1.863670217",
      "0.934064416",
      "0.133816563"
    ],
    [
      "0.136321909",
      "0.934167906",
      "0.133816584"
    ],
    [
      "0.702901945",
      "1.816684545",
      "0.130954028"
    ],
    [
      "1.297196057",
      "1.816648894",
      "0.130954021"
    ],
    [
      "1.613426937",
      "0.383995154",
      "0.130658537"
    ],
    [
      "0.386499334",
      "0.384068603",
      "0.130658551"
    ]
  ]
}

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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