P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 30

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 = 30

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	[5.019, 5.020]	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

123456789101112131415161718192021222324252627282930

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": [
    [
      "0.834345096",
      "1.089533805",
      "0.217633682"
    ],
    [
      "0.256670095",
      "0.706259024",
      "0.200735895"
    ],
    [
      "1.740762714",
      "0.687873566",
      "0.196164003"
    ],
    [
      "0.484147836",
      "0.382226392",
      "0.195172262"
    ],
    [
      "1.748374078",
      "1.307800705",
      "0.190799756"
    ],
    [
      "0.645015989",
      "0.730389643",
      "0.188358977"
    ],
    [
      "0.622753985",
      "1.719374851",
      "0.187710192"
    ],
    [
      "1.517368183",
      "0.374766989",
      "0.188466787"
    ],
    [
      "1.534429541",
      "1.616064828",
      "0.184432218"
    ],
    [
      "0.983026664",
      "1.814829742",
      "0.184993495"
    ],
    [
      "0.442999656",
      "1.035351559",
      "0.177444752"
    ],
    [
      "1.012458692",
      "0.735015330",
      "0.179112839"
    ],
    [
      "1.367881679",
      "0.708582859",
      "0.177291671"
    ],
    [
      "0.558784491",
      "1.365231163",
      "0.172164573"
    ],
    [
      "1.553169106",
      "1.003339809",
      "0.170865031"
    ],
    [
      "1.214005191",
      "1.018398816",
      "0.168633031"
    ],
    [
      "0.224710455",
      "1.302711084",
      "0.167709257"
    ],
    [
      "0.883204695",
      "1.474117185",
      "0.170040969"
    ],
    [
      "0.841277413",
      "0.435460699",
      "0.165903111"
    ],
    [
      "1.394122979",
      "1.297568586",
      "0.163599083"
    ],
    [
      "1.170316401",
      "0.431316481",
      "0.163161972"
    ],
    [
      "1.200024459",
      "1.551632347",
      "0.156123655"
    ],
    [
      "1.004995815",
      "0.157072518",
      "0.157057713"
    ],
    [
      "1.862003462",
      "0.999263576",
      "0.137996223"
    ],
    [
      "0.133146912",
      "1.016287422",
      "0.132993912"
    ],
    [
      "1.296535695",
      "1.819446465",
      "0.128549526"
    ],
    [
      "0.342938873",
      "1.573559591",
      "0.127818866"
    ],
    [
      "0.723387267",
      "0.169697993",
      "0.124833714"
    ],
    [
      "1.108875854",
      "1.289108548",
      "0.121773470"
    ],
    [
      "1.284055705",
      "0.169514731",
      "0.122279414"
    ]
  ]
}

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": 30
}

起步布局答案

{
  "circles": [
    [
      "0.834345096",
      "1.089533805",
      "0.217633682"
    ],
    [
      "0.256670095",
      "0.706259024",
      "0.200735895"
    ],
    [
      "1.740762714",
      "0.687873566",
      "0.196164003"
    ],
    [
      "0.484147836",
      "0.382226392",
      "0.195172262"
    ],
    [
      "1.748374078",
      "1.307800705",
      "0.190799756"
    ],
    [
      "0.645015989",
      "0.730389643",
      "0.188358977"
    ],
    [
      "0.622753985",
      "1.719374851",
      "0.187710192"
    ],
    [
      "1.517368183",
      "0.374766989",
      "0.188466787"
    ],
    [
      "1.534429541",
      "1.616064828",
      "0.184432218"
    ],
    [
      "0.983026664",
      "1.814829742",
      "0.184993495"
    ],
    [
      "0.442999656",
      "1.035351559",
      "0.177444752"
    ],
    [
      "1.012458692",
      "0.735015330",
      "0.179112839"
    ],
    [
      "1.367881679",
      "0.708582859",
      "0.177291671"
    ],
    [
      "0.558784491",
      "1.365231163",
      "0.172164573"
    ],
    [
      "1.553169106",
      "1.003339809",
      "0.170865031"
    ],
    [
      "1.214005191",
      "1.018398816",
      "0.168633031"
    ],
    [
      "0.224710455",
      "1.302711084",
      "0.167709257"
    ],
    [
      "0.883204695",
      "1.474117185",
      "0.170040969"
    ],
    [
      "0.841277413",
      "0.435460699",
      "0.165903111"
    ],
    [
      "1.394122979",
      "1.297568586",
      "0.163599083"
    ],
    [
      "1.170316401",
      "0.431316481",
      "0.163161972"
    ],
    [
      "1.200024459",
      "1.551632347",
      "0.156123655"
    ],
    [
      "1.004995815",
      "0.157072518",
      "0.157057713"
    ],
    [
      "1.862003462",
      "0.999263576",
      "0.137996223"
    ],
    [
      "0.133146912",
      "1.016287422",
      "0.132993912"
    ],
    [
      "1.296535695",
      "1.819446465",
      "0.128549526"
    ],
    [
      "0.342938873",
      "1.573559591",
      "0.127818866"
    ],
    [
      "0.723387267",
      "0.169697993",
      "0.124833714"
    ],
    [
      "1.108875854",
      "1.289108548",
      "0.121773470"
    ],
    [
      "1.284055705",
      "0.169514731",
      "0.122279414"
    ]
  ]
}

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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