P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 47

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 = 47

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	[6.327785, 6.327795]	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

1234567891011121314151617181920212223242526272829303132333435363738394041424344454647

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": [
    [
      "1.532043714",
      "1.281811778",
      "0.181126454"
    ],
    [
      "0.481919834",
      "1.177321697",
      "0.172006027"
    ],
    [
      "0.867102015",
      "0.406576433",
      "0.171529436"
    ],
    [
      "1.563276761",
      "1.622208535",
      "0.160700189"
    ],
    [
      "1.088308516",
      "0.162508855",
      "0.157865917"
    ],
    [
      "0.261483688",
      "1.415579847",
      "0.152584545"
    ],
    [
      "0.161703112",
      "1.127781534",
      "0.152020169"
    ],
    [
      "1.279055234",
      "1.495931719",
      "0.150310598"
    ],
    [
      "1.305588661",
      "1.794148913",
      "0.149084654"
    ],
    [
      "1.182998756",
      "0.453113800",
      "0.147776808"
    ],
    [
      "1.383349101",
      "0.237496272",
      "0.146554941"
    ],
    [
      "0.369063943",
      "0.425249389",
      "0.146525586"
    ],
    [
      "1.214093288",
      "1.207437846",
      "0.145406782"
    ],
    [
      "1.399443745",
      "0.983786362",
      "0.145066587"
    ],
    [
      "0.596466536",
      "0.244816911",
      "0.143763610"
    ],
    [
      "1.010114063",
      "0.684870007",
      "0.141359935"
    ],
    [
      "0.209023714",
      "0.663349894",
      "0.140362414"
    ],
    [
      "1.289334596",
      "0.720683141",
      "0.140147944"
    ],
    [
      "0.585921588",
      "1.753524629",
      "0.140197523"
    ],
    [
      "0.848669357",
      "1.847685160",
      "0.138912840"
    ],
    [
      "1.120409889",
      "0.940738266",
      "0.137268375"
    ],
    [
      "1.564815517",
      "0.754750700",
      "0.137431478"
    ],
    [
      "1.007816157",
      "1.400602112",
      "0.137193090"
    ],
    [
      "0.543577925",
      "1.479845884",
      "0.136737561"
    ],
    [
      "1.726354781",
      "0.533331440",
      "0.136651395"
    ],
    [
      "0.798776459",
      "1.577002469",
      "0.136329634"
    ],
    [
      "1.836499630",
      "0.782364452",
      "0.135652348"
    ],
    [
      "0.341670959",
      "0.904438370",
      "0.134808340"
    ],
    [
      "1.678707254",
      "1.002002441",
      "0.134790399"
    ],
    [
      "1.052975897",
      "1.667499965",
      "0.133498368"
    ],
    [
      "1.839421600",
      "1.216378287",
      "0.133138889"
    ],
    [
      "1.458347133",
      "0.506488855",
      "0.132697120"
    ],
    [
      "0.753686818",
      "1.312824501",
      "0.131668639"
    ],
    [
      "0.738060304",
      "0.679915786",
      "0.130738929"
    ],
    [
      "0.605298931",
      "0.902956647",
      "0.128823795"
    ],
    [
      "0.478143837",
      "0.678460832",
      "0.129181609"
    ],
    [
      "0.949035229",
      "1.142537711",
      "0.127481109"
    ],
    [
      "0.859803818",
      "0.905607666",
      "0.125694898"
    ],
    [
      "0.822284500",
      "0.128281761",
      "0.110350918"
    ],
    [
      "1.769741411",
      "1.448439430",
      "0.109157835"
    ],
    [
      "0.607786643",
      "0.488839538",
      "0.100521442"
    ],
    [
      "0.371698439",
      "1.643531653",
      "0.100613631"
    ],
    [
      "0.107911539",
      "0.881492650",
      "0.100074550"
    ],
    [
      "1.080744550",
      "1.897853625",
      "0.098522981"
    ],
    [
      "1.607746548",
      "0.332071494",
      "0.096958275"
    ],
    [
      "0.733385614",
      "1.086814181",
      "0.095251618"
    ],
    [
      "1.906745356",
      "1.000226648",
      "0.093254615"
    ]
  ]
}

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": 47
}

起步布局答案

{
  "circles": [
    [
      "1.532043714",
      "1.281811778",
      "0.181126454"
    ],
    [
      "0.481919834",
      "1.177321697",
      "0.172006027"
    ],
    [
      "0.867102015",
      "0.406576433",
      "0.171529436"
    ],
    [
      "1.563276761",
      "1.622208535",
      "0.160700189"
    ],
    [
      "1.088308516",
      "0.162508855",
      "0.157865917"
    ],
    [
      "0.261483688",
      "1.415579847",
      "0.152584545"
    ],
    [
      "0.161703112",
      "1.127781534",
      "0.152020169"
    ],
    [
      "1.279055234",
      "1.495931719",
      "0.150310598"
    ],
    [
      "1.305588661",
      "1.794148913",
      "0.149084654"
    ],
    [
      "1.182998756",
      "0.453113800",
      "0.147776808"
    ],
    [
      "1.383349101",
      "0.237496272",
      "0.146554941"
    ],
    [
      "0.369063943",
      "0.425249389",
      "0.146525586"
    ],
    [
      "1.214093288",
      "1.207437846",
      "0.145406782"
    ],
    [
      "1.399443745",
      "0.983786362",
      "0.145066587"
    ],
    [
      "0.596466536",
      "0.244816911",
      "0.143763610"
    ],
    [
      "1.010114063",
      "0.684870007",
      "0.141359935"
    ],
    [
      "0.209023714",
      "0.663349894",
      "0.140362414"
    ],
    [
      "1.289334596",
      "0.720683141",
      "0.140147944"
    ],
    [
      "0.585921588",
      "1.753524629",
      "0.140197523"
    ],
    [
      "0.848669357",
      "1.847685160",
      "0.138912840"
    ],
    [
      "1.120409889",
      "0.940738266",
      "0.137268375"
    ],
    [
      "1.564815517",
      "0.754750700",
      "0.137431478"
    ],
    [
      "1.007816157",
      "1.400602112",
      "0.137193090"
    ],
    [
      "0.543577925",
      "1.479845884",
      "0.136737561"
    ],
    [
      "1.726354781",
      "0.533331440",
      "0.136651395"
    ],
    [
      "0.798776459",
      "1.577002469",
      "0.136329634"
    ],
    [
      "1.836499630",
      "0.782364452",
      "0.135652348"
    ],
    [
      "0.341670959",
      "0.904438370",
      "0.134808340"
    ],
    [
      "1.678707254",
      "1.002002441",
      "0.134790399"
    ],
    [
      "1.052975897",
      "1.667499965",
      "0.133498368"
    ],
    [
      "1.839421600",
      "1.216378287",
      "0.133138889"
    ],
    [
      "1.458347133",
      "0.506488855",
      "0.132697120"
    ],
    [
      "0.753686818",
      "1.312824501",
      "0.131668639"
    ],
    [
      "0.738060304",
      "0.679915786",
      "0.130738929"
    ],
    [
      "0.605298931",
      "0.902956647",
      "0.128823795"
    ],
    [
      "0.478143837",
      "0.678460832",
      "0.129181609"
    ],
    [
      "0.949035229",
      "1.142537711",
      "0.127481109"
    ],
    [
      "0.859803818",
      "0.905607666",
      "0.125694898"
    ],
    [
      "0.822284500",
      "0.128281761",
      "0.110350918"
    ],
    [
      "1.769741411",
      "1.448439430",
      "0.109157835"
    ],
    [
      "0.607786643",
      "0.488839538",
      "0.100521442"
    ],
    [
      "0.371698439",
      "1.643531653",
      "0.100613631"
    ],
    [
      "0.107911539",
      "0.881492650",
      "0.100074550"
    ],
    [
      "1.080744550",
      "1.897853625",
      "0.098522981"
    ],
    [
      "1.607746548",
      "0.332071494",
      "0.096958275"
    ],
    [
      "0.733385614",
      "1.086814181",
      "0.095251618"
    ],
    [
      "1.906745356",
      "1.000226648",
      "0.093254615"
    ]
  ]
}

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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