P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 45

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 = 45

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	[6.187995, 6.188005]	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

123456789101112131415161718192021222324252627282930313233343536373839404142434445

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": [
    [
      "1.196550042",
      "0.750409060",
      "0.180371576"
    ],
    [
      "1.004938704",
      "1.826348637",
      "0.173636604"
    ],
    [
      "0.383029885",
      "0.991967408",
      "0.166063357"
    ],
    [
      "0.678947411",
      "0.851130974",
      "0.161659211"
    ],
    [
      "0.334415425",
      "0.481785074",
      "0.156466043"
    ],
    [
      "1.329963491",
      "1.777942660",
      "0.154972966"
    ],
    [
      "0.681771699",
      "1.785979260",
      "0.152042071"
    ],
    [
      "1.306820385",
      "0.208518294",
      "0.151128961"
    ],
    [
      "1.105374461",
      "0.432639634",
      "0.150219400"
    ],
    [
      "0.185176394",
      "1.239389227",
      "0.150738726"
    ],
    [
      "1.691024499",
      "0.505440809",
      "0.150233178"
    ],
    [
      "1.526839428",
      "0.756778729",
      "0.149979224"
    ],
    [
      "1.013332651",
      "0.148530969",
      "0.148426591"
    ],
    [
      "0.186108702",
      "0.747509545",
      "0.147843632"
    ],
    [
      "0.555973165",
      "0.272896057",
      "0.148037574"
    ],
    [
      "1.146056988",
      "1.538480900",
      "0.146960032"
    ],
    [
      "1.385269566",
      "1.016955697",
      "0.146220169"
    ],
    [
      "1.822596459",
      "0.770984198",
      "0.146118760"
    ],
    [
      "1.395352229",
      "0.491888955",
      "0.145749496"
    ],
    [
      "0.855322381",
      "1.546292514",
      "0.143879498"
    ],
    [
      "0.894325466",
      "0.635924312",
      "0.142810185"
    ],
    [
      "0.307244356",
      "1.505759065",
      "0.142268915"
    ],
    [
      "1.705446932",
      "1.487590927",
      "0.142445170"
    ],
    [
      "1.827041694",
      "1.231307231",
      "0.141221216"
    ],
    [
      "0.622815823",
      "0.553909494",
      "0.140816181"
    ],
    [
      "1.266904326",
      "1.277880548",
      "0.140297033"
    ],
    [
      "1.546444730",
      "1.253817045",
      "0.140277180"
    ],
    [
      "0.826178278",
      "0.364318241",
      "0.137214628"
    ],
    [
      "1.105188077",
      "1.053554986",
      "0.136242479"
    ],
    [
      "0.991223648",
      "1.301201380",
      "0.136368281"
    ],
    [
      "1.427138698",
      "1.503348342",
      "0.136308787"
    ],
    [
      "1.666807664",
      "1.005642266",
      "0.135545148"
    ],
    [
      "0.467208209",
      "1.280225259",
      "0.134234125"
    ],
    [
      "0.580868361",
      "1.520716775",
      "0.131763617"
    ],
    [
      "0.727494391",
      "1.306989657",
      "0.127424487"
    ],
    [
      "0.845821546",
      "1.084110217",
      "0.124917662"
    ],
    [
      "0.108502213",
      "0.991786083",
      "0.108464373"
    ],
    [
      "0.441001765",
      "0.723968187",
      "0.108134236"
    ],
    [
      "1.548763024",
      "0.293260166",
      "0.105225139"
    ],
    [
      "0.942729807",
      "0.877353337",
      "0.103423350"
    ],
    [
      "1.572119074",
      "1.692323038",
      "0.101873380"
    ],
    [
      "0.763468789",
      "0.133590109",
      "0.101883574"
    ],
    [
      "0.442581538",
      "1.706652790",
      "0.099959162"
    ],
    [
      "0.622668101",
      "1.105951380",
      "0.099302088"
    ],
    [
      "1.901166416",
      "1.002988476",
      "0.098828628"
    ]
  ]
}

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": 45
}

起步布局答案

{
  "circles": [
    [
      "1.196550042",
      "0.750409060",
      "0.180371576"
    ],
    [
      "1.004938704",
      "1.826348637",
      "0.173636604"
    ],
    [
      "0.383029885",
      "0.991967408",
      "0.166063357"
    ],
    [
      "0.678947411",
      "0.851130974",
      "0.161659211"
    ],
    [
      "0.334415425",
      "0.481785074",
      "0.156466043"
    ],
    [
      "1.329963491",
      "1.777942660",
      "0.154972966"
    ],
    [
      "0.681771699",
      "1.785979260",
      "0.152042071"
    ],
    [
      "1.306820385",
      "0.208518294",
      "0.151128961"
    ],
    [
      "1.105374461",
      "0.432639634",
      "0.150219400"
    ],
    [
      "0.185176394",
      "1.239389227",
      "0.150738726"
    ],
    [
      "1.691024499",
      "0.505440809",
      "0.150233178"
    ],
    [
      "1.526839428",
      "0.756778729",
      "0.149979224"
    ],
    [
      "1.013332651",
      "0.148530969",
      "0.148426591"
    ],
    [
      "0.186108702",
      "0.747509545",
      "0.147843632"
    ],
    [
      "0.555973165",
      "0.272896057",
      "0.148037574"
    ],
    [
      "1.146056988",
      "1.538480900",
      "0.146960032"
    ],
    [
      "1.385269566",
      "1.016955697",
      "0.146220169"
    ],
    [
      "1.822596459",
      "0.770984198",
      "0.146118760"
    ],
    [
      "1.395352229",
      "0.491888955",
      "0.145749496"
    ],
    [
      "0.855322381",
      "1.546292514",
      "0.143879498"
    ],
    [
      "0.894325466",
      "0.635924312",
      "0.142810185"
    ],
    [
      "0.307244356",
      "1.505759065",
      "0.142268915"
    ],
    [
      "1.705446932",
      "1.487590927",
      "0.142445170"
    ],
    [
      "1.827041694",
      "1.231307231",
      "0.141221216"
    ],
    [
      "0.622815823",
      "0.553909494",
      "0.140816181"
    ],
    [
      "1.266904326",
      "1.277880548",
      "0.140297033"
    ],
    [
      "1.546444730",
      "1.253817045",
      "0.140277180"
    ],
    [
      "0.826178278",
      "0.364318241",
      "0.137214628"
    ],
    [
      "1.105188077",
      "1.053554986",
      "0.136242479"
    ],
    [
      "0.991223648",
      "1.301201380",
      "0.136368281"
    ],
    [
      "1.427138698",
      "1.503348342",
      "0.136308787"
    ],
    [
      "1.666807664",
      "1.005642266",
      "0.135545148"
    ],
    [
      "0.467208209",
      "1.280225259",
      "0.134234125"
    ],
    [
      "0.580868361",
      "1.520716775",
      "0.131763617"
    ],
    [
      "0.727494391",
      "1.306989657",
      "0.127424487"
    ],
    [
      "0.845821546",
      "1.084110217",
      "0.124917662"
    ],
    [
      "0.108502213",
      "0.991786083",
      "0.108464373"
    ],
    [
      "0.441001765",
      "0.723968187",
      "0.108134236"
    ],
    [
      "1.548763024",
      "0.293260166",
      "0.105225139"
    ],
    [
      "0.942729807",
      "0.877353337",
      "0.103423350"
    ],
    [
      "1.572119074",
      "1.692323038",
      "0.101873380"
    ],
    [
      "0.763468789",
      "0.133590109",
      "0.101883574"
    ],
    [
      "0.442581538",
      "1.706652790",
      "0.099959162"
    ],
    [
      "0.622668101",
      "1.105951380",
      "0.099302088"
    ],
    [
      "1.901166416",
      "1.002988476",
      "0.098828628"
    ]
  ]
}

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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