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P02 · 装箱与覆盖 · 经典问题 · 基线易突破 · 难

单位圆内的等圆装箱 · n = 51

文献中也称circles in a circlepacking equal circles in a circlecci

在半径 1 的圆内放置 n 个半径相同、互不重叠的小圆，使共同半径尽可能大。

子题n = 51
目标最大化 共同半径

纪录对比

共同半径 · 越大越好
纪录数值 / 区间作者 / 持有人来源
外部已知最好[0.1245716766023650168004218760095, 0.1245716766023650168004218760105]见来源署名packomania.com ↗
本站纪录暂无选手纪录——

严格定义

  • 容器半径 1 的圆，圆心在 (1, 1)，两个坐标都在 0 到 2 之间
  • 提交恰好 n 个圆：一个共同半径 radius 与 n 个圆心 centers
  • 约束每个圆完整落在容器内；两两内部不重叠，相切允许
  • 目标让共同半径尽可能大
挑战这个纪录 ↗

当前展示：起步布局

2y0
0x2
已验证构造r = 0.124571676

帮助理解

哪里有优化空间

最优构形是「卡死」的接触结构：圆彼此顶住、顶住边界，常出现斜排、错位、以及不碰任何邻居的游离圆。规整的网格摆法几乎从不最优。

前沿在哪里

n = 1..14 与 19 已证明；其余规模的最优性仍未知。本站接入 Packomania 中 n = 1..300 的公开构型，大规模部分集中成一张表供算法批量挑战。

查看来源
单位圆内的等圆装箱 n = 51 的当前构型
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

容器是半径 1 的圆，圆心在 (1, 1)，所以两个坐标都在 0 到 2 之间。坐标和半径用同一个单位，直接写成小数，例如 "0.5"，最多九位小数。

提交 radius 与 centers。每个数写成十进制字符串，例如 "0.2"。

起步布局答案

{
  "centers": [
    [
      "0.894298469",
      "0.130976447"
    ],
    [
      "1.143382923",
      "0.136393558"
    ],
    [
      "0.653775294",
      "0.195945771"
    ],
    [
      "1.380854098",
      "0.211758346"
    ],
    [
      "0.441294539",
      "0.32603935"
    ],
    [
      "1.587478052",
      "0.350966651"
    ],
    [
      "0.830301967",
      "0.371760288"
    ],
    [
      "1.10825184",
      "0.383417371"
    ],
    [
      "1.34627924",
      "0.458490974"
    ],
    [
      "0.617821212",
      "0.501853867"
    ],
    [
      "0.274066051",
      "0.510720273"
    ],
    [
      "1.746519307",
      "0.542743317"
    ],
    [
      "0.960095005",
      "0.584424761"
    ],
    [
      "1.517086542",
      "0.639867036"
    ],
    [
      "1.19792499",
      "0.658649478"
    ],
    [
      "0.450592724",
      "0.68653479"
    ],
    [
      "0.74761425",
      "0.714518339"
    ],
    [
      "0.165634477",
      "0.735030335"
    ],
    [
      "1.84509634",
      "0.771555421"
    ],
    [
      "0.966518971",
      "0.833485282"
    ],
    [
      "1.368106076",
      "0.840613234"
    ],
    [
      "1.615663575",
      "0.86867914"
    ],
    [
      "0.591293647",
      "0.90851906"
    ],
    [
      "0.342161149",
      "0.910844852"
    ],
    [
      "1.164695769",
      "0.984476563"
    ],
    [
      "1.874574605",
      "1.027163814"
    ],
    [
      "0.810198368",
      "1.027486002"
    ],
    [
      "0.125206462",
      "1.033331888"
    ],
    [
      "1.390990221",
      "1.088703391"
    ],
    [
      "1.63854772",
      "1.116769298"
    ],
    [
      "0.597717613",
      "1.15757958"
    ],
    [
      "0.33976069",
      "1.159976641"
    ],
    [
      "1.008375166",
      "1.178477284"
    ],
    [
      "1.831194387",
      "1.274755601"
    ],
    [
      "1.255545977",
      "1.297814049"
    ],
    [
      "0.793564459",
      "1.312479921"
    ],
    [
      "1.526373558",
      "1.339879701"
    ],
    [
      "0.470719816",
      "1.371925026"
    ],
    [
      "0.213492449",
      "1.382320986"
    ],
    [
      "1.041551224",
      "1.425401887"
    ],
    [
      "1.720134539",
      "1.49777605"
    ],
    [
      "1.355126294",
      "1.526191305"
    ],
    [
      "0.663889571",
      "1.529271319"
    ],
    [
      "0.356165178",
      "1.593170693"
    ],
    [
      "0.893560878",
      "1.625829599"
    ],
    [
      "1.141131541",
      "1.653779142"
    ],
    [
      "1.550747499",
      "1.6804792"
    ],
    [
      "0.549334934",
      "1.750516987"
    ],
    [
      "1.336752746",
      "1.808067038"
    ],
    [
      "0.77900624",
      "1.847075267"
    ],
    [
      "1.026576904",
      "1.87502481"
    ]
  ],
  "radius": "0.124571676"
}
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": 51
}

起步布局答案

{
  "centers": [
    [
      "0.894298469",
      "0.130976447"
    ],
    [
      "1.143382923",
      "0.136393558"
    ],
    [
      "0.653775294",
      "0.195945771"
    ],
    [
      "1.380854098",
      "0.211758346"
    ],
    [
      "0.441294539",
      "0.32603935"
    ],
    [
      "1.587478052",
      "0.350966651"
    ],
    [
      "0.830301967",
      "0.371760288"
    ],
    [
      "1.10825184",
      "0.383417371"
    ],
    [
      "1.34627924",
      "0.458490974"
    ],
    [
      "0.617821212",
      "0.501853867"
    ],
    [
      "0.274066051",
      "0.510720273"
    ],
    [
      "1.746519307",
      "0.542743317"
    ],
    [
      "0.960095005",
      "0.584424761"
    ],
    [
      "1.517086542",
      "0.639867036"
    ],
    [
      "1.19792499",
      "0.658649478"
    ],
    [
      "0.450592724",
      "0.68653479"
    ],
    [
      "0.74761425",
      "0.714518339"
    ],
    [
      "0.165634477",
      "0.735030335"
    ],
    [
      "1.84509634",
      "0.771555421"
    ],
    [
      "0.966518971",
      "0.833485282"
    ],
    [
      "1.368106076",
      "0.840613234"
    ],
    [
      "1.615663575",
      "0.86867914"
    ],
    [
      "0.591293647",
      "0.90851906"
    ],
    [
      "0.342161149",
      "0.910844852"
    ],
    [
      "1.164695769",
      "0.984476563"
    ],
    [
      "1.874574605",
      "1.027163814"
    ],
    [
      "0.810198368",
      "1.027486002"
    ],
    [
      "0.125206462",
      "1.033331888"
    ],
    [
      "1.390990221",
      "1.088703391"
    ],
    [
      "1.63854772",
      "1.116769298"
    ],
    [
      "0.597717613",
      "1.15757958"
    ],
    [
      "0.33976069",
      "1.159976641"
    ],
    [
      "1.008375166",
      "1.178477284"
    ],
    [
      "1.831194387",
      "1.274755601"
    ],
    [
      "1.255545977",
      "1.297814049"
    ],
    [
      "0.793564459",
      "1.312479921"
    ],
    [
      "1.526373558",
      "1.339879701"
    ],
    [
      "0.470719816",
      "1.371925026"
    ],
    [
      "0.213492449",
      "1.382320986"
    ],
    [
      "1.041551224",
      "1.425401887"
    ],
    [
      "1.720134539",
      "1.49777605"
    ],
    [
      "1.355126294",
      "1.526191305"
    ],
    [
      "0.663889571",
      "1.529271319"
    ],
    [
      "0.356165178",
      "1.593170693"
    ],
    [
      "0.893560878",
      "1.625829599"
    ],
    [
      "1.141131541",
      "1.653779142"
    ],
    [
      "1.550747499",
      "1.6804792"
    ],
    [
      "0.549334934",
      "1.750516987"
    ],
    [
      "1.336752746",
      "1.808067038"
    ],
    [
      "0.77900624",
      "1.847075267"
    ],
    [
      "1.026576904",
      "1.87502481"
    ]
  ],
  "radius": "0.124571676"
}

提交 radius 与 centers。每个数写成十进制字符串，例如 "0.2"。 · 验证器 v1.0.0

DISCUSSION

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