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 = 268

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 = 268
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[0.0560793522505, 0.0560793522515]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 = 0.056079351

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 = 268 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": [
    [
      "0.929019486",
      "0.058751916"
    ],
    [
      "1.092382925",
      "0.060611058"
    ],
    [
      "1.22732673",
      "0.083862048"
    ],
    [
      "0.822035858",
      "0.092427995"
    ],
    [
      "0.710256818",
      "0.101648689"
    ],
    [
      "1.334547091",
      "0.117353733"
    ],
    [
      "1.009826612",
      "0.136532269"
    ],
    [
      "0.582654081",
      "0.153354517"
    ],
    [
      "1.144770418",
      "0.159783259"
    ],
    [
      "0.902842984",
      "0.170208348"
    ],
    [
      "1.460397509",
      "0.175973106"
    ],
    [
      "1.25199105",
      "0.193275239"
    ],
    [
      "0.79150997",
      "0.20035381"
    ],
    [
      "0.679709228",
      "0.209567268"
    ],
    [
      "0.483618482",
      "0.209850446"
    ],
    [
      "1.360275466",
      "0.226521612"
    ],
    [
      "1.062214106",
      "0.235704471"
    ],
    [
      "0.580997088",
      "0.265500981"
    ],
    [
      "1.595622743",
      "0.267729738"
    ],
    [
      "1.169434738",
      "0.26919645"
    ],
    [
      "0.872371667",
      "0.278148488"
    ],
    [
      "1.484869441",
      "0.285429487"
    ],
    [
      "0.376147919",
      "0.291625262"
    ],
    [
      "1.277719425",
      "0.302443117"
    ],
    [
      "0.979478588",
      "0.311430352"
    ],
    [
      "0.678052236",
      "0.321713732"
    ],
    [
      "0.484113008",
      "0.322008061"
    ],
    [
      "1.384747398",
      "0.335977993"
    ],
    [
      "1.086878426",
      "0.345117661"
    ],
    [
      "0.786505494",
      "0.35030509"
    ],
    [
      "1.664945238",
      "0.355899839"
    ],
    [
      "0.296529963",
      "0.370622453"
    ],
    [
      "1.554191936",
      "0.373599589"
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      "0.378364329"
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      "0.539389157"
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      "0.566667339"
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      "0.185994428",
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  "radius": "0.056079351"
}
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": 268
}

起步布局答案

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      "1.299187618",
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      "0.503341585",
      "1.802693346"
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      "0.909399523",
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      "1.016427377",
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      "0.608529847",
      "1.858339073"
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    [
      "1.308076899",
      "1.892230247"
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    [
      "0.71415781",
      "1.899582964"
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    [
      "1.200072604",
      "1.922473383"
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    [
      "0.835535209",
      "1.929482395"
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    [
      "1.089243542",
      "1.939692386"
    ],
    [
      "0.946944257",
      "1.942428394"
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  ],
  "radius": "0.056079351"
}

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

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聊思路、贴方法、问为什么卡住。所有登录用户都可以发帖;发言公开署名,与纪录使用同一个名字,署名后的 #编号是账号注册序号,冒不了名。新发言经自动审核后公开。

还没有帖子。第一个聊聊这道题的思路?