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 = 271

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 = 271
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[0.0557734621205, 0.0557734621215]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 = 0.055773461

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 = 271 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": [
    [
      "1.011641667",
      "0.055845232"
    ],
    [
      "0.900216735",
      "0.061060676"
    ],
    [
      "1.154090283",
      "0.068445568"
    ],
    [
      "0.790184384",
      "0.079380012"
    ],
    [
      "1.262885366",
      "0.093109249"
    ],
    [
      "0.669096042",
      "0.115654861"
    ],
    [
      "1.374662951",
      "0.133286998"
    ],
    [
      "1.075232113",
      "0.147491134"
    ],
    [
      "0.567114723",
      "0.160849184"
    ],
    [
      "0.86604935",
      "0.16724592"
    ],
    [
      "0.756016999",
      "0.185565257"
    ],
    [
      "0.974165898",
      "0.194696393"
    ],
    [
      "1.175484517",
      "0.196400968"
    ],
    [
      "1.286884835",
      "0.202118086"
    ],
    [
      "1.456153454",
      "0.209457952"
    ],
    [
      "0.471174776",
      "0.217754746"
    ],
    [
      "1.562881085",
      "0.241891407"
    ],
    [
      "0.580121945",
      "0.271635409"
    ],
    [
      "0.691558226",
      "0.276602518"
    ],
    [
      "0.806260792",
      "0.285155807"
    ],
    [
      "0.91766111",
      "0.290872925"
    ],
    [
      "1.36301326",
      "0.293069812"
    ],
    [
      "1.029188649",
      "0.294455307"
    ],
    [
      "1.140461746",
      "0.302307161"
    ],
    [
      "1.251862064",
      "0.308024279"
    ],
    [
      "1.648356649",
      "0.313561794"
    ],
    [
      "0.422670311",
      "0.318203903"
    ],
    [
      "1.4746627",
      "0.319458516"
    ],
    [
      "0.315601046",
      "0.349491102"
    ],
    [
      "0.522814048",
      "0.367335849"
    ],
    [
      "0.634250339",
      "0.372302687"
    ],
    [
      "0.745609463",
      "0.378772753"
    ],
    [
      "0.857009781",
      "0.384489871"
    ],
    [
      "0.968410099",
      "0.390206989"
    ],
    [
      "1.079810417",
      "0.395924108"
    ],
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      "1.191210735",
      "0.401641226"
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      "1.302611053",
      "0.407358344"
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      "0.424509699"
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      "1.749078748",
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      "0.460794557"
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      "0.684958134",
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      "0.478106817"
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      "0.907758771",
      "0.483823936"
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      "1.019159089",
      "0.489541054"
    ],
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      "1.130559407",
      "0.495258172"
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      "1.241959725",
      "0.50097529"
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      "1.353360043",
      "0.506692408"
    ],
    [
      "1.464760361",
      "0.512409527"
    ],
    [
      "1.811647061",
      "0.517528286"
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      "1.576160679",
      "0.518126645"
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      "1.687560997",
      "0.523843763"
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    [
      "0.282932611",
      "0.544583413"
    ],
    [
      "0.171416752",
      "0.547215774"
    ],
    [
      "0.40150617",
      "0.554572409"
    ],
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      "0.512906488",
      "0.560289527"
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      "0.624306806",
      "0.566006646"
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      "0.735707124",
      "0.571723764"
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      "1.181308396",
      "0.594592237"
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      "1.292708714",
      "0.600309355"
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      "1.51550935",
      "0.611743591"
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      "1.754302203",
      "0.613221208"
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      "1.626909669",
      "0.617460709"
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      "0.837205103",
      "0.770391893"
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    [
      "0.128886234",
      "0.770796674"
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    [
      "0.948605421",
      "0.776109011"
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    [
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      "1.171406057",
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      "1.282806375",
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    [
      "1.394206693",
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    [
      "1.617007329",
      "0.81041172"
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    [
      "1.728407647",
      "0.816128839"
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      "1.83992917",
      "0.818509239"
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      "0.21953334",
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}
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": 271
}

起步布局答案

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  "radius": "0.055773461"
}

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?