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 = 239

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 = 239
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[0.0592996511187544813034375058315, 0.0592996511187544813034375058325]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 = 0.059299650

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 = 239 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": [
    [
      "1.294924123",
      "0.31265502"
    ],
    [
      "1.614898782",
      "0.570478735"
    ],
    [
      "1.852124918",
      "1.299387181"
    ],
    [
      "1.78629293",
      "0.48360812"
    ],
    [
      "1.359780016",
      "1.050491792"
    ],
    [
      "0.568892723",
      "0.287554611"
    ],
    [
      "1.918894822",
      "1.201368946"
    ],
    [
      "1.264684176",
      "1.65937046"
    ],
    [
      "0.720104716",
      "0.534765462"
    ],
    [
      "1.577522468",
      "1.602039317"
    ],
    [
      "1.700274821",
      "0.791805465"
    ],
    [
      "1.695268296",
      "1.633655382"
    ],
    [
      "1.445260157",
      "1.271752551"
    ],
    [
      "1.104263341",
      "1.847821187"
    ],
    [
      "0.223890929",
      "0.752079077"
    ],
    [
      "1.625836138",
      "0.884134642"
    ],
    [
      "0.56161165",
      "1.549563243"
    ],
    [
      "1.210902648",
      "1.235150145"
    ],
    [
      "0.308836781",
      "0.530781795"
    ],
    [
      "0.911296394",
      "1.936508766"
    ],
    [
      "0.571078167",
      "0.162776489"
    ],
    [
      "0.637515134",
      "1.77406189"
    ],
    [
      "0.94338786",
      "0.257751411"
    ],
    [
      "1.147505422",
      "1.641069257"
    ],
    [
      "0.731146174",
      "0.848355398"
    ],
    [
      "0.636050334",
      "1.457234066"
    ],
    [
      "1.15712112",
      "0.81092983"
    ],
    [
      "0.064435931",
      "0.901831678"
    ],
    [
      "0.679830506",
      "0.11546134"
    ],
    [
      "1.138603389",
      "0.167889216"
    ],
    [
      "0.721530475",
      "1.678494825"
    ],
    [
      "0.889478283",
      "0.152112631"
    ],
    [
      "0.383275465",
      "0.438452619"
    ],
    [
      "0.965503682",
      "0.884957804"
    ],
    [
      "1.188819734",
      "0.607970274"
    ],
    [
      "1.402520086",
      "1.161122171"
    ],
    [
      "1.030326667",
      "1.622768054"
    ],
    [
      "0.680255204",
      "1.88469227"
    ],
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      "1.317039945",
      "0.939861412"
    ],
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      "0.454048594",
      "0.701122613"
    ],
    [
      "1.252184053",
      "0.202024641"
    ],
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      "0.316212684",
      "1.199370901"
    ],
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      "1.008243753",
      "0.995588183"
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      "0.342750308",
      "0.326991836"
    ],
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      "1.497615926",
      "0.552243503"
    ],
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      "1.125422507",
      "1.013889386"
    ],
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      "0.678790404",
      "1.567864446"
    ],
    [
      "1.413019712",
      "0.32357411"
    ],
    [
      "1.22048544",
      "0.404984197"
    ],
    [
      "0.913147913",
      "1.604466851"
    ],
    [
      "1.370821473",
      "1.364081727"
    ],
    [
      "1.136463964",
      "1.327479322"
    ],
    [
      "1.340801724",
      "0.12320394"
    ],
    [
      "0.337472859",
      "1.667813547"
    ],
    [
      "1.732608347",
      "0.589361418"
    ],
    [
      "0.059519337",
      "1.020329026"
    ],
    [
      "0.539528735",
      "0.922383372"
    ],
    [
      "0.465090052",
      "1.014712548"
    ],
    [
      "0.773886244",
      "0.958985777"
    ],
    [
      "0.848324928",
      "0.866656601"
    ],
    [
      "0.25964054",
      "0.419725536"
    ],
    [
      "1.198783242",
      "1.919457649"
    ],
    [
      "0.65670749",
      "0.940684574"
    ],
    [
      "1.465917313",
      "0.755203059"
    ],
    [
      "1.348738558",
      "0.736901856"
    ],
    [
      "0.880023541",
      "0.663697045"
    ],
    [
      "0.422349981",
      "0.904082169"
    ],
    [
      "0.667748947",
      "1.25427451"
    ],
    [
      "0.17560002",
      "0.86049961"
    ],
    [
      "1.519698841",
      "1.179423374"
    ],
    [
      "0.336490123",
      "0.685444596"
    ],
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      "0.784927701",
      "1.272575713"
    ],
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      "1.083213097",
      "0.062987415"
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      "1.684600296",
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      "0.718600653"
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      "0.710489018",
      "1.364904889"
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      "1.475535269"
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      "1.887138381"
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      "1.607137232",
      "1.399907541"
    ],
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      "0.795969159",
      "1.586165648"
    ],
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      "0.870407842",
      "1.493836472"
    ],
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      "1.395134434",
      "1.68496995"
    ],
    [
      "1.285341332",
      "1.142820968"
    ],
    [
      "1.551397454",
      "0.976463818"
    ],
    [
      "1.781298098",
      "1.523918341"
    ],
    [
      "1.850834998",
      "0.598755745"
    ],
    [
      "1.029854255",
      "1.940196197"
    ],
    [
      "0.69944756",
      "1.051314954"
    ],
    [
      "1.61001308",
      "1.716101381"
    ],
    [
      "1.380437172",
      "0.5339423"
    ],
    [
      "1.724315986",
      "1.418208744"
    ],
    [
      "0.902106456",
      "1.290876916"
    ],
    [
      "1.541875467",
      "0.231047385"
    ],
    [
      "0.327268363",
      "1.549654067"
    ],
    [
      "1.296382789",
      "1.456410904"
    ],
    [
      "0.364062738",
      "0.800794262"
    ],
    [
      "1.662777683",
      "1.519593318"
    ],
    [
      "0.742187631",
      "1.161945333"
    ],
    [
      "1.104765351",
      "1.530438878"
    ],
    [
      "1.179204035",
      "1.438109701"
    ],
    [
      "0.688406103",
      "0.737725018"
    ],
    [
      "1.168162578",
      "1.124519766"
    ],
    [
      "1.508657383",
      "0.865833439"
    ],
    [
      "1.305998488",
      "0.626271477"
    ],
    [
      "1.062025281",
      "1.419808498"
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    [
      "0.944846526",
      "1.401507295"
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    [
      "1.413561543",
      "1.474712107"
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      "1.199861191",
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      "1.050983823",
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      "1.423177242",
      "0.64457268"
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      "1.242601261",
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      "0.976545139",
      "1.198547739"
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      "1.01928521",
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      "0.582268806",
      "1.033013751"
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      "0.604351721",
      "1.660193622"
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      "1.829670978",
      "1.123235086"
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      "0.550570193",
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      "0.602925962",
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      "0.838191506",
      "1.699935931"
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      "0.987084587",
      "1.829519984"
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      "1.4342187",
      "0.958162615"
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      "0.390651368",
      "1.107041725"
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      "1.636877595",
      "1.197724577"
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      "1.454842948",
      "0.441586602"
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      "0.922763612",
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      "0.214248272",
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      "1.422031245",
      "1.840718011"
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      "0.891064998",
      "0.97728698"
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    [
      "0.444432896",
      "1.53126204"
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      "0.987586597",
      "1.512137675"
    ],
    [
      "0.518871579",
      "1.438932863"
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      "0.868949177",
      "0.350080588"
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    [
      "1.02005002",
      "0.164588719"
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    [
      "0.485747208",
      "0.498163057"
    ],
    [
      "1.743014892",
      "0.902435845"
    ],
    [
      "1.253642719",
      "1.345780525"
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    [
      "0.560185892",
      "0.40583388"
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    [
      "0.827667772",
      "1.383206092"
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      "0.401692825",
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      "1.426018813",
      "0.205689345"
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      "1.712241343",
      "1.106133804"
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      "0.497339895"
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    [
      "0.15482741",
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  ],
  "radius": "0.05929965"
}
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": 239
}

起步布局答案

{
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  "radius": "0.05929965"
}

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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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?