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 = 199

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 = 199
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[0.0649255351511510426047208321935, 0.0649255351511510426047208321945]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 = 0.064925534

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 = 199 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": [
    [
      "1.01077289",
      "0.064987594"
    ],
    [
      "0.881139929",
      "0.072510626"
    ],
    [
      "1.215945335",
      "0.090202293"
    ],
    [
      "0.750088424",
      "0.103310189"
    ],
    [
      "0.618739476",
      "0.146182298"
    ],
    [
      "1.103771324",
      "0.155610945"
    ],
    [
      "1.438624861",
      "0.174183746"
    ],
    [
      "1.308943769",
      "0.180825644"
    ],
    [
      "0.973948424",
      "0.189507709"
    ],
    [
      "0.844315463",
      "0.197030742"
    ],
    [
      "0.504134575",
      "0.2072316"
    ],
    [
      "0.718529669",
      "0.229267903"
    ],
    [
      "1.548797558",
      "0.24290985"
    ],
    [
      "1.196769758",
      "0.246234296"
    ],
    [
      "1.066946858",
      "0.28013106"
    ],
    [
      "0.603924769",
      "0.290317205"
    ],
    [
      "1.416565714",
      "0.302147391"
    ],
    [
      "0.94193476",
      "0.315350571"
    ],
    [
      "0.445046082",
      "0.32285967"
    ],
    [
      "0.8123018",
      "0.322873603"
    ],
    [
      "1.651602076",
      "0.335231307"
    ],
    [
      "1.291071304",
      "0.335500856"
    ],
    [
      "0.320064741",
      "0.358087001"
    ],
    [
      "1.159945292",
      "0.370754411"
    ],
    [
      "1.52673841",
      "0.370873495"
    ],
    [
      "0.696417049",
      "0.38145709"
    ],
    [
      "1.034933194",
      "0.405973922"
    ],
    [
      "0.570062513",
      "0.415675284"
    ],
    [
      "1.383882398",
      "0.427817997"
    ],
    [
      "1.743799324",
      "0.433319473"
    ],
    [
      "0.909951853",
      "0.441201253"
    ],
    [
      "0.445604631",
      "0.452709539"
    ],
    [
      "1.254246838",
      "0.460020972"
    ],
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      "1.617549508",
      "0.463688607"
    ],
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      "0.784970512",
      "0.476428584"
    ],
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      "0.320623289",
      "0.48793687"
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      "1.127931628",
      "0.496597273"
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      "1.492559561",
      "0.498885392"
    ],
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      "0.659487863",
      "0.509826265"
    ],
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      "0.195641948",
      "0.523164201"
    ],
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      "1.002950287",
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      "1.345911403",
      "0.551993293"
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      "0.877968946",
      "0.567051935"
    ],
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      "1.688881125",
      "0.572192528"
    ],
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      "0.410048639",
      "0.582087851"
    ],
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      "1.220930062",
      "0.587220624"
    ],
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      "0.752987605",
      "0.602279265"
    ],
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      "1.563891179",
      "0.607389313"
    ],
    [
      "0.285067298",
      "0.617315182"
    ],
    [
      "1.095948721",
      "0.622447955"
    ],
    [
      "0.628006264",
      "0.637506596"
    ],
    [
      "0.137340605",
      "0.639191154"
    ],
    [
      "1.438909837",
      "0.642616644"
    ],
    [
      "0.97096738",
      "0.657675285"
    ],
    [
      "1.783148034",
      "0.667674075"
    ],
    [
      "0.503024923",
      "0.672733927"
    ],
    [
      "1.313928496",
      "0.677843975"
    ],
    [
      "0.845986039",
      "0.692902616"
    ],
    [
      "1.656898218",
      "0.698043209"
    ],
    [
      "0.378043582",
      "0.707961258"
    ],
    [
      "1.188947155",
      "0.713071306"
    ],
    [
      "0.721004698",
      "0.728129947"
    ],
    [
      "1.895880594",
      "0.732115293"
    ],
    [
      "1.531908271",
      "0.733239995"
    ],
    [
      "0.226765955",
      "0.733342135"
    ],
    [
      "1.063965814",
      "0.748298636"
    ],
    [
      "0.596023357",
      "0.763357278"
    ],
    [
      "0.098490472",
      "0.768082373"
    ],
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      "1.40692693",
      "0.768467326"
    ],
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      "0.938984473",
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      "0.814003132",
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    ],
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      "1.624915311",
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      "0.319742239",
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      "0.56404045",
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      "1.374944023",
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      "0.069941918",
      "0.903272449"
    ],
    [
      "0.907001566",
      "0.909376649"
    ],
    [
      "1.721008791",
      "0.919825572"
    ],
    [
      "0.439059109",
      "0.924435291"
    ],
    [
      "1.249962682",
      "0.929545338"
    ],
    [
      "0.782020225",
      "0.94460398"
    ],
    [
      "1.594758975",
      "0.950194706"
    ],
    [
      "0.312534651",
      "0.953639093"
    ],
    [
      "1.124981341",
      "0.964772669"
    ],
    [
      "1.934490286",
      "0.966952145"
    ],
    [
      "0.657038884",
      "0.979831311"
    ],
    [
      "1.467942457",
      "0.984941358"
    ],
    [
      "0.187359055",
      "0.988169814"
    ],
    [
      "1",
      "1"
    ],
    [
      "1.812640945",
      "1.011830186"
    ],
    [
      "0.532057543",
      "1.015058642"
    ],
    [
      "1.342961116",
      "1.020168689"
    ],
    [
      "0.065509714",
      "1.033047855"
    ],
    [
      "0.875018659",
      "1.035227331"
    ],
    [
      "1.687465349",
      "1.046360907"
    ],
    [
      "0.405241025",
      "1.049805294"
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    [
      "1.217979775",
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    [
      "0.750037318",
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      "0.936034186",
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    ],
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      "1.773234045",
      "1.266657865"
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      "0.468091729",
      "1.266760005"
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      "0.104119406",
      "1.267884707"
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    [
      "1.621956418",
      "1.292038742"
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    [
      "0.343101782",
      "1.301956791"
    ],
    [
      "1.154013961",
      "1.307097384"
    ],
    [
      "0.686071504",
      "1.322156025"
    ],
    [
      "1.496975077",
      "1.327266073"
    ],
    [
      "0.216851966",
      "1.332325925"
    ],
    [
      "1.02903262",
      "1.342324715"
    ],
    [
      "0.561090163",
      "1.357383356"
    ],
    [
      "1.862659395",
      "1.360808846"
    ],
    [
      "1.371993736",
      "1.362493404"
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    [
      "0.904051279",
      "1.377552045"
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    [
      "1.714932702",
      "1.382684818"
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    [
      "0.436108821",
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    [
      "1.247012395",
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    [
      "0.779069938",
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    [
      "1.589951361",
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      "0.654088597",
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    [
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      "1.679935259",
      "1.641912999"
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      "1.664499144"
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    [
      "0.348394818",
      "1.664757808"
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    [
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      "1.677126397"
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    [
      "1.554953918",
      "1.67714033"
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    [
      "1.05806524",
      "1.684649429"
    ],
    [
      "0.583434286",
      "1.697852609"
    ],
    [
      "1.396075231",
      "1.709682795"
    ],
    [
      "0.933053142",
      "1.71986894"
    ],
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      "0.803230242",
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      "1.281470331",
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      "1.495865425",
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      "1.802969258"
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      "1.026051576",
      "1.810492291"
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    [
      "0.561375139",
      "1.825816254"
    ],
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      "0.896228676",
      "1.844389055"
    ],
    [
      "1.381260524",
      "1.853817702"
    ],
    [
      "1.249890913",
      "1.896684632"
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    [
      "0.784054665",
      "1.909797707"
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      "1.118860071",
      "1.927489374"
    ],
    [
      "0.98922711",
      "1.935012406"
    ]
  ],
  "radius": "0.064925534"
}
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": 199
}

起步布局答案

{
  "centers": [
    [
      "1.01077289",
      "0.064987594"
    ],
    [
      "0.881139929",
      "0.072510626"
    ],
    [
      "1.215945335",
      "0.090202293"
    ],
    [
      "0.750088424",
      "0.103310189"
    ],
    [
      "0.618739476",
      "0.146182298"
    ],
    [
      "1.103771324",
      "0.155610945"
    ],
    [
      "1.438624861",
      "0.174183746"
    ],
    [
      "1.308943769",
      "0.180825644"
    ],
    [
      "0.973948424",
      "0.189507709"
    ],
    [
      "0.844315463",
      "0.197030742"
    ],
    [
      "0.504134575",
      "0.2072316"
    ],
    [
      "0.718529669",
      "0.229267903"
    ],
    [
      "1.548797558",
      "0.24290985"
    ],
    [
      "1.196769758",
      "0.246234296"
    ],
    [
      "1.066946858",
      "0.28013106"
    ],
    [
      "0.603924769",
      "0.290317205"
    ],
    [
      "1.416565714",
      "0.302147391"
    ],
    [
      "0.94193476",
      "0.315350571"
    ],
    [
      "0.445046082",
      "0.32285967"
    ],
    [
      "0.8123018",
      "0.322873603"
    ],
    [
      "1.651602076",
      "0.335231307"
    ],
    [
      "1.291071304",
      "0.335500856"
    ],
    [
      "0.320064741",
      "0.358087001"
    ],
    [
      "1.159945292",
      "0.370754411"
    ],
    [
      "1.52673841",
      "0.370873495"
    ],
    [
      "0.696417049",
      "0.38145709"
    ],
    [
      "1.034933194",
      "0.405973922"
    ],
    [
      "0.570062513",
      "0.415675284"
    ],
    [
      "1.383882398",
      "0.427817997"
    ],
    [
      "1.743799324",
      "0.433319473"
    ],
    [
      "0.909951853",
      "0.441201253"
    ],
    [
      "0.445604631",
      "0.452709539"
    ],
    [
      "1.254246838",
      "0.460020972"
    ],
    [
      "1.617549508",
      "0.463688607"
    ],
    [
      "0.784970512",
      "0.476428584"
    ],
    [
      "0.320623289",
      "0.48793687"
    ],
    [
      "1.127931628",
      "0.496597273"
    ],
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      "1.492559561",
      "0.498885392"
    ],
    [
      "0.659487863",
      "0.509826265"
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    [
      "0.195641948",
      "0.523164201"
    ],
    [
      "1.002950287",
      "0.531824604"
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  "radius": "0.064925534"
}

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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还没有帖子。第一个聊聊这道题的思路?