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 = 152

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 = 152
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[0.0739002798461323339223435963045, 0.0739002798461323339223435963055]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 = 0.073900279

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 = 152 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": [
    [
      "0.762014982",
      "1.648354082"
    ],
    [
      "0.884238781",
      "1.731458524"
    ],
    [
      "0.68681907",
      "0.573373673"
    ],
    [
      "0.774376597",
      "0.101804714"
    ],
    [
      "1.160423659",
      "1.62610537"
    ],
    [
      "1.648394116",
      "1.033062387"
    ],
    [
      "0.638293426",
      "0.864964625"
    ],
    [
      "1.192746022",
      "0.094180116"
    ],
    [
      "0.575314553",
      "0.476359508"
    ],
    [
      "1.240395059",
      "0.365831601"
    ],
    [
      "0.752424602",
      "0.958874584"
    ],
    [
      "0.703898958",
      "1.250465536"
    ],
    [
      "0.232365389",
      "0.877969928"
    ],
    [
      "1.070555305",
      "1.386399935"
    ],
    [
      "1.11660309",
      "1.918702535"
    ],
    [
      "0.890818599",
      "0.906989066"
    ],
    [
      "0.277398217",
      "1.579229968"
    ],
    [
      "1.004949774",
      "1.000899024"
    ],
    [
      "1.721397609",
      "1.445505925"
    ],
    [
      "0.18069795",
      "0.568253729"
    ],
    [
      "1.101682205",
      "0.416858734"
    ],
    [
      "1.672656938",
      "0.887266911"
    ],
    [
      "0.963607065",
      "0.469602637"
    ],
    [
      "1.140494938",
      "1.77255622"
    ],
    [
      "1.281737769",
      "0.897127988"
    ],
    [
      "1.787174122",
      "1.128976925"
    ],
    [
      "1.604165883",
      "1.325303716"
    ],
    [
      "1.69691976",
      "0.741471434"
    ],
    [
      "0.589738385",
      "1.156591316"
    ],
    [
      "0.327858475",
      "0.554513807"
    ],
    [
      "0.084870119",
      "1.14211964"
    ],
    [
      "1.786788114",
      "0.981176869"
    ],
    [
      "1.184686481",
      "1.480309893"
    ],
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      "1.734052391",
      "0.598411401"
    ],
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      "1.167606593",
      "0.80321803"
    ],
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      "0.800950246",
      "0.667283631"
    ],
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      "1.827370357",
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    ],
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      "0.730303688",
      "0.261563754"
    ],
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      "0.132026413",
      "1.322928079"
    ],
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      "1.895461642",
      "1.229576977"
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      "1.558525763",
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    ],
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      "1.33759107"
    ],
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      "0.614030604",
      "1.010760102"
    ],
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      "1.029212596",
      "0.855103548"
    ],
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      "0.565524043",
      "1.302401924"
    ],
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      "0.269054218",
      "0.728737436"
    ],
    [
      "1.412948936",
      "1.668129683"
    ],
    [
      "1.07773824",
      "0.563512595"
    ],
    [
      "0.93216131",
      "1.438285458"
    ],
    [
      "0.122447281",
      "0.704091371"
    ],
    [
      "1.330263413",
      "0.605537036"
    ],
    [
      "0.662556248",
      "0.719169149"
    ],
    [
      "1.323080478",
      "1.428424375"
    ],
    [
      "0.499899429",
      "0.916850143"
    ],
    [
      "1.485737297",
      "1.230743381"
    ],
    [
      "1.261331776",
      "1.888450356"
    ],
    [
      "0.194453499",
      "1.456897719"
    ],
    [
      "0.542336645",
      "1.45098011"
    ],
    [
      "1.534262941",
      "0.939152429"
    ],
    [
      "1.208949303",
      "1.334514417"
    ],
    [
      "0.495921214",
      "1.776894632"
    ],
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      "0.920140005",
      "0.077349973"
    ],
    [
      "0.353376139",
      "0.967724217"
    ],
    [
      "0.219267051",
      "1.20362143"
    ],
    [
      "1.306000591",
      "0.751332512"
    ],
    [
      "1.268956902",
      "0.220817027"
    ],
    [
      "0.547826816",
      "0.624941329"
    ],
    [
      "0.793786393",
      "1.490221841"
    ],
    [
      "0.714121667",
      "0.4240012"
    ],
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      "1.437354798",
      "1.522358083"
    ],
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      "0.679655218",
      "1.396311883"
    ],
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      "0.403942648",
      "1.502865627"
    ],
    [
      "1.233212125",
      "1.188718941"
    ],
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      "0.599600029",
      "0.330567803"
    ],
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      "1.094818127",
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    ],
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      "0.888828579",
      "1.919402853"
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      "0.915081421",
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    [
      "0.37874793",
      "1.686808967"
    ],
    [
      "0.980686952",
      "1.1466945"
    ],
    [
      "1.257474947",
      "1.042923464"
    ],
    [
      "1.052959576",
      "0.142187979"
    ],
    [
      "0.623314659",
      "1.699415085"
    ],
    [
      "1.711373208",
      "1.592966146"
    ],
    [
      "1.528129479",
      "1.760749594"
    ],
    [
      "0.524162251",
      "0.771054667"
    ],
    [
      "0.337285853",
      "1.114646333"
    ],
    [
      "1.534594133",
      "0.243926486"
    ],
    [
      "1.740974417",
      "1.269371343"
    ],
    [
      "1.510227896",
      "0.390095891"
    ],
    [
      "1.285198691",
      "1.742459223"
    ],
    [
      "0.842292955",
      "1.198580018"
    ],
    [
      "0.866555777",
      "1.052784542"
    ],
    [
      "0.98635346",
      "0.309406433"
    ],
    [
      "0.449520684",
      "1.210814665"
    ],
    [
      "0.385368823",
      "0.823427738"
    ],
    [
      "1.191869415",
      "0.657422553"
    ],
    [
      "1.510000119",
      "1.084947905"
    ],
    [
      "1.461474475",
      "1.376538858"
    ],
    [
      "1.444394588",
      "0.699446994"
    ],
    [
      "1.745891309",
      "0.451085756"
    ],
    [
      "0.475636607",
      "1.06264562"
    ],
    [
      "1.216132237",
      "0.511627077"
    ],
    [
      "1.407401379",
      "0.168362904"
    ],
    [
      "1.864502987",
      "0.667892673"
    ],
    [
      "1.022041228",
      "1.678021729"
    ],
    [
      "0.625933612",
      "1.847192439"
    ],
    [
      "0.656467821",
      "1.544890068"
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      "1.053475418",
      "0.709308071"
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    [
      "1.420131766",
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      "1.3473433",
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      "0.634359919",
      "0.149136895"
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      "1.56567339",
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      "1.371606122",
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      "1.610444626"
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      "0.216799603"
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      "1.470336723"
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      "0.772510799",
      "1.828215247"
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      "1.624131294",
      "1.178857863"
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      "1.46865741",
      "0.553651518"
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      "1.119080949",
      "1.094808982"
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      "0.939344243",
      "0.615398113"
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      "0.430270854",
      "0.447945941"
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      "0.825213068",
      "0.521488155"
    ],
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      "0.852515665",
      "0.372115682"
    ],
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      "1.002112507",
      "1.824472579"
    ],
    [
      "1.298817656",
      "1.574219852"
    ],
    [
      "0.218092711",
      "1.027250678"
    ],
    [
      "0.72816178",
      "1.10467006"
    ],
    [
      "1.354526234",
      "0.459741559"
    ],
    [
      "0.086548276",
      "0.847465939"
    ],
    [
      "1.848755506",
      "1.370506116"
    ],
    [
      "1.582788585",
      "0.647561476"
    ],
    [
      "0.073915299",
      "0.994725619"
    ],
    [
      "0.776687424",
      "0.813079107"
    ],
    [
      "0.50365626",
      "0.218140944"
    ],
    [
      "0.42820547",
      "1.35707015"
    ],
    [
      "0.95642413",
      "1.292489977"
    ],
    [
      "0.818030133",
      "1.344375495"
    ]
  ],
  "radius": "0.073900279"
}
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": 152
}

起步布局答案

{
  "centers": [
    [
      "0.762014982",
      "1.648354082"
    ],
    [
      "0.884238781",
      "1.731458524"
    ],
    [
      "0.68681907",
      "0.573373673"
    ],
    [
      "0.774376597",
      "0.101804714"
    ],
    [
      "1.160423659",
      "1.62610537"
    ],
    [
      "1.648394116",
      "1.033062387"
    ],
    [
      "0.638293426",
      "0.864964625"
    ],
    [
      "1.192746022",
      "0.094180116"
    ],
    [
      "0.575314553",
      "0.476359508"
    ],
    [
      "1.240395059",
      "0.365831601"
    ],
    [
      "0.752424602",
      "0.958874584"
    ],
    [
      "0.703898958",
      "1.250465536"
    ],
    [
      "0.232365389",
      "0.877969928"
    ],
    [
      "1.070555305",
      "1.386399935"
    ],
    [
      "1.11660309",
      "1.918702535"
    ],
    [
      "0.890818599",
      "0.906989066"
    ],
    [
      "0.277398217",
      "1.579229968"
    ],
    [
      "1.004949774",
      "1.000899024"
    ],
    [
      "1.721397609",
      "1.445505925"
    ],
    [
      "0.18069795",
      "0.568253729"
    ],
    [
      "1.101682205",
      "0.416858734"
    ],
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      "1.672656938",
      "0.887266911"
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      "1.77255622"
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      "1.456897719"
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      "1.45098011"
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      "0.939152429"
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      "1.208949303",
      "1.334514417"
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      "1.776894632"
    ],
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      "0.920140005",
      "0.077349973"
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      "0.967724217"
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      "1.20362143"
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      "0.751332512"
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      "0.761193589"
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      "1.686808967"
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      "1.1466945"
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      "1.042923464"
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      "1.592966146"
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  "radius": "0.073900279"
}

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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