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 = 148

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 = 148
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[0.0746996735751545854300305171975, 0.0746996735751545854300305171985]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 = 0.074699673

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 = 148 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": [
    [
      "1.022186361",
      "0.074965698"
    ],
    [
      "0.872064963",
      "0.083592275"
    ],
    [
      "1.170766061",
      "0.090593795"
    ],
    [
      "0.723937094",
      "0.116840917"
    ],
    [
      "1.391533452",
      "0.161619269"
    ],
    [
      "0.954534675",
      "0.208170106"
    ],
    [
      "0.511088915",
      "0.214413184"
    ],
    [
      "1.25229862",
      "0.215784077"
    ],
    [
      "1.103114376",
      "0.223798203"
    ],
    [
      "0.808557765",
      "0.239964866"
    ],
    [
      "0.656188911",
      "0.249996272"
    ],
    [
      "1.473547361",
      "0.286494742"
    ],
    [
      "1.620473758",
      "0.313564999"
    ],
    [
      "1.334312528",
      "0.34065955"
    ],
    [
      "0.443437229",
      "0.347617592"
    ],
    [
      "1.185160242",
      "0.349247942"
    ],
    [
      "1.035975998",
      "0.357262067"
    ],
    [
      "0.886905714",
      "0.367172429"
    ],
    [
      "0.73772147",
      "0.375186554"
    ],
    [
      "0.588537225",
      "0.38320068"
    ],
    [
      "0.3012389",
      "0.393441166"
    ],
    [
      "1.722856354",
      "0.42236743"
    ],
    [
      "1.563449306",
      "0.451653293"
    ],
    [
      "1.414792406",
      "0.466529142"
    ],
    [
      "1.26564012",
      "0.475117534"
    ],
    [
      "1.116455875",
      "0.48313166"
    ],
    [
      "0.967385591",
      "0.493042021"
    ],
    [
      "0.818201347",
      "0.501056147"
    ],
    [
      "0.669017103",
      "0.509070272"
    ],
    [
      "0.212731499",
      "0.513801479"
    ],
    [
      "0.518505869",
      "0.515522806"
    ],
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      "0.36681994",
      "0.527677123"
    ],
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      "1.665831902",
      "0.560455724"
    ],
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      "0.562973393"
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      "0.58783187"
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      "0.608321942"
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      "0.618232303"
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      "0.899733906",
      "0.626246429"
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      "0.750549661",
      "0.634260555"
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      "0.601365417",
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      "0.131929968",
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      "1.725978694",
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    ],
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      "1.577836649",
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    ],
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      "1.428705237",
      "0.725498099"
    ],
    [
      "1.279520993",
      "0.733512225"
    ],
    [
      "1.130450708",
      "0.743422586"
    ],
    [
      "0.981266464",
      "0.751436712"
    ],
    [
      "0.83208222",
      "0.759450837"
    ],
    [
      "0.682897976",
      "0.767464963"
    ],
    [
      "0.533713731",
      "0.775479089"
    ],
    [
      "0.384529487",
      "0.783493214"
    ],
    [
      "0.235324869",
      "0.791118619"
    ],
    [
      "0.09070908",
      "0.828621132"
    ],
    [
      "1.808553452",
      "0.83373072"
    ],
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      "1.659369207",
      "0.841744846"
    ],
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      "1.510237795",
      "0.850688381"
    ],
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      "1.361053551",
      "0.858702507"
    ],
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      "1.211983267",
      "0.868612868"
    ],
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      "1.062799023",
      "0.876626994"
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      "0.913614778",
      "0.88464112"
    ],
    [
      "0.764430534",
      "0.892655246"
    ],
    [
      "0.61524629",
      "0.900669371"
    ],
    [
      "0.466062046",
      "0.908683497"
    ],
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      "0.316877802",
      "0.916697623"
    ],
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      "1.922661244",
      "0.930165032"
    ],
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      "1.740901766",
      "0.966935128"
    ],
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      "1.591770354",
      "0.975878664"
    ],
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      "1.44258611",
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    ],
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      "1.293515825",
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      "0.215900893",
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    ],
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      "1.063369441"
    ],
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      "1.675276238",
      "1.10168425"
    ],
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      "1.524118668",
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      "1.12635404"
    ],
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    ],
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      "1.076679895",
      "1.135021685"
    ],
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      "0.927495651",
      "1.14303581"
    ],
    [
      "0.778311407",
      "1.151049936"
    ],
    [
      "0.297433451",
      "1.151996164"
    ],
    [
      "0.629127163",
      "1.159064062"
    ],
    [
      "0.479942919",
      "1.167078188"
    ],
    [
      "1.902284922",
      "1.205091722"
    ],
    [
      "1.754992023",
      "1.230091059"
    ],
    [
      "1.605651227",
      "1.234273355"
    ],
    [
      "1.456580942",
      "1.244183716"
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    [
      "1.307396698",
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    [
      "1.158212454",
      "1.260211967"
    ],
    [
      "1.00902821",
      "1.268226093"
    ],
    [
      "0.115650024",
      "1.272223832"
    ],
    [
      "0.859843966",
      "1.276240219"
    ],
    [
      "0.710659721",
      "1.284254344"
    ],
    [
      "0.561475477",
      "1.29226847"
    ],
    [
      "0.262920584",
      "1.297354435"
    ],
    [
      "0.412291233",
      "1.300282596"
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    [
      "1.85751778",
      "1.347626166"
    ],
    [
      "1.539698709",
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    [
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      "1.675237544",
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      "1.104441641",
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      "1.292798903",
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      "1.675415104"
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      "1.701189356"
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      "1.733063203"
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      "1.19527219",
      "1.770427997"
    ],
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      "0.516964391",
      "1.789213086"
    ],
    [
      "1.047653447",
      "1.793425205"
    ],
    [
      "0.898322651",
      "1.797950496"
    ],
    [
      "1.43928887",
      "1.814374596"
    ],
    [
      "0.66208999",
      "1.824691605"
    ],
    [
      "1.302502805",
      "1.874455686"
    ],
    [
      "0.790385544",
      "1.901244958"
    ],
    [
      "1.140198456",
      "1.914570558"
    ],
    [
      "0.976907065",
      "1.925012114"
    ]
  ],
  "radius": "0.074699673"
}
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": 148
}

起步布局答案

{
  "centers": [
    [
      "1.022186361",
      "0.074965698"
    ],
    [
      "0.872064963",
      "0.083592275"
    ],
    [
      "1.170766061",
      "0.090593795"
    ],
    [
      "0.723937094",
      "0.116840917"
    ],
    [
      "1.391533452",
      "0.161619269"
    ],
    [
      "0.954534675",
      "0.208170106"
    ],
    [
      "0.511088915",
      "0.214413184"
    ],
    [
      "1.25229862",
      "0.215784077"
    ],
    [
      "1.103114376",
      "0.223798203"
    ],
    [
      "0.808557765",
      "0.239964866"
    ],
    [
      "0.656188911",
      "0.249996272"
    ],
    [
      "1.473547361",
      "0.286494742"
    ],
    [
      "1.620473758",
      "0.313564999"
    ],
    [
      "1.334312528",
      "0.34065955"
    ],
    [
      "0.443437229",
      "0.347617592"
    ],
    [
      "1.185160242",
      "0.349247942"
    ],
    [
      "1.035975998",
      "0.357262067"
    ],
    [
      "0.886905714",
      "0.367172429"
    ],
    [
      "0.73772147",
      "0.375186554"
    ],
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      "0.38320068"
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      "0.42236743"
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  "radius": "0.074699673"
}

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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