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 = 161

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 = 161
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[0.0715850344824458584605674055645, 0.0715850344824458584605674055655]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 = 0.071585034

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 = 161 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": [
    [
      "1.066914655",
      "0.074094593"
    ],
    [
      "0.923322232",
      "0.07475686"
    ],
    [
      "1.208713095",
      "0.095349132"
    ],
    [
      "0.781977763",
      "0.097547424"
    ],
    [
      "1.34532157",
      "0.138195288"
    ],
    [
      "0.645817963",
      "0.141798723"
    ],
    [
      "0.994975699",
      "0.198706242"
    ],
    [
      "1.136688269",
      "0.219083093"
    ],
    [
      "0.85363123",
      "0.221496806"
    ],
    [
      "1.56149918",
      "0.260626604"
    ],
    [
      "1.418330943",
      "0.261350881"
    ],
    [
      "1.273296744",
      "0.261929249"
    ],
    [
      "0.57410091",
      "0.265711325"
    ],
    [
      "0.71747143",
      "0.265748105"
    ],
    [
      "0.430932673",
      "0.266435602"
    ],
    [
      "1.068252402",
      "0.344837619"
    ],
    [
      "0.925084165",
      "0.345561897"
    ],
    [
      "1.668501528",
      "0.355748453"
    ],
    [
      "0.324913446",
      "0.362651984"
    ],
    [
      "1.491190062",
      "0.38534347"
    ],
    [
      "1.348021825",
      "0.386067747"
    ],
    [
      "1.204860877",
      "0.387683775"
    ],
    [
      "0.645754377",
      "0.389660707"
    ],
    [
      "0.788924365",
      "0.389813195"
    ],
    [
      "0.50258614",
      "0.390384985"
    ],
    [
      "1.759606603",
      "0.466190899"
    ],
    [
      "0.997295526",
      "0.469187088"
    ],
    [
      "0.234948086",
      "0.474024794"
    ],
    [
      "1.59819241",
      "0.480465319"
    ],
    [
      "0.396566913",
      "0.486601367"
    ],
    [
      "1.420233187",
      "0.509692939"
    ],
    [
      "1.277072238",
      "0.511308966"
    ],
    [
      "1.133904002",
      "0.512033244"
    ],
    [
      "0.861135726",
      "0.513438387"
    ],
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      "0.574797501",
      "0.514010176"
    ],
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      "0.717967489",
      "0.514162664"
    ],
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      "1.689297486",
      "0.590907765"
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      "1.83386424",
      "0.592024787"
    ],
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      "0.306601553",
      "0.597974176"
    ],
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      "0.162671184",
      "0.599183584"
    ],
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      "1.527235534",
      "0.604814788"
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      "0.639313256"
    ],
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      "1.069774676",
      "0.640037534"
    ],
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      "0.926606439",
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    ],
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      "0.640268214",
      "0.6413336"
    ],
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      "0.783438202",
      "0.641486089"
    ],
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      "1.758703838",
      "0.716129276"
    ],
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      "1.615540414",
      "0.717508696"
    ],
    [
      "0.381641958",
      "0.723826431"
    ],
    [
      "0.238471906",
      "0.723894865"
    ],
    [
      "1.463106208",
      "0.732819078"
    ],
    [
      "0.534248987",
      "0.737549982"
    ],
    [
      "1.895986075",
      "0.756764925"
    ],
    [
      "1.285917856",
      "0.762489562"
    ],
    [
      "1.141986037",
      "0.763662725"
    ],
    [
      "0.9988178",
      "0.764387003"
    ],
    [
      "0.855649563",
      "0.76511128"
    ],
    [
      "0.711727262",
      "0.76540223"
    ],
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      "0.101592423",
      "0.765867187"
    ],
    [
      "1.785447765",
      "0.856779307"
    ],
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      "1.642284341",
      "0.858158726"
    ],
    [
      "0.356326297",
      "0.864740539"
    ],
    [
      "0.213156244",
      "0.864808973"
    ],
    [
      "1.499712245",
      "0.871230301"
    ],
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      "0.499027334",
      "0.87631995"
    ],
    [
      "1.357365635",
      "0.886563639"
    ],
    [
      "1.214197398",
      "0.887287916"
    ],
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      "1.071029161",
      "0.888012194"
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    ],
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      "1.927530988",
      "1.040504505"
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      "1.049912187"
    ],
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      "1.102316843"
    ],
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      "1.110429447"
    ],
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      "1.64918133",
      "1.115388418"
    ],
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      "0.352085564",
      "1.122008857"
    ],
    [
      "1.215451884",
      "1.135262577"
    ],
    [
      "1.072283647",
      "1.135986854"
    ],
    [
      "0.92911541",
      "1.136711132"
    ],
    [
      "0.785947173",
      "1.13743541"
    ],
    [
      "1.365628168",
      "1.138808047"
    ],
    [
      "0.635875628",
      "1.142459534"
    ],
    [
      "1.508731881",
      "1.143166482"
    ],
    [
      "0.492824055",
      "1.148283287"
    ],
    [
      "1.910274862",
      "1.182630839"
    ],
    [
      "0.091624765",
      "1.191856144"
    ],
    [
      "1.752850548",
      "1.240100132"
    ],
    [
      "0.249683209",
      "1.247810977"
    ],
    [
      "1.610278452",
      "1.253171707"
    ],
    [
      "1.287663245",
      "1.258887768"
    ],
    [
      "0.392384246",
      "1.259390387"
    ],
    [
      "1.144495008",
      "1.259612046"
    ],
    [
      "1.001326771",
      "1.260336323"
    ],
    [
      "0.858158534",
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      "0.714990297",
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      "1.455724524",
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      "0.54710755",
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      "1.871371984",
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    [
      "0.131923447",
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      "1.577431659"
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      "1.624563498"
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      "0.503547317",
      "1.629651337"
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    [
      "1.217960855",
      "1.631211898"
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    [
      "1.074792618",
      "1.631936175"
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      "0.931624381",
      "1.632660453"
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      "0.788456144",
      "1.63338473"
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      "1.360721603",
      "1.64203028"
    ],
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      "0.645816003",
      "1.645691586"
    ],
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      "1.636464247",
      "1.675919825"
    ],
    [
      "0.370425687",
      "1.682341947"
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      "1.279972235",
      "1.760255523"
    ],
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      "1.760979801"
    ],
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      "0.870958152",
      "1.762341898"
    ],
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      "0.72778994",
      "1.763071066"
    ],
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      "1.524464499",
      "1.766088335"
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    ],
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      "1.837615816"
    ],
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      "0.608112655",
      "1.841652337"
    ],
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    [
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    ]
  ],
  "radius": "0.071585034"
}
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": 161
}

起步布局答案

{
  "centers": [
    [
      "1.066914655",
      "0.074094593"
    ],
    [
      "0.923322232",
      "0.07475686"
    ],
    [
      "1.208713095",
      "0.095349132"
    ],
    [
      "0.781977763",
      "0.097547424"
    ],
    [
      "1.34532157",
      "0.138195288"
    ],
    [
      "0.645817963",
      "0.141798723"
    ],
    [
      "0.994975699",
      "0.198706242"
    ],
    [
      "1.136688269",
      "0.219083093"
    ],
    [
      "0.85363123",
      "0.221496806"
    ],
    [
      "1.56149918",
      "0.260626604"
    ],
    [
      "1.418330943",
      "0.261350881"
    ],
    [
      "1.273296744",
      "0.261929249"
    ],
    [
      "0.57410091",
      "0.265711325"
    ],
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    ],
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      "0.266435602"
    ],
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      "1.068252402",
      "0.344837619"
    ],
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      "0.345561897"
    ],
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      "0.355748453"
    ],
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      "0.324913446",
      "0.362651984"
    ],
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      "1.491190062",
      "0.38534347"
    ],
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      "0.386067747"
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      "0.756764925"
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      "0.763662725"
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      "0.855649563",
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  "radius": "0.071585034"
}

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?