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 = 202

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 = 202
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[0.0642342078415, 0.0642342078425]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 = 0.064234207

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 = 202 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": [
    [
      "0.974344273",
      "0.064585973"
    ],
    [
      "1.176946107",
      "0.081116061"
    ],
    [
      "0.742871369",
      "0.100254211"
    ],
    [
      "1.301131794",
      "0.114010575"
    ],
    [
      "0.866653773",
      "0.13463517"
    ],
    [
      "0.622062531",
      "0.143950418"
    ],
    [
      "1.069255607",
      "0.151165258"
    ],
    [
      "1.422040807",
      "0.165123524"
    ],
    [
      "0.508376946",
      "0.203781192"
    ],
    [
      "1.183765241",
      "0.209403369"
    ],
    [
      "0.961565107",
      "0.221214455"
    ],
    [
      "0.73155413",
      "0.228223168"
    ],
    [
      "1.535392882",
      "0.232528776"
    ],
    [
      "1.307950928",
      "0.242297883"
    ],
    [
      "0.613164365",
      "0.278102997"
    ],
    [
      "0.403957325",
      "0.278618861"
    ],
    [
      "1.076074741",
      "0.279452565"
    ],
    [
      "0.848300098",
      "0.281837649"
    ],
    [
      "1.424227926",
      "0.296921993"
    ],
    [
      "1.635462663",
      "0.313090384"
    ],
    [
      "1.190584375",
      "0.337690676"
    ],
    [
      "0.968355993",
      "0.349503261"
    ],
    [
      "0.508744744",
      "0.352940666"
    ],
    [
      "0.743495611",
      "0.356135383"
    ],
    [
      "0.31077174",
      "0.367052907"
    ],
    [
      "1.314770061",
      "0.37058519"
    ],
    [
      "1.524297708",
      "0.377483601"
    ],
    [
      "0.625409888",
      "0.406730796"
    ],
    [
      "1.082865627",
      "0.407741371"
    ],
    [
      "0.857755701",
      "0.414861575"
    ],
    [
      "1.671563476",
      "0.436382172"
    ],
    [
      "0.415559158",
      "0.441374712"
    ],
    [
      "1.417089806",
      "0.44826919"
    ],
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      "0.465979482"
    ],
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      "0.230576523",
      "0.467416551"
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      "0.975255077",
      "0.477913326"
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      "0.749767282",
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      "0.498416131"
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      "1.64774897",
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    ],
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      "0.428223792",
      "0.569217353"
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      "0.299912424",
      "0.575567679"
    ],
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      "0.16488317",
      "0.577818169"
    ],
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      "1.427696727",
      "0.579859056"
    ],
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      "1.205644325",
      "0.594181496"
    ],
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      "0.982123443",
      "0.606301885"
    ],
    [
      "1.762909703",
      "0.619563803"
    ],
    [
      "0.761671078",
      "0.620047482"
    ],
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      "0.540285878",
      "0.633039945"
    ],
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      "1.542857461",
      "0.636798844"
    ],
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      "1.320805059",
      "0.651121283"
    ],
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      "1.098033775",
      "0.664353451"
    ],
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      "1.878070436",
      "0.676503591"
    ],
    [
      "0.875981373",
      "0.67867589"
    ],
    [
      "0.234219072",
      "0.685969297"
    ],
    [
      "0.654779409",
      "0.691309709"
    ],
    [
      "1.658018194",
      "0.693738631"
    ],
    [
      "0.429265173",
      "0.697681548"
    ],
    [
      "1.435965792",
      "0.708061071"
    ],
    [
      "0.108913925",
      "0.714302214"
    ],
    [
      "1.21391339",
      "0.72238351"
    ],
    [
      "0.991142106",
      "0.735615677"
    ],
    [
      "0.769089704",
      "0.749938117"
    ],
    [
      "1.773178927",
      "0.750678418"
    ],
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      "0.547069548",
      "0.761329133"
    ],
    [
      "1.551126525",
      "0.765000858"
    ],
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      "0.326741802",
      "0.775096612"
    ],
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      "1.329074123",
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    ],
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      "1.106302839",
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      "1.914102705",
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      "1.222182454",
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      "1.559395589",
      "0.893202872"
    ],
    [
      "0.333020309",
      "0.903411514"
    ],
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      "1.337343187",
      "0.907525311"
    ],
    [
      "1.114571903",
      "0.920757479"
    ],
    [
      "1.932901984",
      "0.928469586"
    ],
    [
      "0.892519501",
      "0.935079918"
    ],
    [
      "0.190605284",
      "0.940679666"
    ],
    [
      "1.702319527",
      "0.945252483"
    ],
    [
      "0.670467099",
      "0.949402358"
    ],
    [
      "0.447339039",
      "0.962638076"
    ],
    [
      "1.45250392",
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      "0.0648229",
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    ],
    [
      "1.230451518",
      "0.978787538"
    ],
    [
      "1.007680235",
      "0.992019705"
    ],
    [
      "1.819210759",
      "1.002068914"
    ],
    [
      "0.785627833",
      "1.006342145"
    ],
    [
      "0.563575431",
      "1.020664584"
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    [
      "1.567664653",
      "1.021404886"
    ],
    [
      "0.288468737",
      "1.023907536"
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    [
      "1.345612251",
      "1.035727325"
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      "1.122840968",
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      "1.933835042",
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      "1.248423734"
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      "0.346747622",
      "1.270444463"
    ],
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      "0.580113559",
      "1.277068612"
    ],
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      "1.584363946",
      "1.277872917"
    ],
    [
      "1.36215038",
      "1.292131353"
    ],
    [
      "1.139379096",
      "1.305363521"
    ],
    [
      "0.917326694",
      "1.31968596"
    ],
    [
      "1.702401896",
      "1.328992332"
    ],
    [
      "0.695274292",
      "1.3340084"
    ],
    [
      "0.12985082",
      "1.344235416"
    ],
    [
      "0.452160509",
      "1.347992433"
    ],
    [
      "1.477311113",
      "1.349071141"
    ],
    [
      "0.25702939",
      "1.362394301"
    ],
    [
      "1.255258711",
      "1.36339358"
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    [
      "1.032487427",
      "1.376625748"
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    [
      "1.85655481",
      "1.376791024"
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    [
      "0.810435025",
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    [
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    [
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      "0.353903536",
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      "1.590412428"
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      "1.638642619"
    ],
    [
      "0.494772907",
      "1.648076458"
    ],
    [
      "0.815027729",
      "1.667051377"
    ],
    [
      "1.651441552",
      "1.671774905"
    ],
    [
      "0.619320754",
      "1.679571978"
    ],
    [
      "1.173273757",
      "1.689058101"
    ],
    [
      "1.406246562",
      "1.693633454"
    ],
    [
      "0.374532232",
      "1.696022764"
    ],
    [
      "0.937948004",
      "1.704493499"
    ],
    [
      "1.536177831",
      "1.728505923"
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      "1.282812432",
      "1.75618047"
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      "0.844093733",
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      "1.447746304",
      "1.8216939"
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      "1.330985059",
      "1.875275104"
    ],
    [
      "0.700806594",
      "1.886645884"
    ],
    [
      "1.207985506",
      "1.912359385"
    ],
    [
      "0.825063924",
      "1.919268724"
    ],
    [
      "1.081065905",
      "1.932247787"
    ],
    [
      "0.952618397",
      "1.934565461"
    ]
  ],
  "radius": "0.064234207"
}
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": 202
}

起步布局答案

{
  "centers": [
    [
      "0.974344273",
      "0.064585973"
    ],
    [
      "1.176946107",
      "0.081116061"
    ],
    [
      "0.742871369",
      "0.100254211"
    ],
    [
      "1.301131794",
      "0.114010575"
    ],
    [
      "0.866653773",
      "0.13463517"
    ],
    [
      "0.622062531",
      "0.143950418"
    ],
    [
      "1.069255607",
      "0.151165258"
    ],
    [
      "1.422040807",
      "0.165123524"
    ],
    [
      "0.508376946",
      "0.203781192"
    ],
    [
      "1.183765241",
      "0.209403369"
    ],
    [
      "0.961565107",
      "0.221214455"
    ],
    [
      "0.73155413",
      "0.228223168"
    ],
    [
      "1.535392882",
      "0.232528776"
    ],
    [
      "1.307950928",
      "0.242297883"
    ],
    [
      "0.613164365",
      "0.278102997"
    ],
    [
      "0.403957325",
      "0.278618861"
    ],
    [
      "1.076074741",
      "0.279452565"
    ],
    [
      "0.848300098",
      "0.281837649"
    ],
    [
      "1.424227926",
      "0.296921993"
    ],
    [
      "1.635462663",
      "0.313090384"
    ],
    [
      "1.190584375",
      "0.337690676"
    ],
    [
      "0.968355993",
      "0.349503261"
    ],
    [
      "0.508744744",
      "0.352940666"
    ],
    [
      "0.743495611",
      "0.356135383"
    ],
    [
      "0.31077174",
      "0.367052907"
    ],
    [
      "1.314770061",
      "0.37058519"
    ],
    [
      "1.524297708",
      "0.377483601"
    ],
    [
      "0.625409888",
      "0.406730796"
    ],
    [
      "1.082865627",
      "0.407741371"
    ],
    [
      "0.857755701",
      "0.414861575"
    ],
    [
      "1.671563476",
      "0.436382172"
    ],
    [
      "0.415559158",
      "0.441374712"
    ],
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      "1.417089806",
      "0.44826919"
    ],
    [
      "1.197375261",
      "0.465979482"
    ],
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      "0.230576523",
      "0.467416551"
    ],
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      "0.975255077",
      "0.477913326"
    ],
    [
      "0.749767282",
      "0.48445062"
    ],
    [
      "1.786724209",
      "0.49332196"
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  "radius": "0.064234207"
}

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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