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 = 128

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 = 128
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[0.0799857804565, 0.0799857804575]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 = 0.079985780

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 = 128 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": [
    [
      "0.947028857",
      "0.081511991"
    ],
    [
      "1.169039025",
      "0.095648314"
    ],
    [
      "0.788311487",
      "0.107041544"
    ],
    [
      "1.323136416",
      "0.138600545"
    ],
    [
      "0.60308256",
      "0.170010416"
    ],
    [
      "1.050727915",
      "0.203320881"
    ],
    [
      "1.467464082",
      "0.20759638"
    ],
    [
      "0.892204757",
      "0.231795784"
    ],
    [
      "0.465311096",
      "0.251311854"
    ],
    [
      "0.734865635",
      "0.260697358"
    ],
    [
      "1.317687644",
      "0.298479284"
    ],
    [
      "1.59765841",
      "0.300549795"
    ],
    [
      "1.159284012",
      "0.320821835"
    ],
    [
      "0.995903815",
      "0.353604674"
    ],
    [
      "1.457100484",
      "0.376932841"
    ],
    [
      "0.838564693",
      "0.382506248"
    ],
    [
      "0.37452303",
      "0.383025284"
    ],
    [
      "0.681225571",
      "0.411407821"
    ],
    [
      "0.523886449",
      "0.440309395"
    ],
    [
      "1.257835045",
      "0.446832129"
    ],
    [
      "1.656220347",
      "0.449416849"
    ],
    [
      "1.100276014",
      "0.474962603"
    ],
    [
      "0.244064426",
      "0.475607425"
    ],
    [
      "0.942367867",
      "0.504352153"
    ],
    [
      "1.515662421",
      "0.525799895"
    ],
    [
      "1.792648968",
      "0.532952061"
    ],
    [
      "0.785028745",
      "0.533253727"
    ],
    [
      "0.627689623",
      "0.562155301"
    ],
    [
      "1.361534103",
      "0.568641019"
    ],
    [
      "0.470350501",
      "0.591056875"
    ],
    [
      "1.203406048",
      "0.597259469"
    ],
    [
      "1.652091042",
      "0.609335107"
    ],
    [
      "0.313011379",
      "0.619958449"
    ],
    [
      "1.046066926",
      "0.626161043"
    ],
    [
      "0.153561475",
      "0.640905268"
    ],
    [
      "0.888727804",
      "0.655062617"
    ],
    [
      "1.86156901",
      "0.677315935"
    ],
    [
      "1.476120468",
      "0.680882013"
    ],
    [
      "0.731388681",
      "0.683964191"
    ],
    [
      "0.574049559",
      "0.712865765"
    ],
    [
      "1.307105106",
      "0.719068359"
    ],
    [
      "0.416710437",
      "0.741767338"
    ],
    [
      "1.149765984",
      "0.747969933"
    ],
    [
      "1.717711493",
      "0.755228413"
    ],
    [
      "0.259371315",
      "0.770668912"
    ],
    [
      "0.992426862",
      "0.776871507"
    ],
    [
      "0.102055354",
      "0.799696289"
    ],
    [
      "0.83508774",
      "0.805773081"
    ],
    [
      "1.568143286",
      "0.811975675"
    ],
    [
      "0.677748618",
      "0.834674654"
    ],
    [
      "1.855110886",
      "0.837157084"
    ],
    [
      "1.410804164",
      "0.840877249"
    ],
    [
      "0.520409496",
      "0.863576228"
    ],
    [
      "1.253465042",
      "0.869778823"
    ],
    [
      "0.363070373",
      "0.892477802"
    ],
    [
      "1.09612592",
      "0.898680396"
    ],
    [
      "1.693688099",
      "0.915467419"
    ],
    [
      "0.205754412",
      "0.921505179"
    ],
    [
      "0.938786798",
      "0.92758197"
    ],
    [
      "0.781447676",
      "0.956483544"
    ],
    [
      "1.514503222",
      "0.962686139"
    ],
    [
      "1.919865114",
      "0.983436919"
    ],
    [
      "0.624108554",
      "0.985385118"
    ],
    [
      "1.3571641",
      "0.991587712"
    ],
    [
      "0.466769432",
      "1.014286692"
    ],
    [
      "1.199824978",
      "1.020489286"
    ],
    [
      "0.080217756",
      "1.020658839"
    ],
    [
      "0.306237",
      "1.045980677"
    ],
    [
      "1.042485856",
      "1.04939086"
    ],
    [
      "1.776137499",
      "1.053674181"
    ],
    [
      "0.885146734",
      "1.078292434"
    ],
    [
      "1.619026059",
      "1.083788892"
    ],
    [
      "0.727807612",
      "1.107194008"
    ],
    [
      "1.461883849",
      "1.113760321"
    ],
    [
      "0.57046849",
      "1.136095582"
    ],
    [
      "1.303524036",
      "1.142298176"
    ],
    [
      "1.90882856",
      "1.143027316"
    ],
    [
      "0.413129368",
      "1.164997156"
    ],
    [
      "1.146184914",
      "1.17119975"
    ],
    [
      "0.097700678",
      "1.179672196"
    ],
    [
      "0.256611891",
      "1.198060422"
    ],
    [
      "0.988845792",
      "1.200101324"
    ],
    [
      "1.729338699",
      "1.208430408"
    ],
    [
      "0.83150667",
      "1.229002898"
    ],
    [
      "1.56888512",
      "1.235699347"
    ],
    [
      "0.674167548",
      "1.257904471"
    ],
    [
      "1.407223094",
      "1.264107066"
    ],
    [
      "0.516828426",
      "1.286806045"
    ],
    [
      "1.249883972",
      "1.29300864"
    ],
    [
      "1.870314427",
      "1.298293416"
    ],
    [
      "1.09254485",
      "1.321910214"
    ],
    [
      "0.142463773",
      "1.333253333"
    ],
    [
      "0.309483194",
      "1.349042294"
    ],
    [
      "0.935205728",
      "1.350811787"
    ],
    [
      "1.669777173",
      "1.359843192"
    ],
    [
      "0.777866606",
      "1.379713361"
    ],
    [
      "1.511819423",
      "1.38514635"
    ],
    [
      "0.612164621",
      "1.415897227"
    ],
    [
      "1.354680336",
      "1.416080465"
    ],
    [
      "0.447018912",
      "1.430741906"
    ],
    [
      "1.196243909",
      "1.443719103"
    ],
    [
      "1.805487155",
      "1.444540895"
    ],
    [
      "1.039176654",
      "1.474142033"
    ],
    [
      "0.213153671",
      "1.476758868"
    ],
    [
      "0.881565664",
      "1.501522251"
    ],
    [
      "0.724226542",
      "1.530423825"
    ],
    [
      "1.561448682",
      "1.537224741"
    ],
    [
      "0.350689389",
      "1.55845848"
    ],
    [
      "1.298031053",
      "1.567130114"
    ],
    [
      "1.716306735",
      "1.577348098"
    ],
    [
      "0.509459948",
      "1.578024007"
    ],
    [
      "1.141295357",
      "1.599142957"
    ],
    [
      "0.983408866",
      "1.624887005"
    ],
    [
      "1.439842012",
      "1.641160866"
    ],
    [
      "0.641978745",
      "1.667632426"
    ],
    [
      "0.805047385",
      "1.669641294"
    ],
    [
      "1.595224117",
      "1.701464647"
    ],
    [
      "0.410894307",
      "1.706668696"
    ],
    [
      "1.24738714",
      "1.718873629"
    ],
    [
      "1.089932884",
      "1.759082425"
    ],
    [
      "0.92691386",
      "1.774550677"
    ],
    [
      "0.542209737",
      "1.798031478"
    ],
    [
      "1.453352422",
      "1.800560894"
    ],
    [
      "0.701139038",
      "1.816262705"
    ],
    [
      "1.307825071",
      "1.866988979"
    ],
    [
      "0.834267116",
      "1.904963411"
    ],
    [
      "1.152990918",
      "1.907204466"
    ],
    [
      "0.993531228",
      "1.919991478"
    ]
  ],
  "radius": "0.07998578"
}
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": 128
}

起步布局答案

{
  "centers": [
    [
      "0.947028857",
      "0.081511991"
    ],
    [
      "1.169039025",
      "0.095648314"
    ],
    [
      "0.788311487",
      "0.107041544"
    ],
    [
      "1.323136416",
      "0.138600545"
    ],
    [
      "0.60308256",
      "0.170010416"
    ],
    [
      "1.050727915",
      "0.203320881"
    ],
    [
      "1.467464082",
      "0.20759638"
    ],
    [
      "0.892204757",
      "0.231795784"
    ],
    [
      "0.465311096",
      "0.251311854"
    ],
    [
      "0.734865635",
      "0.260697358"
    ],
    [
      "1.317687644",
      "0.298479284"
    ],
    [
      "1.59765841",
      "0.300549795"
    ],
    [
      "1.159284012",
      "0.320821835"
    ],
    [
      "0.995903815",
      "0.353604674"
    ],
    [
      "1.457100484",
      "0.376932841"
    ],
    [
      "0.838564693",
      "0.382506248"
    ],
    [
      "0.37452303",
      "0.383025284"
    ],
    [
      "0.681225571",
      "0.411407821"
    ],
    [
      "0.523886449",
      "0.440309395"
    ],
    [
      "1.257835045",
      "0.446832129"
    ],
    [
      "1.656220347",
      "0.449416849"
    ],
    [
      "1.100276014",
      "0.474962603"
    ],
    [
      "0.244064426",
      "0.475607425"
    ],
    [
      "0.942367867",
      "0.504352153"
    ],
    [
      "1.515662421",
      "0.525799895"
    ],
    [
      "1.792648968",
      "0.532952061"
    ],
    [
      "0.785028745",
      "0.533253727"
    ],
    [
      "0.627689623",
      "0.562155301"
    ],
    [
      "1.361534103",
      "0.568641019"
    ],
    [
      "0.470350501",
      "0.591056875"
    ],
    [
      "1.203406048",
      "0.597259469"
    ],
    [
      "1.652091042",
      "0.609335107"
    ],
    [
      "0.313011379",
      "0.619958449"
    ],
    [
      "1.046066926",
      "0.626161043"
    ],
    [
      "0.153561475",
      "0.640905268"
    ],
    [
      "0.888727804",
      "0.655062617"
    ],
    [
      "1.86156901",
      "0.677315935"
    ],
    [
      "1.476120468",
      "0.680882013"
    ],
    [
      "0.731388681",
      "0.683964191"
    ],
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      "0.574049559",
      "0.712865765"
    ],
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      "1.307105106",
      "0.719068359"
    ],
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      "0.741767338"
    ],
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      "0.747969933"
    ],
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      "0.755228413"
    ],
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      "0.259371315",
      "0.770668912"
    ],
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      "0.776871507"
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      "0.805773081"
    ],
    [
      "1.568143286",
      "0.811975675"
    ],
    [
      "0.677748618",
      "0.834674654"
    ],
    [
      "1.855110886",
      "0.837157084"
    ],
    [
      "1.410804164",
      "0.840877249"
    ],
    [
      "0.520409496",
      "0.863576228"
    ],
    [
      "1.253465042",
      "0.869778823"
    ],
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      "0.363070373",
      "0.892477802"
    ],
    [
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      "0.898680396"
    ],
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      "0.915467419"
    ],
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      "0.921505179"
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      "0.956483544"
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      "0.962686139"
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      "1.235699347"
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      "1.257904471"
    ],
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      "1.264107066"
    ],
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      "1.286806045"
    ],
    [
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      "1.29300864"
    ],
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      "1.870314427",
      "1.298293416"
    ],
    [
      "1.09254485",
      "1.321910214"
    ],
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      "1.333253333"
    ],
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      "1.349042294"
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      "1.350811787"
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      "1.359843192"
    ],
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      "1.379713361"
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      "1.38514635"
    ],
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      "1.415897227"
    ],
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      "1.354680336",
      "1.416080465"
    ],
    [
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      "1.430741906"
    ],
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      "1.196243909",
      "1.443719103"
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    [
      "1.805487155",
      "1.444540895"
    ],
    [
      "1.039176654",
      "1.474142033"
    ],
    [
      "0.213153671",
      "1.476758868"
    ],
    [
      "0.881565664",
      "1.501522251"
    ],
    [
      "0.724226542",
      "1.530423825"
    ],
    [
      "1.561448682",
      "1.537224741"
    ],
    [
      "0.350689389",
      "1.55845848"
    ],
    [
      "1.298031053",
      "1.567130114"
    ],
    [
      "1.716306735",
      "1.577348098"
    ],
    [
      "0.509459948",
      "1.578024007"
    ],
    [
      "1.141295357",
      "1.599142957"
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    [
      "0.983408866",
      "1.624887005"
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    [
      "1.439842012",
      "1.641160866"
    ],
    [
      "0.641978745",
      "1.667632426"
    ],
    [
      "0.805047385",
      "1.669641294"
    ],
    [
      "1.595224117",
      "1.701464647"
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    [
      "0.410894307",
      "1.706668696"
    ],
    [
      "1.24738714",
      "1.718873629"
    ],
    [
      "1.089932884",
      "1.759082425"
    ],
    [
      "0.92691386",
      "1.774550677"
    ],
    [
      "0.542209737",
      "1.798031478"
    ],
    [
      "1.453352422",
      "1.800560894"
    ],
    [
      "0.701139038",
      "1.816262705"
    ],
    [
      "1.307825071",
      "1.866988979"
    ],
    [
      "0.834267116",
      "1.904963411"
    ],
    [
      "1.152990918",
      "1.907204466"
    ],
    [
      "0.993531228",
      "1.919991478"
    ]
  ],
  "radius": "0.07998578"
}

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

DISCUSSION

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聊思路、贴方法、问为什么卡住。所有登录用户都可以发帖;发言公开署名,与纪录使用同一个名字,署名后的 #编号是账号注册序号,冒不了名。新发言经自动审核后公开。

还没有帖子。第一个聊聊这道题的思路?