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 = 103

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 = 103
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[0.0887693968706772630214989574745, 0.0887693968706772630214989574755]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 = 0.088769396

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 = 103 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": [
    [
      "1.014703163",
      "0.088888026"
    ],
    [
      "0.837752741",
      "0.103330028"
    ],
    [
      "1.191095448",
      "0.109032132"
    ],
    [
      "0.666961278",
      "0.151809914"
    ],
    [
      "1.360233688",
      "0.162997669"
    ],
    [
      "0.508812073",
      "0.232487373"
    ],
    [
      "0.935113133",
      "0.251792068"
    ],
    [
      "1.111916332",
      "0.267936792"
    ],
    [
      "1.458259602",
      "0.311021124"
    ],
    [
      "1.281054572",
      "0.321902329"
    ],
    [
      "0.369308522",
      "0.342299862"
    ],
    [
      "1.63267002",
      "0.344202884"
    ],
    [
      "0.641891883",
      "0.350002425"
    ],
    [
      "0.816215357",
      "0.38363797"
    ],
    [
      "0.989846146",
      "0.420683514"
    ],
    [
      "1.163966031",
      "0.455357553"
    ],
    [
      "0.502388333",
      "0.459814913"
    ],
    [
      "0.253746226",
      "0.477078862"
    ],
    [
      "1.515290946",
      "0.479150388"
    ],
    [
      "1.747825754",
      "0.479329419"
    ],
    [
      "1.338085916",
      "0.490031592"
    ],
    [
      "0.698099887",
      "0.51840873"
    ],
    [
      "0.872120592",
      "0.553577145"
    ],
    [
      "1.04687749",
      "0.588812777"
    ],
    [
      "0.377542361",
      "0.606392588"
    ],
    [
      "1.63145505",
      "0.613411045"
    ],
    [
      "1.220997375",
      "0.623486816"
    ],
    [
      "0.558596336",
      "0.628221218"
    ],
    [
      "0.166511966",
      "0.631708119"
    ],
    [
      "1.83493336",
      "0.635424004"
    ],
    [
      "1.395122654",
      "0.658165571"
    ],
    [
      "0.754653498",
      "0.686699289"
    ],
    [
      "0.929151936",
      "0.721706408"
    ],
    [
      "1.103908834",
      "0.75694204"
    ],
    [
      "0.288684632",
      "0.760525288"
    ],
    [
      "1.537598305",
      "0.764112521"
    ],
    [
      "1.716366436",
      "0.769327946"
    ],
    [
      "0.465235321",
      "0.779230305"
    ],
    [
      "1.278028719",
      "0.791616079"
    ],
    [
      "0.110917184",
      "0.800317857"
    ],
    [
      "1.890205445",
      "0.805383766"
    ],
    [
      "0.637919305",
      "0.820464573"
    ],
    [
      "0.812038888",
      "0.855140127"
    ],
    [
      "0.98618328",
      "0.889835671"
    ],
    [
      "1.420495789",
      "0.897555482"
    ],
    [
      "1.1609093",
      "0.925081774"
    ],
    [
      "1.594612041",
      "0.932247756"
    ],
    [
      "0.331462281",
      "0.932833431"
    ],
    [
      "1.772060683",
      "0.937904854"
    ],
    [
      "0.515867388",
      "0.949396156"
    ],
    [
      "0.089072276",
      "0.976507593"
    ],
    [
      "0.694949186",
      "0.988594333"
    ],
    [
      "0.869070232",
      "1.02326939"
    ],
    [
      "1.303393272",
      "1.030998444"
    ],
    [
      "1.909866555",
      "1.049840394"
    ],
    [
      "1.043183746",
      "1.057975405"
    ],
    [
      "1.477509525",
      "1.065690717"
    ],
    [
      "0.232944851",
      "1.080530213"
    ],
    [
      "1.678437172",
      "1.088751337"
    ],
    [
      "0.410112731",
      "1.092000419"
    ],
    [
      "0.584110184",
      "1.127283694"
    ],
    [
      "0.758230053",
      "1.161957811"
    ],
    [
      "1.185667718",
      "1.163892075"
    ],
    [
      "0.932343568",
      "1.196663826"
    ],
    [
      "1.360405444",
      "1.199134209"
    ],
    [
      "0.114067229",
      "1.212622974"
    ],
    [
      "1.882932718",
      "1.225324273"
    ],
    [
      "1.534521696",
      "1.233826482"
    ],
    [
      "0.289865743",
      "1.248696902"
    ],
    [
      "0.466997136",
      "1.260717413"
    ],
    [
      "1.709556079",
      "1.263541602"
    ],
    [
      "0.64111065",
      "1.295423428"
    ],
    [
      "1.07482754",
      "1.302580496"
    ],
    [
      "0.81523052",
      "1.330097545"
    ],
    [
      "1.248935052",
      "1.337316607"
    ],
    [
      "0.172773739",
      "1.382149088"
    ],
    [
      "1.43861404",
      "1.383231116"
    ],
    [
      "1.81594904",
      "1.40567028"
    ],
    [
      "1.613426398",
      "1.41477143"
    ],
    [
      "0.346892724",
      "1.416827645"
    ],
    [
      "0.524022049",
      "1.428878599"
    ],
    [
      "0.957714492",
      "1.436014215"
    ],
    [
      "0.698141919",
      "1.463552716"
    ],
    [
      "1.134183472",
      "1.472786522"
    ],
    [
      "1.30887965",
      "1.504429349"
    ],
    [
      "1.478707593",
      "1.556183509"
    ],
    [
      "1.721799725",
      "1.556189148"
    ],
    [
      "0.840625891",
      "1.569469386"
    ],
    [
      "0.305018871",
      "1.589357653"
    ],
    [
      "1.01474576",
      "1.604143503"
    ],
    [
      "0.573905039",
      "1.60472967"
    ],
    [
      "1.314311823",
      "1.681885019"
    ],
    [
      "1.600250657",
      "1.685594895"
    ],
    [
      "0.72353729",
      "1.702924558"
    ],
    [
      "0.432490536",
      "1.712933532"
    ],
    [
      "1.142667597",
      "1.72725332"
    ],
    [
      "0.897657159",
      "1.737598675"
    ],
    [
      "1.455915879",
      "1.788975236"
    ],
    [
      "0.581505046",
      "1.809446221"
    ],
    [
      "1.293839137",
      "1.862406918"
    ],
    [
      "0.765211699",
      "1.87922109"
    ],
    [
      "1.119830282",
      "1.903317173"
    ],
    [
      "0.942395999",
      "1.909408044"
    ]
  ],
  "radius": "0.088769396"
}
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": 103
}

起步布局答案

{
  "centers": [
    [
      "1.014703163",
      "0.088888026"
    ],
    [
      "0.837752741",
      "0.103330028"
    ],
    [
      "1.191095448",
      "0.109032132"
    ],
    [
      "0.666961278",
      "0.151809914"
    ],
    [
      "1.360233688",
      "0.162997669"
    ],
    [
      "0.508812073",
      "0.232487373"
    ],
    [
      "0.935113133",
      "0.251792068"
    ],
    [
      "1.111916332",
      "0.267936792"
    ],
    [
      "1.458259602",
      "0.311021124"
    ],
    [
      "1.281054572",
      "0.321902329"
    ],
    [
      "0.369308522",
      "0.342299862"
    ],
    [
      "1.63267002",
      "0.344202884"
    ],
    [
      "0.641891883",
      "0.350002425"
    ],
    [
      "0.816215357",
      "0.38363797"
    ],
    [
      "0.989846146",
      "0.420683514"
    ],
    [
      "1.163966031",
      "0.455357553"
    ],
    [
      "0.502388333",
      "0.459814913"
    ],
    [
      "0.253746226",
      "0.477078862"
    ],
    [
      "1.515290946",
      "0.479150388"
    ],
    [
      "1.747825754",
      "0.479329419"
    ],
    [
      "1.338085916",
      "0.490031592"
    ],
    [
      "0.698099887",
      "0.51840873"
    ],
    [
      "0.872120592",
      "0.553577145"
    ],
    [
      "1.04687749",
      "0.588812777"
    ],
    [
      "0.377542361",
      "0.606392588"
    ],
    [
      "1.63145505",
      "0.613411045"
    ],
    [
      "1.220997375",
      "0.623486816"
    ],
    [
      "0.558596336",
      "0.628221218"
    ],
    [
      "0.166511966",
      "0.631708119"
    ],
    [
      "1.83493336",
      "0.635424004"
    ],
    [
      "1.395122654",
      "0.658165571"
    ],
    [
      "0.754653498",
      "0.686699289"
    ],
    [
      "0.929151936",
      "0.721706408"
    ],
    [
      "1.103908834",
      "0.75694204"
    ],
    [
      "0.288684632",
      "0.760525288"
    ],
    [
      "1.537598305",
      "0.764112521"
    ],
    [
      "1.716366436",
      "0.769327946"
    ],
    [
      "0.465235321",
      "0.779230305"
    ],
    [
      "1.278028719",
      "0.791616079"
    ],
    [
      "0.110917184",
      "0.800317857"
    ],
    [
      "1.890205445",
      "0.805383766"
    ],
    [
      "0.637919305",
      "0.820464573"
    ],
    [
      "0.812038888",
      "0.855140127"
    ],
    [
      "0.98618328",
      "0.889835671"
    ],
    [
      "1.420495789",
      "0.897555482"
    ],
    [
      "1.1609093",
      "0.925081774"
    ],
    [
      "1.594612041",
      "0.932247756"
    ],
    [
      "0.331462281",
      "0.932833431"
    ],
    [
      "1.772060683",
      "0.937904854"
    ],
    [
      "0.515867388",
      "0.949396156"
    ],
    [
      "0.089072276",
      "0.976507593"
    ],
    [
      "0.694949186",
      "0.988594333"
    ],
    [
      "0.869070232",
      "1.02326939"
    ],
    [
      "1.303393272",
      "1.030998444"
    ],
    [
      "1.909866555",
      "1.049840394"
    ],
    [
      "1.043183746",
      "1.057975405"
    ],
    [
      "1.477509525",
      "1.065690717"
    ],
    [
      "0.232944851",
      "1.080530213"
    ],
    [
      "1.678437172",
      "1.088751337"
    ],
    [
      "0.410112731",
      "1.092000419"
    ],
    [
      "0.584110184",
      "1.127283694"
    ],
    [
      "0.758230053",
      "1.161957811"
    ],
    [
      "1.185667718",
      "1.163892075"
    ],
    [
      "0.932343568",
      "1.196663826"
    ],
    [
      "1.360405444",
      "1.199134209"
    ],
    [
      "0.114067229",
      "1.212622974"
    ],
    [
      "1.882932718",
      "1.225324273"
    ],
    [
      "1.534521696",
      "1.233826482"
    ],
    [
      "0.289865743",
      "1.248696902"
    ],
    [
      "0.466997136",
      "1.260717413"
    ],
    [
      "1.709556079",
      "1.263541602"
    ],
    [
      "0.64111065",
      "1.295423428"
    ],
    [
      "1.07482754",
      "1.302580496"
    ],
    [
      "0.81523052",
      "1.330097545"
    ],
    [
      "1.248935052",
      "1.337316607"
    ],
    [
      "0.172773739",
      "1.382149088"
    ],
    [
      "1.43861404",
      "1.383231116"
    ],
    [
      "1.81594904",
      "1.40567028"
    ],
    [
      "1.613426398",
      "1.41477143"
    ],
    [
      "0.346892724",
      "1.416827645"
    ],
    [
      "0.524022049",
      "1.428878599"
    ],
    [
      "0.957714492",
      "1.436014215"
    ],
    [
      "0.698141919",
      "1.463552716"
    ],
    [
      "1.134183472",
      "1.472786522"
    ],
    [
      "1.30887965",
      "1.504429349"
    ],
    [
      "1.478707593",
      "1.556183509"
    ],
    [
      "1.721799725",
      "1.556189148"
    ],
    [
      "0.840625891",
      "1.569469386"
    ],
    [
      "0.305018871",
      "1.589357653"
    ],
    [
      "1.01474576",
      "1.604143503"
    ],
    [
      "0.573905039",
      "1.60472967"
    ],
    [
      "1.314311823",
      "1.681885019"
    ],
    [
      "1.600250657",
      "1.685594895"
    ],
    [
      "0.72353729",
      "1.702924558"
    ],
    [
      "0.432490536",
      "1.712933532"
    ],
    [
      "1.142667597",
      "1.72725332"
    ],
    [
      "0.897657159",
      "1.737598675"
    ],
    [
      "1.455915879",
      "1.788975236"
    ],
    [
      "0.581505046",
      "1.809446221"
    ],
    [
      "1.293839137",
      "1.862406918"
    ],
    [
      "0.765211699",
      "1.87922109"
    ],
    [
      "1.119830282",
      "1.903317173"
    ],
    [
      "0.942395999",
      "1.909408044"
    ]
  ],
  "radius": "0.088769396"
}

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

讨论区

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。

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