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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 = 86

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 = 86
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[0.0970996240053012834578522643245, 0.0970996240053012834578522643255]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 = 0.097099623

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 = 86 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": [
    [
      "1.125165798",
      "0.10581735"
    ],
    [
      "0.789129494",
      "0.12206907"
    ],
    [
      "1.31347948",
      "0.153265269"
    ],
    [
      "0.606273416",
      "0.187467868"
    ],
    [
      "0.961835425",
      "0.210872274"
    ],
    [
      "1.487291275",
      "0.23988402"
    ],
    [
      "0.441631541",
      "0.29045525"
    ],
    [
      "1.138827089",
      "0.299535488"
    ],
    [
      "0.797238599",
      "0.316098939"
    ],
    [
      "1.326627967",
      "0.348974118"
    ],
    [
      "1.638560467",
      "0.36166653"
    ],
    [
      "0.614382521",
      "0.381497737"
    ],
    [
      "0.9704908",
      "0.404878543"
    ],
    [
      "0.302820373",
      "0.426266912"
    ],
    [
      "1.477897159",
      "0.470756627"
    ],
    [
      "1.143194744",
      "0.493685614"
    ],
    [
      "0.807229649",
      "0.51004101"
    ],
    [
      "0.474783727",
      "0.516499681"
    ],
    [
      "1.652703609",
      "0.555350084"
    ],
    [
      "1.317233851",
      "0.579846725"
    ],
    [
      "0.979933593",
      "0.598848081"
    ],
    [
      "0.642956157",
      "0.613614983"
    ],
    [
      "0.31065818",
      "0.62030793"
    ],
    [
      "1.827768935",
      "0.639406491"
    ],
    [
      "1.492040301",
      "0.664440182"
    ],
    [
      "1.1563877",
      "0.688914366"
    ],
    [
      "0.815067456",
      "0.704082028"
    ],
    [
      "0.142467647",
      "0.717391875"
    ],
    [
      "0.47883061",
      "0.717423232"
    ],
    [
      "1.666846751",
      "0.749033639"
    ],
    [
      "1.331376993",
      "0.77353028"
    ],
    [
      "0.9877714",
      "0.792889099"
    ],
    [
      "0.650793964",
      "0.807656001"
    ],
    [
      "0.310640076",
      "0.814507177"
    ],
    [
      "1.885730239",
      "0.824754367"
    ],
    [
      "1.506183443",
      "0.858123737"
    ],
    [
      "1.170713685",
      "0.882620378"
    ],
    [
      "0.824510249",
      "0.898051566"
    ],
    [
      "0.101870903",
      "0.907300412"
    ],
    [
      "0.478812506",
      "0.911622479"
    ],
    [
      "1.72532247",
      "0.934219865"
    ],
    [
      "1.345520135",
      "0.967213835"
    ],
    [
      "1.006436971",
      "0.98618924"
    ],
    [
      "0.650775861",
      "1.001855248"
    ],
    [
      "0.271148658",
      "1.00464864"
    ],
    [
      "1.902716739",
      "1.018209288"
    ],
    [
      "1.523828975",
      "1.051519661"
    ],
    [
      "1.186078546",
      "1.080864305"
    ],
    [
      "0.822658338",
      "1.092241984"
    ],
    [
      "0.102822547",
      "1.101497328"
    ],
    [
      "0.486476535",
      "1.105670438"
    ],
    [
      "1.702717516",
      "1.127099009"
    ],
    [
      "1.363165668",
      "1.160609758"
    ],
    [
      "0.995847244",
      "1.180099543"
    ],
    [
      "0.313604667",
      "1.194150184"
    ],
    [
      "0.65843989",
      "1.195903207"
    ],
    [
      "1.877942621",
      "1.21082183"
    ],
    [
      "1.537972118",
      "1.245203215"
    ],
    [
      "1.215705884",
      "1.28697811"
    ],
    [
      "0.145278557",
      "1.290998872"
    ],
    [
      "0.476666308",
      "1.29962174"
    ],
    [
      "0.840555136",
      "1.3141897"
    ],
    [
      "1.717823979",
      "1.320709811"
    ],
    [
      "1.039799398",
      "1.369259676"
    ],
    [
      "0.648629663",
      "1.389854509"
    ],
    [
      "0.308340198",
      "1.396470429"
    ],
    [
      "1.37196711",
      "1.402285427"
    ],
    [
      "1.562891908",
      "1.437796964"
    ],
    [
      "1.201370257",
      "1.504203484"
    ],
    [
      "0.804839918",
      "1.505230868"
    ],
    [
      "0.495178814",
      "1.513161745"
    ],
    [
      "1.742803093",
      "1.513295874"
    ],
    [
      "0.991885258",
      "1.557455274"
    ],
    [
      "1.436675615",
      "1.585386921"
    ],
    [
      "0.316945304",
      "1.590478934"
    ],
    [
      "0.651693923",
      "1.628179947"
    ],
    [
      "1.615860309",
      "1.66026144"
    ],
    [
      "0.853134925",
      "1.693329087"
    ],
    [
      "1.278250156",
      "1.697702199"
    ],
    [
      "0.459010627",
      "1.722882831"
    ],
    [
      "1.086298059",
      "1.727159637"
    ],
    [
      "1.454021208",
      "1.780425548"
    ],
    [
      "0.626102682",
      "1.821845414"
    ],
    [
      "1.207784998",
      "1.87866631"
    ],
    [
      "0.810491618",
      "1.882788572"
    ],
    [
      "1.003647409",
      "1.902893009"
    ]
  ],
  "radius": "0.097099623"
}
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": 86
}

起步布局答案

{
  "centers": [
    [
      "1.125165798",
      "0.10581735"
    ],
    [
      "0.789129494",
      "0.12206907"
    ],
    [
      "1.31347948",
      "0.153265269"
    ],
    [
      "0.606273416",
      "0.187467868"
    ],
    [
      "0.961835425",
      "0.210872274"
    ],
    [
      "1.487291275",
      "0.23988402"
    ],
    [
      "0.441631541",
      "0.29045525"
    ],
    [
      "1.138827089",
      "0.299535488"
    ],
    [
      "0.797238599",
      "0.316098939"
    ],
    [
      "1.326627967",
      "0.348974118"
    ],
    [
      "1.638560467",
      "0.36166653"
    ],
    [
      "0.614382521",
      "0.381497737"
    ],
    [
      "0.9704908",
      "0.404878543"
    ],
    [
      "0.302820373",
      "0.426266912"
    ],
    [
      "1.477897159",
      "0.470756627"
    ],
    [
      "1.143194744",
      "0.493685614"
    ],
    [
      "0.807229649",
      "0.51004101"
    ],
    [
      "0.474783727",
      "0.516499681"
    ],
    [
      "1.652703609",
      "0.555350084"
    ],
    [
      "1.317233851",
      "0.579846725"
    ],
    [
      "0.979933593",
      "0.598848081"
    ],
    [
      "0.642956157",
      "0.613614983"
    ],
    [
      "0.31065818",
      "0.62030793"
    ],
    [
      "1.827768935",
      "0.639406491"
    ],
    [
      "1.492040301",
      "0.664440182"
    ],
    [
      "1.1563877",
      "0.688914366"
    ],
    [
      "0.815067456",
      "0.704082028"
    ],
    [
      "0.142467647",
      "0.717391875"
    ],
    [
      "0.47883061",
      "0.717423232"
    ],
    [
      "1.666846751",
      "0.749033639"
    ],
    [
      "1.331376993",
      "0.77353028"
    ],
    [
      "0.9877714",
      "0.792889099"
    ],
    [
      "0.650793964",
      "0.807656001"
    ],
    [
      "0.310640076",
      "0.814507177"
    ],
    [
      "1.885730239",
      "0.824754367"
    ],
    [
      "1.506183443",
      "0.858123737"
    ],
    [
      "1.170713685",
      "0.882620378"
    ],
    [
      "0.824510249",
      "0.898051566"
    ],
    [
      "0.101870903",
      "0.907300412"
    ],
    [
      "0.478812506",
      "0.911622479"
    ],
    [
      "1.72532247",
      "0.934219865"
    ],
    [
      "1.345520135",
      "0.967213835"
    ],
    [
      "1.006436971",
      "0.98618924"
    ],
    [
      "0.650775861",
      "1.001855248"
    ],
    [
      "0.271148658",
      "1.00464864"
    ],
    [
      "1.902716739",
      "1.018209288"
    ],
    [
      "1.523828975",
      "1.051519661"
    ],
    [
      "1.186078546",
      "1.080864305"
    ],
    [
      "0.822658338",
      "1.092241984"
    ],
    [
      "0.102822547",
      "1.101497328"
    ],
    [
      "0.486476535",
      "1.105670438"
    ],
    [
      "1.702717516",
      "1.127099009"
    ],
    [
      "1.363165668",
      "1.160609758"
    ],
    [
      "0.995847244",
      "1.180099543"
    ],
    [
      "0.313604667",
      "1.194150184"
    ],
    [
      "0.65843989",
      "1.195903207"
    ],
    [
      "1.877942621",
      "1.21082183"
    ],
    [
      "1.537972118",
      "1.245203215"
    ],
    [
      "1.215705884",
      "1.28697811"
    ],
    [
      "0.145278557",
      "1.290998872"
    ],
    [
      "0.476666308",
      "1.29962174"
    ],
    [
      "0.840555136",
      "1.3141897"
    ],
    [
      "1.717823979",
      "1.320709811"
    ],
    [
      "1.039799398",
      "1.369259676"
    ],
    [
      "0.648629663",
      "1.389854509"
    ],
    [
      "0.308340198",
      "1.396470429"
    ],
    [
      "1.37196711",
      "1.402285427"
    ],
    [
      "1.562891908",
      "1.437796964"
    ],
    [
      "1.201370257",
      "1.504203484"
    ],
    [
      "0.804839918",
      "1.505230868"
    ],
    [
      "0.495178814",
      "1.513161745"
    ],
    [
      "1.742803093",
      "1.513295874"
    ],
    [
      "0.991885258",
      "1.557455274"
    ],
    [
      "1.436675615",
      "1.585386921"
    ],
    [
      "0.316945304",
      "1.590478934"
    ],
    [
      "0.651693923",
      "1.628179947"
    ],
    [
      "1.615860309",
      "1.66026144"
    ],
    [
      "0.853134925",
      "1.693329087"
    ],
    [
      "1.278250156",
      "1.697702199"
    ],
    [
      "0.459010627",
      "1.722882831"
    ],
    [
      "1.086298059",
      "1.727159637"
    ],
    [
      "1.454021208",
      "1.780425548"
    ],
    [
      "0.626102682",
      "1.821845414"
    ],
    [
      "1.207784998",
      "1.87866631"
    ],
    [
      "0.810491618",
      "1.882788572"
    ],
    [
      "1.003647409",
      "1.902893009"
    ]
  ],
  "radius": "0.097099623"
}

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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