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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 = 87

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 = 87
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[0.0964952118361805591049828632235, 0.0964952118361805591049828632245]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 = 0.096495211

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 = 87 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": [
    [
      "0.976008461",
      "0.096813802"
    ],
    [
      "1.168374714",
      "0.112322774"
    ],
    [
      "0.784736841",
      "0.122513433"
    ],
    [
      "0.603286777",
      "0.188249102"
    ],
    [
      "1.479596908",
      "0.234292675"
    ],
    [
      "1.288776133",
      "0.263149833"
    ],
    [
      "0.905589105",
      "0.27649802"
    ],
    [
      "1.097955359",
      "0.292006991"
    ],
    [
      "0.723168302",
      "0.339489717"
    ],
    [
      "1.631277044",
      "0.353617609"
    ],
    [
      "0.526967623",
      "0.365507925"
    ],
    [
      "0.335297402",
      "0.388043009"
    ],
    [
      "1.411954652",
      "0.415040645"
    ],
    [
      "1.221133877",
      "0.443897804"
    ],
    [
      "1.030313103",
      "0.472754962"
    ],
    [
      "0.838519368",
      "0.494213455"
    ],
    [
      "1.754154653",
      "0.502434264"
    ],
    [
      "0.646849147",
      "0.51674854"
    ],
    [
      "1.563634788",
      "0.534365579"
    ],
    [
      "0.450648468",
      "0.542766747"
    ],
    [
      "0.220493603",
      "0.54317325"
    ],
    [
      "1.341535297",
      "0.594724863"
    ],
    [
      "1.150714522",
      "0.623582021"
    ],
    [
      "0.959893748",
      "0.652439179"
    ],
    [
      "0.762200214",
      "0.671472278"
    ],
    [
      "1.842623344",
      "0.67395276"
    ],
    [
      "0.570529993",
      "0.694007363"
    ],
    [
      "0.33584467",
      "0.697896988"
    ],
    [
      "1.650931347",
      "0.706483604"
    ],
    [
      "0.141255417",
      "0.719146581"
    ],
    [
      "1.461936717",
      "0.745551922"
    ],
    [
      "1.271115942",
      "0.77440908"
    ],
    [
      "1.080295167",
      "0.803266239"
    ],
    [
      "0.876905493",
      "0.826675379"
    ],
    [
      "0.685235272",
      "0.849210464"
    ],
    [
      "1.774670893",
      "0.854584341"
    ],
    [
      "0.484992461",
      "0.867006306"
    ],
    [
      "0.292944805",
      "0.8860589"
    ],
    [
      "1.585676263",
      "0.893652659"
    ],
    [
      "0.101198149",
      "0.907934078"
    ],
    [
      "1.391517361",
      "0.92523614"
    ],
    [
      "1.200696587",
      "0.954093298"
    ],
    [
      "0.997306912",
      "0.977502438"
    ],
    [
      "1.903503145",
      "0.998276911"
    ],
    [
      "0.796684253",
      "1.014726553"
    ],
    [
      "0.59969774",
      "1.022209407"
    ],
    [
      "1.714775911",
      "1.038631174"
    ],
    [
      "0.408806232",
      "1.052908984"
    ],
    [
      "0.216015937",
      "1.063053964"
    ],
    [
      "1.515256908",
      "1.073336876"
    ],
    [
      "1.318993922",
      "1.126960183"
    ],
    [
      "1.117708332",
      "1.128329497"
    ],
    [
      "0.926887557",
      "1.157186656"
    ],
    [
      "1.883257563",
      "1.190202469"
    ],
    [
      "0.693405856",
      "1.190922455"
    ],
    [
      "0.320894794",
      "1.225059301"
    ],
    [
      "1.692514205",
      "1.230334693"
    ],
    [
      "0.1274655",
      "1.234530271"
    ],
    [
      "0.511491388",
      "1.255361806"
    ],
    [
      "1.48727281",
      "1.264287639"
    ],
    [
      "1.214360822",
      "1.295373203"
    ],
    [
      "0.831990629",
      "1.325233901"
    ],
    [
      "1.024864779",
      "1.331932102"
    ],
    [
      "1.822712638",
      "1.373449886"
    ],
    [
      "0.389950351",
      "1.405272046"
    ],
    [
      "0.197180151",
      "1.414489075"
    ],
    [
      "1.366161253",
      "1.414545066"
    ],
    [
      "0.624075913",
      "1.416298644"
    ],
    [
      "1.55684378",
      "1.444302038"
    ],
    [
      "1.187779229",
      "1.488208211"
    ],
    [
      "0.804851001",
      "1.516306516"
    ],
    [
      "0.997725151",
      "1.523004718"
    ],
    [
      "1.724630781",
      "1.539658348"
    ],
    [
      "0.499763683",
      "1.563974081"
    ],
    [
      "0.307335775",
      "1.580077567"
    ],
    [
      "1.418259492",
      "1.60037046"
    ],
    [
      "0.67018203",
      "1.654543891"
    ],
    [
      "1.24000444",
      "1.674332885"
    ],
    [
      "1.593487057",
      "1.681244461"
    ],
    [
      "0.857233594",
      "1.702051953"
    ],
    [
      "1.050107744",
      "1.708750155"
    ],
    [
      "0.5065371",
      "1.756845604"
    ],
    [
      "1.43340985",
      "1.79276529"
    ],
    [
      "0.685332388",
      "1.846938721"
    ],
    [
      "1.255154798",
      "1.866727714"
    ],
    [
      "0.872383952",
      "1.894446782"
    ],
    [
      "1.065258102",
      "1.901144984"
    ]
  ],
  "radius": "0.096495211"
}
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": 87
}

起步布局答案

{
  "centers": [
    [
      "0.976008461",
      "0.096813802"
    ],
    [
      "1.168374714",
      "0.112322774"
    ],
    [
      "0.784736841",
      "0.122513433"
    ],
    [
      "0.603286777",
      "0.188249102"
    ],
    [
      "1.479596908",
      "0.234292675"
    ],
    [
      "1.288776133",
      "0.263149833"
    ],
    [
      "0.905589105",
      "0.27649802"
    ],
    [
      "1.097955359",
      "0.292006991"
    ],
    [
      "0.723168302",
      "0.339489717"
    ],
    [
      "1.631277044",
      "0.353617609"
    ],
    [
      "0.526967623",
      "0.365507925"
    ],
    [
      "0.335297402",
      "0.388043009"
    ],
    [
      "1.411954652",
      "0.415040645"
    ],
    [
      "1.221133877",
      "0.443897804"
    ],
    [
      "1.030313103",
      "0.472754962"
    ],
    [
      "0.838519368",
      "0.494213455"
    ],
    [
      "1.754154653",
      "0.502434264"
    ],
    [
      "0.646849147",
      "0.51674854"
    ],
    [
      "1.563634788",
      "0.534365579"
    ],
    [
      "0.450648468",
      "0.542766747"
    ],
    [
      "0.220493603",
      "0.54317325"
    ],
    [
      "1.341535297",
      "0.594724863"
    ],
    [
      "1.150714522",
      "0.623582021"
    ],
    [
      "0.959893748",
      "0.652439179"
    ],
    [
      "0.762200214",
      "0.671472278"
    ],
    [
      "1.842623344",
      "0.67395276"
    ],
    [
      "0.570529993",
      "0.694007363"
    ],
    [
      "0.33584467",
      "0.697896988"
    ],
    [
      "1.650931347",
      "0.706483604"
    ],
    [
      "0.141255417",
      "0.719146581"
    ],
    [
      "1.461936717",
      "0.745551922"
    ],
    [
      "1.271115942",
      "0.77440908"
    ],
    [
      "1.080295167",
      "0.803266239"
    ],
    [
      "0.876905493",
      "0.826675379"
    ],
    [
      "0.685235272",
      "0.849210464"
    ],
    [
      "1.774670893",
      "0.854584341"
    ],
    [
      "0.484992461",
      "0.867006306"
    ],
    [
      "0.292944805",
      "0.8860589"
    ],
    [
      "1.585676263",
      "0.893652659"
    ],
    [
      "0.101198149",
      "0.907934078"
    ],
    [
      "1.391517361",
      "0.92523614"
    ],
    [
      "1.200696587",
      "0.954093298"
    ],
    [
      "0.997306912",
      "0.977502438"
    ],
    [
      "1.903503145",
      "0.998276911"
    ],
    [
      "0.796684253",
      "1.014726553"
    ],
    [
      "0.59969774",
      "1.022209407"
    ],
    [
      "1.714775911",
      "1.038631174"
    ],
    [
      "0.408806232",
      "1.052908984"
    ],
    [
      "0.216015937",
      "1.063053964"
    ],
    [
      "1.515256908",
      "1.073336876"
    ],
    [
      "1.318993922",
      "1.126960183"
    ],
    [
      "1.117708332",
      "1.128329497"
    ],
    [
      "0.926887557",
      "1.157186656"
    ],
    [
      "1.883257563",
      "1.190202469"
    ],
    [
      "0.693405856",
      "1.190922455"
    ],
    [
      "0.320894794",
      "1.225059301"
    ],
    [
      "1.692514205",
      "1.230334693"
    ],
    [
      "0.1274655",
      "1.234530271"
    ],
    [
      "0.511491388",
      "1.255361806"
    ],
    [
      "1.48727281",
      "1.264287639"
    ],
    [
      "1.214360822",
      "1.295373203"
    ],
    [
      "0.831990629",
      "1.325233901"
    ],
    [
      "1.024864779",
      "1.331932102"
    ],
    [
      "1.822712638",
      "1.373449886"
    ],
    [
      "0.389950351",
      "1.405272046"
    ],
    [
      "0.197180151",
      "1.414489075"
    ],
    [
      "1.366161253",
      "1.414545066"
    ],
    [
      "0.624075913",
      "1.416298644"
    ],
    [
      "1.55684378",
      "1.444302038"
    ],
    [
      "1.187779229",
      "1.488208211"
    ],
    [
      "0.804851001",
      "1.516306516"
    ],
    [
      "0.997725151",
      "1.523004718"
    ],
    [
      "1.724630781",
      "1.539658348"
    ],
    [
      "0.499763683",
      "1.563974081"
    ],
    [
      "0.307335775",
      "1.580077567"
    ],
    [
      "1.418259492",
      "1.60037046"
    ],
    [
      "0.67018203",
      "1.654543891"
    ],
    [
      "1.24000444",
      "1.674332885"
    ],
    [
      "1.593487057",
      "1.681244461"
    ],
    [
      "0.857233594",
      "1.702051953"
    ],
    [
      "1.050107744",
      "1.708750155"
    ],
    [
      "0.5065371",
      "1.756845604"
    ],
    [
      "1.43340985",
      "1.79276529"
    ],
    [
      "0.685332388",
      "1.846938721"
    ],
    [
      "1.255154798",
      "1.866727714"
    ],
    [
      "0.872383952",
      "1.894446782"
    ],
    [
      "1.065258102",
      "1.901144984"
    ]
  ],
  "radius": "0.096495211"
}

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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