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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 = 89

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 = 89
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[0.0952336345438698529476853810945, 0.0952336345438698529476853810955]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 = 0.095233634

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 = 89 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": [
    [
      "0.992423485",
      "0.095265358"
    ],
    [
      "1.181993947",
      "0.11372669"
    ],
    [
      "0.80318879",
      "0.116898916"
    ],
    [
      "1.363499029",
      "0.171464767"
    ],
    [
      "0.622676127",
      "0.177668637"
    ],
    [
      "1.52889502",
      "0.265920826"
    ],
    [
      "1.06960423",
      "0.269394371"
    ],
    [
      "0.458885228",
      "0.274881403"
    ],
    [
      "0.880369535",
      "0.29102793"
    ],
    [
      "1.251592628",
      "0.325590488"
    ],
    [
      "0.699856871",
      "0.35179765"
    ],
    [
      "1.670852106",
      "0.392908881"
    ],
    [
      "0.319074772",
      "0.404229063"
    ],
    [
      "1.416988618",
      "0.420046547"
    ],
    [
      "0.536065972",
      "0.449010417"
    ],
    [
      "1.105481255",
      "0.456452169"
    ],
    [
      "0.910391041",
      "0.479114319"
    ],
    [
      "0.729878377",
      "0.539884039"
    ],
    [
      "1.273783965",
      "0.545626035"
    ],
    [
      "1.783079214",
      "0.546801235"
    ],
    [
      "1.558945705",
      "0.547034602"
    ],
    [
      "0.209440703",
      "0.559979349"
    ],
    [
      "1.026767264",
      "0.629893479"
    ],
    [
      "0.566087478",
      "0.637096806"
    ],
    [
      "0.376204132",
      "0.651999669"
    ],
    [
      "1.415741052",
      "0.67261409"
    ],
    [
      "0.844564457",
      "0.69195272"
    ],
    [
      "1.671172813",
      "0.700926956"
    ],
    [
      "1.195069974",
      "0.719067346"
    ],
    [
      "1.860602803",
      "0.720777882"
    ],
    [
      "0.134841645",
      "0.735229916"
    ],
    [
      "0.680773558",
      "0.789165486"
    ],
    [
      "0.491488421",
      "0.812347373"
    ],
    [
      "0.988659445",
      "0.816509595"
    ],
    [
      "1.52796816",
      "0.826506445"
    ],
    [
      "0.301605074",
      "0.827250236"
    ],
    [
      "1.337998862",
      "0.846493164"
    ],
    [
      "1.71055729",
      "0.887277818"
    ],
    [
      "1.156962155",
      "0.905683462"
    ],
    [
      "1.899987281",
      "0.907128744"
    ],
    [
      "0.824417524",
      "0.914242229"
    ],
    [
      "0.098583587",
      "0.92221423"
    ],
    [
      "0.644275904",
      "0.976103182"
    ],
    [
      "0.455230362",
      "0.999331687"
    ],
    [
      "0.990791444",
      "1.006964932"
    ],
    [
      "0.265347016",
      "1.01423455"
    ],
    [
      "1.52954714",
      "1.016967169"
    ],
    [
      "1.34013086",
      "1.036948501"
    ],
    [
      "1.710057265",
      "1.077744431"
    ],
    [
      "1.159094153",
      "1.096138798"
    ],
    [
      "1.899487255",
      "1.097595357"
    ],
    [
      "0.829091291",
      "1.107617012"
    ],
    [
      "0.102273366",
      "1.112645756"
    ],
    [
      "0.647965683",
      "1.166534708"
    ],
    [
      "0.458920141",
      "1.189763213"
    ],
    [
      "0.997715965",
      "1.197306287"
    ],
    [
      "1.489184771",
      "1.203108674"
    ],
    [
      "0.269036795",
      "1.204666077"
    ],
    [
      "1.299910245",
      "1.224390929"
    ],
    [
      "1.669694895",
      "1.263885935"
    ],
    [
      "1.859124886",
      "1.283736861"
    ],
    [
      "0.789552709",
      "1.293935237"
    ],
    [
      "1.139616478",
      "1.327267987"
    ],
    [
      "0.556104498",
      "1.35357097"
    ],
    [
      "0.366086876",
      "1.369615446"
    ],
    [
      "0.174304382",
      "1.369903937"
    ],
    [
      "1.409413777",
      "1.380233019"
    ],
    [
      "0.957642309",
      "1.383510161"
    ],
    [
      "1.591104762",
      "1.437383404"
    ],
    [
      "1.780536229",
      "1.457553732"
    ],
    [
      "0.70934262",
      "1.466689751"
    ],
    [
      "1.249120011",
      "1.483110076"
    ],
    [
      "0.270038361",
      "1.534563543"
    ],
    [
      "1.067147665",
      "1.53935815"
    ],
    [
      "0.534759821",
      "1.542838467"
    ],
    [
      "0.87743222",
      "1.556264675"
    ],
    [
      "1.416405888",
      "1.580428655"
    ],
    [
      "1.665852346",
      "1.612570673"
    ],
    [
      "0.693508549",
      "1.658067071"
    ],
    [
      "1.250199065",
      "1.673574289"
    ],
    [
      "0.398121861",
      "1.675533035"
    ],
    [
      "1.068226719",
      "1.729822363"
    ],
    [
      "1.522858923",
      "1.73839063"
    ],
    [
      "0.878511274",
      "1.746728888"
    ],
    [
      "0.552878638",
      "1.786565105"
    ],
    [
      "1.356694097",
      "1.831487521"
    ],
    [
      "0.727450377",
      "1.862739172"
    ],
    [
      "1.17472175",
      "1.887735595"
    ],
    [
      "0.985006306",
      "1.90464212"
    ]
  ],
  "radius": "0.095233634"
}
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": 89
}

起步布局答案

{
  "centers": [
    [
      "0.992423485",
      "0.095265358"
    ],
    [
      "1.181993947",
      "0.11372669"
    ],
    [
      "0.80318879",
      "0.116898916"
    ],
    [
      "1.363499029",
      "0.171464767"
    ],
    [
      "0.622676127",
      "0.177668637"
    ],
    [
      "1.52889502",
      "0.265920826"
    ],
    [
      "1.06960423",
      "0.269394371"
    ],
    [
      "0.458885228",
      "0.274881403"
    ],
    [
      "0.880369535",
      "0.29102793"
    ],
    [
      "1.251592628",
      "0.325590488"
    ],
    [
      "0.699856871",
      "0.35179765"
    ],
    [
      "1.670852106",
      "0.392908881"
    ],
    [
      "0.319074772",
      "0.404229063"
    ],
    [
      "1.416988618",
      "0.420046547"
    ],
    [
      "0.536065972",
      "0.449010417"
    ],
    [
      "1.105481255",
      "0.456452169"
    ],
    [
      "0.910391041",
      "0.479114319"
    ],
    [
      "0.729878377",
      "0.539884039"
    ],
    [
      "1.273783965",
      "0.545626035"
    ],
    [
      "1.783079214",
      "0.546801235"
    ],
    [
      "1.558945705",
      "0.547034602"
    ],
    [
      "0.209440703",
      "0.559979349"
    ],
    [
      "1.026767264",
      "0.629893479"
    ],
    [
      "0.566087478",
      "0.637096806"
    ],
    [
      "0.376204132",
      "0.651999669"
    ],
    [
      "1.415741052",
      "0.67261409"
    ],
    [
      "0.844564457",
      "0.69195272"
    ],
    [
      "1.671172813",
      "0.700926956"
    ],
    [
      "1.195069974",
      "0.719067346"
    ],
    [
      "1.860602803",
      "0.720777882"
    ],
    [
      "0.134841645",
      "0.735229916"
    ],
    [
      "0.680773558",
      "0.789165486"
    ],
    [
      "0.491488421",
      "0.812347373"
    ],
    [
      "0.988659445",
      "0.816509595"
    ],
    [
      "1.52796816",
      "0.826506445"
    ],
    [
      "0.301605074",
      "0.827250236"
    ],
    [
      "1.337998862",
      "0.846493164"
    ],
    [
      "1.71055729",
      "0.887277818"
    ],
    [
      "1.156962155",
      "0.905683462"
    ],
    [
      "1.899987281",
      "0.907128744"
    ],
    [
      "0.824417524",
      "0.914242229"
    ],
    [
      "0.098583587",
      "0.92221423"
    ],
    [
      "0.644275904",
      "0.976103182"
    ],
    [
      "0.455230362",
      "0.999331687"
    ],
    [
      "0.990791444",
      "1.006964932"
    ],
    [
      "0.265347016",
      "1.01423455"
    ],
    [
      "1.52954714",
      "1.016967169"
    ],
    [
      "1.34013086",
      "1.036948501"
    ],
    [
      "1.710057265",
      "1.077744431"
    ],
    [
      "1.159094153",
      "1.096138798"
    ],
    [
      "1.899487255",
      "1.097595357"
    ],
    [
      "0.829091291",
      "1.107617012"
    ],
    [
      "0.102273366",
      "1.112645756"
    ],
    [
      "0.647965683",
      "1.166534708"
    ],
    [
      "0.458920141",
      "1.189763213"
    ],
    [
      "0.997715965",
      "1.197306287"
    ],
    [
      "1.489184771",
      "1.203108674"
    ],
    [
      "0.269036795",
      "1.204666077"
    ],
    [
      "1.299910245",
      "1.224390929"
    ],
    [
      "1.669694895",
      "1.263885935"
    ],
    [
      "1.859124886",
      "1.283736861"
    ],
    [
      "0.789552709",
      "1.293935237"
    ],
    [
      "1.139616478",
      "1.327267987"
    ],
    [
      "0.556104498",
      "1.35357097"
    ],
    [
      "0.366086876",
      "1.369615446"
    ],
    [
      "0.174304382",
      "1.369903937"
    ],
    [
      "1.409413777",
      "1.380233019"
    ],
    [
      "0.957642309",
      "1.383510161"
    ],
    [
      "1.591104762",
      "1.437383404"
    ],
    [
      "1.780536229",
      "1.457553732"
    ],
    [
      "0.70934262",
      "1.466689751"
    ],
    [
      "1.249120011",
      "1.483110076"
    ],
    [
      "0.270038361",
      "1.534563543"
    ],
    [
      "1.067147665",
      "1.53935815"
    ],
    [
      "0.534759821",
      "1.542838467"
    ],
    [
      "0.87743222",
      "1.556264675"
    ],
    [
      "1.416405888",
      "1.580428655"
    ],
    [
      "1.665852346",
      "1.612570673"
    ],
    [
      "0.693508549",
      "1.658067071"
    ],
    [
      "1.250199065",
      "1.673574289"
    ],
    [
      "0.398121861",
      "1.675533035"
    ],
    [
      "1.068226719",
      "1.729822363"
    ],
    [
      "1.522858923",
      "1.73839063"
    ],
    [
      "0.878511274",
      "1.746728888"
    ],
    [
      "0.552878638",
      "1.786565105"
    ],
    [
      "1.356694097",
      "1.831487521"
    ],
    [
      "0.727450377",
      "1.862739172"
    ],
    [
      "1.17472175",
      "1.887735595"
    ],
    [
      "0.985006306",
      "1.90464212"
    ]
  ],
  "radius": "0.095233634"
}

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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