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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 = 91

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 = 91
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[0.0946362785060467409044299419565, 0.0946362785060467409044299419575]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 = 0.094636278

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 = 91 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": [
    [
      "0.905363721",
      "0.099595956"
    ],
    [
      "1.094636279",
      "0.099595956"
    ],
    [
      "0.720227224",
      "0.138947933"
    ],
    [
      "1.279772776",
      "0.138947933"
    ],
    [
      "0.547318139",
      "0.215932018"
    ],
    [
      "1.452681861",
      "0.215932018"
    ],
    [
      "0.885579352",
      "0.287831658"
    ],
    [
      "1.074851909",
      "0.287831658"
    ],
    [
      "0.394193424",
      "0.327183635"
    ],
    [
      "0.700442854",
      "0.327183635"
    ],
    [
      "1.259988407",
      "0.327183635"
    ],
    [
      "1.605806576",
      "0.327183635"
    ],
    [
      "1.432897491",
      "0.40416772"
    ],
    [
      "0.547318139",
      "0.438435253"
    ],
    [
      "0.267545363",
      "0.467840557"
    ],
    [
      "0.944067789",
      "0.467840557"
    ],
    [
      "1.133340346",
      "0.467840557"
    ],
    [
      "1.732454637",
      "0.467840557"
    ],
    [
      "0.758931291",
      "0.507192534"
    ],
    [
      "1.30624943",
      "0.544824641"
    ],
    [
      "1.559545552",
      "0.544824641"
    ],
    [
      "0.420670078",
      "0.579092174"
    ],
    [
      "0.605806576",
      "0.618444152"
    ],
    [
      "0.172909085",
      "0.631755399"
    ],
    [
      "1.827090915",
      "0.631755399"
    ],
    [
      "0.963852158",
      "0.656076259"
    ],
    [
      "1.153124715",
      "0.656076259"
    ],
    [
      "1.432897491",
      "0.685481562"
    ],
    [
      "0.778715661",
      "0.695428236"
    ],
    [
      "1.654181831",
      "0.708739484"
    ],
    [
      "0.3260338",
      "0.743007017"
    ],
    [
      "0.511170297",
      "0.782358994"
    ],
    [
      "1.279772776",
      "0.79673318"
    ],
    [
      "0.114420648",
      "0.811764298"
    ],
    [
      "1.885579352",
      "0.811764298"
    ],
    [
      "0.905363721",
      "0.836085157"
    ],
    [
      "1.094636279",
      "0.836085157"
    ],
    [
      "1.52753377",
      "0.849396405"
    ],
    [
      "0.684079382",
      "0.859343079"
    ],
    [
      "1.712670267",
      "0.888748382"
    ],
    [
      "0.267545363",
      "0.923015916"
    ],
    [
      "1.374409055",
      "0.960648023"
    ],
    [
      "0.452681861",
      "0.962367893"
    ],
    [
      "0.094636279",
      "1"
    ],
    [
      "0.810727443",
      "1"
    ],
    [
      "1",
      "1"
    ],
    [
      "1.189272557",
      "1"
    ],
    [
      "1.905363721",
      "1"
    ],
    [
      "1.547318139",
      "1.037632107"
    ],
    [
      "0.625590945",
      "1.039351977"
    ],
    [
      "1.732454637",
      "1.076984084"
    ],
    [
      "0.287329733",
      "1.111251618"
    ],
    [
      "1.315920618",
      "1.140656921"
    ],
    [
      "0.47246623",
      "1.150603595"
    ],
    [
      "0.905363721",
      "1.163914843"
    ],
    [
      "1.094636279",
      "1.163914843"
    ],
    [
      "0.114420648",
      "1.188235702"
    ],
    [
      "1.885579352",
      "1.188235702"
    ],
    [
      "0.720227224",
      "1.20326682"
    ],
    [
      "1.488829703",
      "1.217641006"
    ],
    [
      "1.6739662",
      "1.256992983"
    ],
    [
      "0.345818169",
      "1.291260516"
    ],
    [
      "1.221284339",
      "1.304571764"
    ],
    [
      "0.567102509",
      "1.314518438"
    ],
    [
      "0.846875285",
      "1.343923741"
    ],
    [
      "1.036147842",
      "1.343923741"
    ],
    [
      "0.172909085",
      "1.368244601"
    ],
    [
      "1.827090915",
      "1.368244601"
    ],
    [
      "1.394193424",
      "1.381555848"
    ],
    [
      "1.579329922",
      "1.420907826"
    ],
    [
      "0.440454448",
      "1.455175359"
    ],
    [
      "0.69375057",
      "1.455175359"
    ],
    [
      "1.241068709",
      "1.492807466"
    ],
    [
      "0.267545363",
      "1.532159443"
    ],
    [
      "0.866659654",
      "1.532159443"
    ],
    [
      "1.055932211",
      "1.532159443"
    ],
    [
      "1.732454637",
      "1.532159443"
    ],
    [
      "1.452681861",
      "1.561564747"
    ],
    [
      "0.567102509",
      "1.59583228"
    ],
    [
      "0.394193424",
      "1.672816365"
    ],
    [
      "0.740011593",
      "1.672816365"
    ],
    [
      "1.299557146",
      "1.672816365"
    ],
    [
      "1.605806576",
      "1.672816365"
    ],
    [
      "0.925148091",
      "1.712168342"
    ],
    [
      "1.114420648",
      "1.712168342"
    ],
    [
      "0.547318139",
      "1.784067982"
    ],
    [
      "1.452681861",
      "1.784067982"
    ],
    [
      "0.720227224",
      "1.861052067"
    ],
    [
      "1.279772776",
      "1.861052067"
    ],
    [
      "0.905363721",
      "1.900404044"
    ],
    [
      "1.094636279",
      "1.900404044"
    ]
  ],
  "radius": "0.094636278"
}
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": 91
}

起步布局答案

{
  "centers": [
    [
      "0.905363721",
      "0.099595956"
    ],
    [
      "1.094636279",
      "0.099595956"
    ],
    [
      "0.720227224",
      "0.138947933"
    ],
    [
      "1.279772776",
      "0.138947933"
    ],
    [
      "0.547318139",
      "0.215932018"
    ],
    [
      "1.452681861",
      "0.215932018"
    ],
    [
      "0.885579352",
      "0.287831658"
    ],
    [
      "1.074851909",
      "0.287831658"
    ],
    [
      "0.394193424",
      "0.327183635"
    ],
    [
      "0.700442854",
      "0.327183635"
    ],
    [
      "1.259988407",
      "0.327183635"
    ],
    [
      "1.605806576",
      "0.327183635"
    ],
    [
      "1.432897491",
      "0.40416772"
    ],
    [
      "0.547318139",
      "0.438435253"
    ],
    [
      "0.267545363",
      "0.467840557"
    ],
    [
      "0.944067789",
      "0.467840557"
    ],
    [
      "1.133340346",
      "0.467840557"
    ],
    [
      "1.732454637",
      "0.467840557"
    ],
    [
      "0.758931291",
      "0.507192534"
    ],
    [
      "1.30624943",
      "0.544824641"
    ],
    [
      "1.559545552",
      "0.544824641"
    ],
    [
      "0.420670078",
      "0.579092174"
    ],
    [
      "0.605806576",
      "0.618444152"
    ],
    [
      "0.172909085",
      "0.631755399"
    ],
    [
      "1.827090915",
      "0.631755399"
    ],
    [
      "0.963852158",
      "0.656076259"
    ],
    [
      "1.153124715",
      "0.656076259"
    ],
    [
      "1.432897491",
      "0.685481562"
    ],
    [
      "0.778715661",
      "0.695428236"
    ],
    [
      "1.654181831",
      "0.708739484"
    ],
    [
      "0.3260338",
      "0.743007017"
    ],
    [
      "0.511170297",
      "0.782358994"
    ],
    [
      "1.279772776",
      "0.79673318"
    ],
    [
      "0.114420648",
      "0.811764298"
    ],
    [
      "1.885579352",
      "0.811764298"
    ],
    [
      "0.905363721",
      "0.836085157"
    ],
    [
      "1.094636279",
      "0.836085157"
    ],
    [
      "1.52753377",
      "0.849396405"
    ],
    [
      "0.684079382",
      "0.859343079"
    ],
    [
      "1.712670267",
      "0.888748382"
    ],
    [
      "0.267545363",
      "0.923015916"
    ],
    [
      "1.374409055",
      "0.960648023"
    ],
    [
      "0.452681861",
      "0.962367893"
    ],
    [
      "0.094636279",
      "1"
    ],
    [
      "0.810727443",
      "1"
    ],
    [
      "1",
      "1"
    ],
    [
      "1.189272557",
      "1"
    ],
    [
      "1.905363721",
      "1"
    ],
    [
      "1.547318139",
      "1.037632107"
    ],
    [
      "0.625590945",
      "1.039351977"
    ],
    [
      "1.732454637",
      "1.076984084"
    ],
    [
      "0.287329733",
      "1.111251618"
    ],
    [
      "1.315920618",
      "1.140656921"
    ],
    [
      "0.47246623",
      "1.150603595"
    ],
    [
      "0.905363721",
      "1.163914843"
    ],
    [
      "1.094636279",
      "1.163914843"
    ],
    [
      "0.114420648",
      "1.188235702"
    ],
    [
      "1.885579352",
      "1.188235702"
    ],
    [
      "0.720227224",
      "1.20326682"
    ],
    [
      "1.488829703",
      "1.217641006"
    ],
    [
      "1.6739662",
      "1.256992983"
    ],
    [
      "0.345818169",
      "1.291260516"
    ],
    [
      "1.221284339",
      "1.304571764"
    ],
    [
      "0.567102509",
      "1.314518438"
    ],
    [
      "0.846875285",
      "1.343923741"
    ],
    [
      "1.036147842",
      "1.343923741"
    ],
    [
      "0.172909085",
      "1.368244601"
    ],
    [
      "1.827090915",
      "1.368244601"
    ],
    [
      "1.394193424",
      "1.381555848"
    ],
    [
      "1.579329922",
      "1.420907826"
    ],
    [
      "0.440454448",
      "1.455175359"
    ],
    [
      "0.69375057",
      "1.455175359"
    ],
    [
      "1.241068709",
      "1.492807466"
    ],
    [
      "0.267545363",
      "1.532159443"
    ],
    [
      "0.866659654",
      "1.532159443"
    ],
    [
      "1.055932211",
      "1.532159443"
    ],
    [
      "1.732454637",
      "1.532159443"
    ],
    [
      "1.452681861",
      "1.561564747"
    ],
    [
      "0.567102509",
      "1.59583228"
    ],
    [
      "0.394193424",
      "1.672816365"
    ],
    [
      "0.740011593",
      "1.672816365"
    ],
    [
      "1.299557146",
      "1.672816365"
    ],
    [
      "1.605806576",
      "1.672816365"
    ],
    [
      "0.925148091",
      "1.712168342"
    ],
    [
      "1.114420648",
      "1.712168342"
    ],
    [
      "0.547318139",
      "1.784067982"
    ],
    [
      "1.452681861",
      "1.784067982"
    ],
    [
      "0.720227224",
      "1.861052067"
    ],
    [
      "1.279772776",
      "1.861052067"
    ],
    [
      "0.905363721",
      "1.900404044"
    ],
    [
      "1.094636279",
      "1.900404044"
    ]
  ],
  "radius": "0.094636278"
}

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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