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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 = 62

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 = 62
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[0.1132532919825803750795878219225, 0.1132532919825803750795878219235]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 = 0.113253291

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 = 62 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": [
    [
      "1.064284521",
      "0.115586508"
    ],
    [
      "0.83812681",
      "0.128153228"
    ],
    [
      "1.28624784",
      "0.160725374"
    ],
    [
      "0.622530886",
      "0.197605588"
    ],
    [
      "1.489534264",
      "0.260624636"
    ],
    [
      "0.4315638",
      "0.319412011"
    ],
    [
      "1.131290073",
      "0.33195538"
    ],
    [
      "0.905132362",
      "0.344522099"
    ],
    [
      "1.660879902",
      "0.408766139"
    ],
    [
      "0.689536438",
      "0.41397446"
    ],
    [
      "1.336574044",
      "0.427682721"
    ],
    [
      "0.277685637",
      "0.485624957"
    ],
    [
      "1.029094315",
      "0.534097062"
    ],
    [
      "0.498569352",
      "0.535780882"
    ],
    [
      "1.507919682",
      "0.575824223"
    ],
    [
      "1.789104917",
      "0.595484049"
    ],
    [
      "0.811641602",
      "0.607376525"
    ],
    [
      "1.234536476",
      "0.630000977"
    ],
    [
      "0.17093653",
      "0.685399481"
    ],
    [
      "0.618204612",
      "0.728115583"
    ],
    [
      "0.391820245",
      "0.735555407"
    ],
    [
      "0.990685753",
      "0.757323438"
    ],
    [
      "1.636144697",
      "0.762542134"
    ],
    [
      "1.410329735",
      "0.780229299"
    ],
    [
      "1.865842973",
      "0.808595531"
    ],
    [
      "1.195899494",
      "0.85320124"
    ],
    [
      "0.794432026",
      "0.870415026"
    ],
    [
      "0.118281569",
      "0.905700838"
    ],
    [
      "0.565549651",
      "0.94841694"
    ],
    [
      "0.339165284",
      "0.955856763"
    ],
    [
      "1.010260011",
      "0.982982652"
    ],
    [
      "1.664495195",
      "0.987267481"
    ],
    [
      "1.438680233",
      "1.004954646"
    ],
    [
      "1.886087119",
      "1.034195635"
    ],
    [
      "1.224249992",
      "1.077926588"
    ],
    [
      "0.782989585",
      "1.096632406"
    ],
    [
      "0.123156348",
      "1.132154959"
    ],
    [
      "0.57042443",
      "1.174871061"
    ],
    [
      "0.344040063",
      "1.182310885"
    ],
    [
      "0.980494225",
      "1.207524921"
    ],
    [
      "1.626924556",
      "1.210636413"
    ],
    [
      "1.401109595",
      "1.228323578"
    ],
    [
      "1.848516481",
      "1.257564567"
    ],
    [
      "1.186679354",
      "1.30129552"
    ],
    [
      "0.1852428",
      "1.349986328"
    ],
    [
      "0.682575268",
      "1.371663902"
    ],
    [
      "0.456190901",
      "1.379103725"
    ],
    [
      "1.533990515",
      "1.417199955"
    ],
    [
      "0.903726722",
      "1.420625797"
    ],
    [
      "1.75558244",
      "1.464128108"
    ],
    [
      "1.109911851",
      "1.514396395"
    ],
    [
      "1.336049886",
      "1.52731235"
    ],
    [
      "0.300489954",
      "1.544982036"
    ],
    [
      "0.747538658",
      "1.588654663"
    ],
    [
      "0.521154291",
      "1.596094487"
    ],
    [
      "1.613348691",
      "1.640408547"
    ],
    [
      "0.954975295",
      "1.679623198"
    ],
    [
      "1.212136725",
      "1.71691921"
    ],
    [
      "1.431095601",
      "1.774904063"
    ],
    [
      "0.618636659",
      "1.800550889"
    ],
    [
      "0.833892582",
      "1.87104997"
    ],
    [
      "1.059986563",
      "1.884715399"
    ]
  ],
  "radius": "0.113253291"
}
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": 62
}

起步布局答案

{
  "centers": [
    [
      "1.064284521",
      "0.115586508"
    ],
    [
      "0.83812681",
      "0.128153228"
    ],
    [
      "1.28624784",
      "0.160725374"
    ],
    [
      "0.622530886",
      "0.197605588"
    ],
    [
      "1.489534264",
      "0.260624636"
    ],
    [
      "0.4315638",
      "0.319412011"
    ],
    [
      "1.131290073",
      "0.33195538"
    ],
    [
      "0.905132362",
      "0.344522099"
    ],
    [
      "1.660879902",
      "0.408766139"
    ],
    [
      "0.689536438",
      "0.41397446"
    ],
    [
      "1.336574044",
      "0.427682721"
    ],
    [
      "0.277685637",
      "0.485624957"
    ],
    [
      "1.029094315",
      "0.534097062"
    ],
    [
      "0.498569352",
      "0.535780882"
    ],
    [
      "1.507919682",
      "0.575824223"
    ],
    [
      "1.789104917",
      "0.595484049"
    ],
    [
      "0.811641602",
      "0.607376525"
    ],
    [
      "1.234536476",
      "0.630000977"
    ],
    [
      "0.17093653",
      "0.685399481"
    ],
    [
      "0.618204612",
      "0.728115583"
    ],
    [
      "0.391820245",
      "0.735555407"
    ],
    [
      "0.990685753",
      "0.757323438"
    ],
    [
      "1.636144697",
      "0.762542134"
    ],
    [
      "1.410329735",
      "0.780229299"
    ],
    [
      "1.865842973",
      "0.808595531"
    ],
    [
      "1.195899494",
      "0.85320124"
    ],
    [
      "0.794432026",
      "0.870415026"
    ],
    [
      "0.118281569",
      "0.905700838"
    ],
    [
      "0.565549651",
      "0.94841694"
    ],
    [
      "0.339165284",
      "0.955856763"
    ],
    [
      "1.010260011",
      "0.982982652"
    ],
    [
      "1.664495195",
      "0.987267481"
    ],
    [
      "1.438680233",
      "1.004954646"
    ],
    [
      "1.886087119",
      "1.034195635"
    ],
    [
      "1.224249992",
      "1.077926588"
    ],
    [
      "0.782989585",
      "1.096632406"
    ],
    [
      "0.123156348",
      "1.132154959"
    ],
    [
      "0.57042443",
      "1.174871061"
    ],
    [
      "0.344040063",
      "1.182310885"
    ],
    [
      "0.980494225",
      "1.207524921"
    ],
    [
      "1.626924556",
      "1.210636413"
    ],
    [
      "1.401109595",
      "1.228323578"
    ],
    [
      "1.848516481",
      "1.257564567"
    ],
    [
      "1.186679354",
      "1.30129552"
    ],
    [
      "0.1852428",
      "1.349986328"
    ],
    [
      "0.682575268",
      "1.371663902"
    ],
    [
      "0.456190901",
      "1.379103725"
    ],
    [
      "1.533990515",
      "1.417199955"
    ],
    [
      "0.903726722",
      "1.420625797"
    ],
    [
      "1.75558244",
      "1.464128108"
    ],
    [
      "1.109911851",
      "1.514396395"
    ],
    [
      "1.336049886",
      "1.52731235"
    ],
    [
      "0.300489954",
      "1.544982036"
    ],
    [
      "0.747538658",
      "1.588654663"
    ],
    [
      "0.521154291",
      "1.596094487"
    ],
    [
      "1.613348691",
      "1.640408547"
    ],
    [
      "0.954975295",
      "1.679623198"
    ],
    [
      "1.212136725",
      "1.71691921"
    ],
    [
      "1.431095601",
      "1.774904063"
    ],
    [
      "0.618636659",
      "1.800550889"
    ],
    [
      "0.833892582",
      "1.87104997"
    ],
    [
      "1.059986563",
      "1.884715399"
    ]
  ],
  "radius": "0.113253291"
}

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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