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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 = 59

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 = 59
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[0.1163805649960470182428063380505, 0.1163805649960470182428063380515]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 = 0.116380564

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 = 59 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": [
    [
      "0.941882206",
      "0.118293911"
    ],
    [
      "1.17413237",
      "0.13370835"
    ],
    [
      "0.713664775",
      "0.164060143"
    ],
    [
      "1.39429967",
      "0.209233867"
    ],
    [
      "0.505315849",
      "0.267831374"
    ],
    [
      "0.957961766",
      "0.355393863"
    ],
    [
      "1.191238921",
      "0.365840014"
    ],
    [
      "1.638159823",
      "0.388824619"
    ],
    [
      "0.701830459",
      "0.396520231"
    ],
    [
      "0.331292543",
      "0.422407027"
    ],
    [
      "1.411406221",
      "0.441365532"
    ],
    [
      "0.526339792",
      "0.549427946"
    ],
    [
      "0.859963612",
      "0.567317332"
    ],
    [
      "1.09253804",
      "0.576638211"
    ],
    [
      "1.775611188",
      "0.576667281"
    ],
    [
      "0.203670153",
      "0.617061257"
    ],
    [
      "1.548441801",
      "0.629727399"
    ],
    [
      "1.31270534",
      "0.652163728"
    ],
    [
      "0.684472945",
      "0.720225048"
    ],
    [
      "0.398717403",
      "0.744082177"
    ],
    [
      "1.859243698",
      "0.793884562"
    ],
    [
      "0.904640244",
      "0.795750565"
    ],
    [
      "1.137214673",
      "0.805071444"
    ],
    [
      "0.131304265",
      "0.83828721"
    ],
    [
      "1.449270402",
      "0.840651731"
    ],
    [
      "0.556850555",
      "0.914879278"
    ],
    [
      "1.666470007",
      "0.924330136"
    ],
    [
      "0.326351515",
      "0.965308129"
    ],
    [
      "0.780280912",
      "0.99250549"
    ],
    [
      "1.012855341",
      "1.001826368"
    ],
    [
      "1.249225256",
      "1.009109119"
    ],
    [
      "1.883254178",
      "1.025403979"
    ],
    [
      "0.119362493",
      "1.07183667"
    ],
    [
      "1.473280883",
      "1.072171148"
    ],
    [
      "0.602771989",
      "1.14306553"
    ],
    [
      "1.690480488",
      "1.155849553"
    ],
    [
      "0.375395897",
      "1.192843604"
    ],
    [
      "0.889096285",
      "1.198959419"
    ],
    [
      "1.124865924",
      "1.205864043"
    ],
    [
      "1.352151373",
      "1.270930765"
    ],
    [
      "0.169090275",
      "1.300619918"
    ],
    [
      "0.711181198",
      "1.349039283"
    ],
    [
      "1.569350977",
      "1.35460917"
    ],
    [
      "1.808532711",
      "1.356452186"
    ],
    [
      "0.482722674",
      "1.399383446"
    ],
    [
      "0.95069133",
      "1.423422766"
    ],
    [
      "1.183323667",
      "1.431164811"
    ],
    [
      "0.276417052",
      "1.50715976"
    ],
    [
      "1.400523272",
      "1.514843216"
    ],
    [
      "1.687403201",
      "1.555211802"
    ],
    [
      "0.64025812",
      "1.570732"
    ],
    [
      "0.862797773",
      "1.638951108"
    ],
    [
      "1.09543011",
      "1.646693153"
    ],
    [
      "0.433952498",
      "1.678508314"
    ],
    [
      "1.518575495",
      "1.715445848"
    ],
    [
      "0.632009478",
      "1.803346925"
    ],
    [
      "1.313764374",
      "1.826035849"
    ],
    [
      "0.854549131",
      "1.871566033"
    ],
    [
      "1.087181468",
      "1.879308079"
    ]
  ],
  "radius": "0.116380564"
}
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": 59
}

起步布局答案

{
  "centers": [
    [
      "0.941882206",
      "0.118293911"
    ],
    [
      "1.17413237",
      "0.13370835"
    ],
    [
      "0.713664775",
      "0.164060143"
    ],
    [
      "1.39429967",
      "0.209233867"
    ],
    [
      "0.505315849",
      "0.267831374"
    ],
    [
      "0.957961766",
      "0.355393863"
    ],
    [
      "1.191238921",
      "0.365840014"
    ],
    [
      "1.638159823",
      "0.388824619"
    ],
    [
      "0.701830459",
      "0.396520231"
    ],
    [
      "0.331292543",
      "0.422407027"
    ],
    [
      "1.411406221",
      "0.441365532"
    ],
    [
      "0.526339792",
      "0.549427946"
    ],
    [
      "0.859963612",
      "0.567317332"
    ],
    [
      "1.09253804",
      "0.576638211"
    ],
    [
      "1.775611188",
      "0.576667281"
    ],
    [
      "0.203670153",
      "0.617061257"
    ],
    [
      "1.548441801",
      "0.629727399"
    ],
    [
      "1.31270534",
      "0.652163728"
    ],
    [
      "0.684472945",
      "0.720225048"
    ],
    [
      "0.398717403",
      "0.744082177"
    ],
    [
      "1.859243698",
      "0.793884562"
    ],
    [
      "0.904640244",
      "0.795750565"
    ],
    [
      "1.137214673",
      "0.805071444"
    ],
    [
      "0.131304265",
      "0.83828721"
    ],
    [
      "1.449270402",
      "0.840651731"
    ],
    [
      "0.556850555",
      "0.914879278"
    ],
    [
      "1.666470007",
      "0.924330136"
    ],
    [
      "0.326351515",
      "0.965308129"
    ],
    [
      "0.780280912",
      "0.99250549"
    ],
    [
      "1.012855341",
      "1.001826368"
    ],
    [
      "1.249225256",
      "1.009109119"
    ],
    [
      "1.883254178",
      "1.025403979"
    ],
    [
      "0.119362493",
      "1.07183667"
    ],
    [
      "1.473280883",
      "1.072171148"
    ],
    [
      "0.602771989",
      "1.14306553"
    ],
    [
      "1.690480488",
      "1.155849553"
    ],
    [
      "0.375395897",
      "1.192843604"
    ],
    [
      "0.889096285",
      "1.198959419"
    ],
    [
      "1.124865924",
      "1.205864043"
    ],
    [
      "1.352151373",
      "1.270930765"
    ],
    [
      "0.169090275",
      "1.300619918"
    ],
    [
      "0.711181198",
      "1.349039283"
    ],
    [
      "1.569350977",
      "1.35460917"
    ],
    [
      "1.808532711",
      "1.356452186"
    ],
    [
      "0.482722674",
      "1.399383446"
    ],
    [
      "0.95069133",
      "1.423422766"
    ],
    [
      "1.183323667",
      "1.431164811"
    ],
    [
      "0.276417052",
      "1.50715976"
    ],
    [
      "1.400523272",
      "1.514843216"
    ],
    [
      "1.687403201",
      "1.555211802"
    ],
    [
      "0.64025812",
      "1.570732"
    ],
    [
      "0.862797773",
      "1.638951108"
    ],
    [
      "1.09543011",
      "1.646693153"
    ],
    [
      "0.433952498",
      "1.678508314"
    ],
    [
      "1.518575495",
      "1.715445848"
    ],
    [
      "0.632009478",
      "1.803346925"
    ],
    [
      "1.313764374",
      "1.826035849"
    ],
    [
      "0.854549131",
      "1.871566033"
    ],
    [
      "1.087181468",
      "1.879308079"
    ]
  ],
  "radius": "0.116380564"
}

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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