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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 = 60

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 = 60
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[0.1156574801328149894922258673595, 0.1156574801328149894922258673605]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 = 0.115657479

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 = 60 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": [
    [
      "1.045658831",
      "0.116836955"
    ],
    [
      "0.815074582",
      "0.135208533"
    ],
    [
      "1.273119223",
      "0.158889078"
    ],
    [
      "0.597142445",
      "0.212746875"
    ],
    [
      "1.481893513",
      "0.258487806"
    ],
    [
      "0.946276959",
      "0.325714561"
    ],
    [
      "0.406772766",
      "0.344147013"
    ],
    [
      "1.173737351",
      "0.367766684"
    ],
    [
      "0.728344822",
      "0.403252904"
    ],
    [
      "1.657697913",
      "0.408818853"
    ],
    [
      "1.382511641",
      "0.467365412"
    ],
    [
      "0.256990139",
      "0.520418892"
    ],
    [
      "0.537975143",
      "0.534653041"
    ],
    [
      "0.90548418",
      "0.553404175"
    ],
    [
      "1.132944572",
      "0.595456298"
    ],
    [
      "1.790872857",
      "0.604295797"
    ],
    [
      "1.559993586",
      "0.618486389"
    ],
    [
      "0.714586192",
      "0.684035598"
    ],
    [
      "1.341718863",
      "0.695055025"
    ],
    [
      "0.388192516",
      "0.710924921"
    ],
    [
      "0.157710497",
      "0.730537412"
    ],
    [
      "0.923554275",
      "0.784037858"
    ],
    [
      "1.866432362",
      "0.822921898"
    ],
    [
      "1.151076689",
      "0.826059501"
    ],
    [
      "1.635553091",
      "0.83711249"
    ],
    [
      "0.564803565",
      "0.860307478"
    ],
    [
      "1.417278368",
      "0.913681126"
    ],
    [
      "0.347128952",
      "0.938565854"
    ],
    [
      "0.116646933",
      "0.958178345"
    ],
    [
      "0.773478733",
      "0.960113716"
    ],
    [
      "1.000960923",
      "1.00204776"
    ],
    [
      "1.227949473",
      "1.046576501"
    ],
    [
      "1.882712836",
      "1.053663218"
    ],
    [
      "1.651833565",
      "1.06785381"
    ],
    [
      "0.585704272",
      "1.095196516"
    ],
    [
      "1.435276566",
      "1.149153729"
    ],
    [
      "0.366502088",
      "1.169068116"
    ],
    [
      "0.854153647",
      "1.180805235"
    ],
    [
      "0.136020069",
      "1.188680607"
    ],
    [
      "1.081142198",
      "1.225333975"
    ],
    [
      "1.838600412",
      "1.280733042"
    ],
    [
      "1.608325924",
      "1.302648127"
    ],
    [
      "0.664188651",
      "1.312789745"
    ],
    [
      "1.325477729",
      "1.352748489"
    ],
    [
      "0.444986466",
      "1.386661344"
    ],
    [
      "0.214504447",
      "1.406273835"
    ],
    [
      "0.889448571",
      "1.412470897"
    ],
    [
      "1.117397861",
      "1.453789962"
    ],
    [
      "1.72974474",
      "1.499534089"
    ],
    [
      "1.498527088",
      "1.506242888"
    ],
    [
      "0.63481158",
      "1.542231674"
    ],
    [
      "0.346730372",
      "1.596070873"
    ],
    [
      "1.29327011",
      "1.612902082"
    ],
    [
      "0.84432303",
      "1.640270253"
    ],
    [
      "1.072090786",
      "1.680624439"
    ],
    [
      "1.48826817",
      "1.737330242"
    ],
    [
      "0.523651293",
      "1.745086305"
    ],
    [
      "0.733162743",
      "1.843124884"
    ],
    [
      "1.257269169",
      "1.843966684"
    ],
    [
      "0.960930499",
      "1.88347907"
    ]
  ],
  "radius": "0.115657479"
}
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": 60
}

起步布局答案

{
  "centers": [
    [
      "1.045658831",
      "0.116836955"
    ],
    [
      "0.815074582",
      "0.135208533"
    ],
    [
      "1.273119223",
      "0.158889078"
    ],
    [
      "0.597142445",
      "0.212746875"
    ],
    [
      "1.481893513",
      "0.258487806"
    ],
    [
      "0.946276959",
      "0.325714561"
    ],
    [
      "0.406772766",
      "0.344147013"
    ],
    [
      "1.173737351",
      "0.367766684"
    ],
    [
      "0.728344822",
      "0.403252904"
    ],
    [
      "1.657697913",
      "0.408818853"
    ],
    [
      "1.382511641",
      "0.467365412"
    ],
    [
      "0.256990139",
      "0.520418892"
    ],
    [
      "0.537975143",
      "0.534653041"
    ],
    [
      "0.90548418",
      "0.553404175"
    ],
    [
      "1.132944572",
      "0.595456298"
    ],
    [
      "1.790872857",
      "0.604295797"
    ],
    [
      "1.559993586",
      "0.618486389"
    ],
    [
      "0.714586192",
      "0.684035598"
    ],
    [
      "1.341718863",
      "0.695055025"
    ],
    [
      "0.388192516",
      "0.710924921"
    ],
    [
      "0.157710497",
      "0.730537412"
    ],
    [
      "0.923554275",
      "0.784037858"
    ],
    [
      "1.866432362",
      "0.822921898"
    ],
    [
      "1.151076689",
      "0.826059501"
    ],
    [
      "1.635553091",
      "0.83711249"
    ],
    [
      "0.564803565",
      "0.860307478"
    ],
    [
      "1.417278368",
      "0.913681126"
    ],
    [
      "0.347128952",
      "0.938565854"
    ],
    [
      "0.116646933",
      "0.958178345"
    ],
    [
      "0.773478733",
      "0.960113716"
    ],
    [
      "1.000960923",
      "1.00204776"
    ],
    [
      "1.227949473",
      "1.046576501"
    ],
    [
      "1.882712836",
      "1.053663218"
    ],
    [
      "1.651833565",
      "1.06785381"
    ],
    [
      "0.585704272",
      "1.095196516"
    ],
    [
      "1.435276566",
      "1.149153729"
    ],
    [
      "0.366502088",
      "1.169068116"
    ],
    [
      "0.854153647",
      "1.180805235"
    ],
    [
      "0.136020069",
      "1.188680607"
    ],
    [
      "1.081142198",
      "1.225333975"
    ],
    [
      "1.838600412",
      "1.280733042"
    ],
    [
      "1.608325924",
      "1.302648127"
    ],
    [
      "0.664188651",
      "1.312789745"
    ],
    [
      "1.325477729",
      "1.352748489"
    ],
    [
      "0.444986466",
      "1.386661344"
    ],
    [
      "0.214504447",
      "1.406273835"
    ],
    [
      "0.889448571",
      "1.412470897"
    ],
    [
      "1.117397861",
      "1.453789962"
    ],
    [
      "1.72974474",
      "1.499534089"
    ],
    [
      "1.498527088",
      "1.506242888"
    ],
    [
      "0.63481158",
      "1.542231674"
    ],
    [
      "0.346730372",
      "1.596070873"
    ],
    [
      "1.29327011",
      "1.612902082"
    ],
    [
      "0.84432303",
      "1.640270253"
    ],
    [
      "1.072090786",
      "1.680624439"
    ],
    [
      "1.48826817",
      "1.737330242"
    ],
    [
      "0.523651293",
      "1.745086305"
    ],
    [
      "0.733162743",
      "1.843124884"
    ],
    [
      "1.257269169",
      "1.843966684"
    ],
    [
      "0.960930499",
      "1.88347907"
    ]
  ],
  "radius": "0.115657479"
}

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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