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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 = 43

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 = 43
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[0.1347718910802124099360443648095, 0.1347718910802124099360443648105]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 = 0.134771890

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 = 43 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": [
    [
      "0.998270637",
      "0.134773619"
    ],
    [
      "1.26460781",
      "0.176226738"
    ],
    [
      "0.7321013",
      "0.17729108"
    ],
    [
      "1.505264685",
      "0.297627393"
    ],
    [
      "0.491931643",
      "0.299652783"
    ],
    [
      "0.869566817",
      "0.409146816"
    ],
    [
      "1.139052512",
      "0.414742393"
    ],
    [
      "1.696885412",
      "0.487193601"
    ],
    [
      "0.30107023",
      "0.489983473"
    ],
    [
      "1.379709387",
      "0.536143048"
    ],
    [
      "0.562185758",
      "0.55988001"
    ],
    [
      "0.807550613",
      "0.671459303"
    ],
    [
      "1.077036308",
      "0.67705488"
    ],
    [
      "1.820873122",
      "0.726527883"
    ],
    [
      "1.55186705",
      "0.743545048"
    ],
    [
      "0.325207847",
      "0.758444319"
    ],
    [
      "0.588237683",
      "0.828161854"
    ],
    [
      "1.297364341",
      "0.832326926"
    ],
    [
      "0.834185225",
      "0.939683925"
    ],
    [
      "0.136183317",
      "0.950599407"
    ],
    [
      "1.865194754",
      "0.992402747"
    ],
    [
      "1.596188682",
      "1.009419913"
    ],
    [
      "1.09445966",
      "1.009762945"
    ],
    [
      "0.397106769",
      "1.018221908"
    ],
    [
      "1.345413631",
      "1.108375484"
    ],
    [
      "0.643357796",
      "1.130048688"
    ],
    [
      "0.903632231",
      "1.200127708"
    ],
    [
      "0.162898319",
      "1.218816034"
    ],
    [
      "1.825548877",
      "1.259014928"
    ],
    [
      "1.556542805",
      "1.276032093"
    ],
    [
      "0.423821771",
      "1.286438535"
    ],
    [
      "1.155273996",
      "1.301340149"
    ],
    [
      "1.36828052",
      "1.468934044"
    ],
    [
      "0.891071088",
      "1.469378647"
    ],
    [
      "0.621527442",
      "1.469649666"
    ],
    [
      "1.705783138",
      "1.500489602"
    ],
    [
      "0.320222746",
      "1.535278027"
    ],
    [
      "1.129792622",
      "1.594542058"
    ],
    [
      "1.517520854",
      "1.693391553"
    ],
    [
      "0.787472063",
      "1.718218138"
    ],
    [
      "0.517928417",
      "1.718489157"
    ],
    [
      "1.279032956",
      "1.818999567"
    ],
    [
      "1.013464794",
      "1.865123332"
    ]
  ],
  "radius": "0.13477189"
}
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": 43
}

起步布局答案

{
  "centers": [
    [
      "0.998270637",
      "0.134773619"
    ],
    [
      "1.26460781",
      "0.176226738"
    ],
    [
      "0.7321013",
      "0.17729108"
    ],
    [
      "1.505264685",
      "0.297627393"
    ],
    [
      "0.491931643",
      "0.299652783"
    ],
    [
      "0.869566817",
      "0.409146816"
    ],
    [
      "1.139052512",
      "0.414742393"
    ],
    [
      "1.696885412",
      "0.487193601"
    ],
    [
      "0.30107023",
      "0.489983473"
    ],
    [
      "1.379709387",
      "0.536143048"
    ],
    [
      "0.562185758",
      "0.55988001"
    ],
    [
      "0.807550613",
      "0.671459303"
    ],
    [
      "1.077036308",
      "0.67705488"
    ],
    [
      "1.820873122",
      "0.726527883"
    ],
    [
      "1.55186705",
      "0.743545048"
    ],
    [
      "0.325207847",
      "0.758444319"
    ],
    [
      "0.588237683",
      "0.828161854"
    ],
    [
      "1.297364341",
      "0.832326926"
    ],
    [
      "0.834185225",
      "0.939683925"
    ],
    [
      "0.136183317",
      "0.950599407"
    ],
    [
      "1.865194754",
      "0.992402747"
    ],
    [
      "1.596188682",
      "1.009419913"
    ],
    [
      "1.09445966",
      "1.009762945"
    ],
    [
      "0.397106769",
      "1.018221908"
    ],
    [
      "1.345413631",
      "1.108375484"
    ],
    [
      "0.643357796",
      "1.130048688"
    ],
    [
      "0.903632231",
      "1.200127708"
    ],
    [
      "0.162898319",
      "1.218816034"
    ],
    [
      "1.825548877",
      "1.259014928"
    ],
    [
      "1.556542805",
      "1.276032093"
    ],
    [
      "0.423821771",
      "1.286438535"
    ],
    [
      "1.155273996",
      "1.301340149"
    ],
    [
      "1.36828052",
      "1.468934044"
    ],
    [
      "0.891071088",
      "1.469378647"
    ],
    [
      "0.621527442",
      "1.469649666"
    ],
    [
      "1.705783138",
      "1.500489602"
    ],
    [
      "0.320222746",
      "1.535278027"
    ],
    [
      "1.129792622",
      "1.594542058"
    ],
    [
      "1.517520854",
      "1.693391553"
    ],
    [
      "0.787472063",
      "1.718218138"
    ],
    [
      "0.517928417",
      "1.718489157"
    ],
    [
      "1.279032956",
      "1.818999567"
    ],
    [
      "1.013464794",
      "1.865123332"
    ]
  ],
  "radius": "0.13477189"
}

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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