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 = 139

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 = 139
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[0.0768699347224580998917412756185, 0.0768699347224580998917412756195]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 = 0.076869934

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 = 139 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": [
    [
      "0.940182017",
      "0.078810049"
    ],
    [
      "1.093895513",
      "0.081657607"
    ],
    [
      "1.245004703",
      "0.10997651"
    ],
    [
      "0.721339719",
      "0.120013928"
    ],
    [
      "1.389318389",
      "0.162981297"
    ],
    [
      "0.576653238",
      "0.179753355"
    ],
    [
      "0.859874799",
      "0.209908096"
    ],
    [
      "1.144925219",
      "0.226681456"
    ],
    [
      "0.442707897",
      "0.264068869"
    ],
    [
      "1.492735993",
      "0.2767387"
    ],
    [
      "0.722707533",
      "0.279342156"
    ],
    [
      "1.289238905",
      "0.279686243"
    ],
    [
      "0.99530647",
      "0.285130034"
    ],
    [
      "1.63645119",
      "0.331345381"
    ],
    [
      "0.579272572",
      "0.334680745"
    ],
    [
      "0.852480246",
      "0.363470032"
    ],
    [
      "1.118362804",
      "0.378109279"
    ],
    [
      "1.392656509",
      "0.393443646"
    ],
    [
      "0.296917258",
      "0.401799553"
    ],
    [
      "0.449838556",
      "0.417643285"
    ],
    [
      "0.71531298",
      "0.432904092"
    ],
    [
      "1.738597095",
      "0.44624604"
    ],
    [
      "0.981195539",
      "0.447543339"
    ],
    [
      "1.251394746",
      "0.455918054"
    ],
    [
      "1.534220335",
      "0.460823966"
    ],
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      "0.578145715",
      "0.502338152"
    ],
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      "0.844028273",
      "0.516977399"
    ],
    [
      "0.207388229",
      "0.526781554"
    ],
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      "1.109910831",
      "0.531616645"
    ],
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      "1.37579339",
      "0.546255892"
    ],
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      "1.645212231",
      "0.568373902"
    ],
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      "0.42713905",
      "0.570361126"
    ],
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    ],
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      "0.09615577",
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    ],
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      "0.827124327",
      "0.823992132"
    ],
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      "1.774110171",
      "0.830757537"
    ],
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    ],
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      "1.358889444",
      "0.853270625"
    ],
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      "1.6250812",
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      "0.423447733",
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    ],
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      "0.961366053"
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    ],
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      "1.787773605",
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    ],
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    ],
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    ],
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      "1.764984237",
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    ],
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      "1.076102939",
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    ],
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      "1.341985498",
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    ],
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      "1.609378886",
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    ],
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      "0.407170557",
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    ],
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    ],
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    ],
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    ],
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    ],
    [
      "0.53588585",
      "1.269874985"
    ],
    [
      "1.717277584",
      "1.282081143"
    ],
    [
      "0.801768408",
      "1.284514232"
    ],
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      "1.067650966",
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    ],
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    [
      "1.044244018",
      "1.922069186"
    ]
  ],
  "radius": "0.076869934"
}
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": 139
}

起步布局答案

{
  "centers": [
    [
      "0.940182017",
      "0.078810049"
    ],
    [
      "1.093895513",
      "0.081657607"
    ],
    [
      "1.245004703",
      "0.10997651"
    ],
    [
      "0.721339719",
      "0.120013928"
    ],
    [
      "1.389318389",
      "0.162981297"
    ],
    [
      "0.576653238",
      "0.179753355"
    ],
    [
      "0.859874799",
      "0.209908096"
    ],
    [
      "1.144925219",
      "0.226681456"
    ],
    [
      "0.442707897",
      "0.264068869"
    ],
    [
      "1.492735993",
      "0.2767387"
    ],
    [
      "0.722707533",
      "0.279342156"
    ],
    [
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      "0.279686243"
    ],
    [
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      "0.285130034"
    ],
    [
      "1.63645119",
      "0.331345381"
    ],
    [
      "0.579272572",
      "0.334680745"
    ],
    [
      "0.852480246",
      "0.363470032"
    ],
    [
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      "0.378109279"
    ],
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  "radius": "0.076869934"
}

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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还没有帖子。第一个聊聊这道题的思路?