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 = 251

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 = 251
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[0.0578188757756518754389769538075, 0.0578188757756518754389769538085]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 = 0.057818875

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 = 251 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": [
    [
      "0.946425164",
      "1.496649083"
    ],
    [
      "0.535357358",
      "0.799328749"
    ],
    [
      "1.060243227",
      "1.295318989"
    ],
    [
      "0.780005039",
      "1.602452392"
    ],
    [
      "1.350233207",
      "1.392840467"
    ],
    [
      "1.519145361",
      "0.890563088"
    ],
    [
      "1.117152259",
      "1.194653942"
    ],
    [
      "1.449740733",
      "0.364381513"
    ],
    [
      "1.688404264",
      "0.571288638"
    ],
    [
      "1.695317278",
      "1.18941466"
    ],
    [
      "0.54080218",
      "1.400175462"
    ],
    [
      "0.723870915",
      "1.70423314"
    ],
    [
      "1.523313354",
      "1.717975459"
    ],
    [
      "0.651474296",
      "0.395403139"
    ],
    [
      "1.056613346",
      "0.894754514"
    ],
    [
      "0.825347338",
      "0.896850227"
    ],
    [
      "0.772068188",
      "1.398079749"
    ],
    [
      "0.882740304",
      "0.393307426"
    ],
    [
      "1.353507213",
      "1.793613874"
    ],
    [
      "0.235586723",
      "1.523319988"
    ],
    [
      "1.227340441",
      "0.592759372"
    ],
    [
      "0.830797195",
      "1.498381896"
    ],
    [
      "1.054798405",
      "0.694472276"
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    [
      "0.599526152",
      "1.499792653"
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    [
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      "1.191510373"
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      "0.417912487",
      "0.600092556"
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      "0.821717457",
      "0.496285751"
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  "radius": "0.057818875"
}
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": 251
}

起步布局答案

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  "radius": "0.057818875"
}

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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