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 = 244

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 = 244
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[0.0587033382017620243708350657905, 0.0587033382017620243708350657915]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 = 0.058703337

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 = 244 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": [
    [
      "0.995248542",
      "0.226242334"
    ],
    [
      "0.688846955",
      "0.266083662"
    ],
    [
      "1.938313146",
      "0.925114446"
    ],
    [
      "1.59122508",
      "0.425487138"
    ],
    [
      "1.135750563",
      "0.847850858"
    ],
    [
      "0.288040702",
      "1.108477208"
    ],
    [
      "1.940222276",
      "1.044840589"
    ],
    [
      "0.882422965",
      "1.292516298"
    ],
    [
      "1.578488393",
      "0.54220091"
    ],
    [
      "0.666976001",
      "1.385888055"
    ],
    [
      "1.071769706",
      "1.431387182"
    ],
    [
      "1.084565877",
      "1.314679917"
    ],
    [
      "1.38138216",
      "1.406278978"
    ],
    [
      "0.800545766",
      "1.106373592"
    ],
    [
      "1.647505929",
      "0.844906273"
    ],
    [
      "1.821875894",
      "0.541132459"
    ],
    [
      "1.304111297",
      "0.216736422"
    ],
    [
      "0.573597928",
      "0.160840053"
    ],
    [
      "0.663144656",
      "0.499647866"
    ],
    [
      "0.186747145",
      "0.804544144"
    ],
    [
      "1.122954391",
      "0.964558123"
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    [
      "1.363485818",
      "0.636599947"
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    [
      "1.350689646",
      "0.753307212"
    ],
    [
      "1.777937351",
      "0.758285825"
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      "0.147377196"
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      "1.189240251",
      "0.241005135"
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      "1.292160791"
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      "1.527240216",
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      "0.895219137",
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  ],
  "radius": "0.058703337"
}
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": 244
}

起步布局答案

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}

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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