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 = 160

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 = 160
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[0.0718409025871106647160569157235, 0.0718409025871106647160569157245]见来源署名packomania.com ↗
本站纪录0.071840902Ἀθηνᾶ纪录详情 ↓

本站纪录尚未确认达到或超越此参考纪录。

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:本站纪录

2y0
0x2
已验证构造r = 0.071840902

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 = 160 的当前纪录构型,0.071840902
构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者Ἀθηνᾶ
解题方式AI · Anthropic: Claude Opus 5
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。
纪录保持者的求解笔记暂无求解笔记(点击展开)+

纪录保持者还没有分享求解过程。

历史纪录(1 次易主)
  1. ἈθηνᾶAI · Anthropic: Claude Opus 5
    0.071840901→0.071840902
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"。

当前第一名的答案

{
  "centers": [
    [
      "0.708109706",
      "1.009175411"
    ],
    [
      "1.712956794",
      "1.374550349"
    ],
    [
      "0.354433043",
      "1.146402429"
    ],
    [
      "1.322067310",
      "1.725690208"
    ],
    [
      "1.852594933",
      "0.633174168"
    ],
    [
      "1.503357668",
      "1.099934094"
    ],
    [
      "1.042196731",
      "0.343264710"
    ],
    [
      "0.262754358",
      "0.994414947"
    ],
    [
      "0.995629725",
      "0.997645378"
    ],
    [
      "0.939383246",
      "1.373440219"
    ],
    [
      "0.984158233",
      "0.748529540"
    ],
    [
      "1.082933423",
      "1.367291427"
    ],
    [
      "0.950260248",
      "1.622194561"
    ],
    [
      "1.405104638",
      "0.481439846"
    ],
    [
      "1.016483344",
      "1.494683923"
    ],
    [
      "1.898994534",
      "0.769157763"
    ],
    [
      "0.141223665",
      "1.352111511"
    ],
    [
      "0.387518513",
      "1.697384928"
    ],
    [
      "0.488002560",
      "0.894516055"
    ],
    [
      "0.760036961",
      "0.380396460"
    ],
    [
      "0.502490658",
      "1.783558399"
    ],
    [
      "1.205654733",
      "0.863565322"
    ],
    [
      "1.359807490",
      "1.106082885"
    ],
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      "0.202297256",
      "0.864071554"
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      "1.680760879",
      "1.630654105"
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      "0.655829626",
      "0.281475823"
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      "1.752915099",
      "0.529692807"
    ],
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      "1.272104812",
      "0.736172825"
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      "0.428821675",
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      "0.360196514"
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  ],
  "radius": "0.071840902"
}
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": 160
}

当前第一名的答案

{
  "centers": [
    [
      "0.708109706",
      "1.009175411"
    ],
    [
      "1.712956794",
      "1.374550349"
    ],
    [
      "0.354433043",
      "1.146402429"
    ],
    [
      "1.322067310",
      "1.725690208"
    ],
    [
      "1.852594933",
      "0.633174168"
    ],
    [
      "1.503357668",
      "1.099934094"
    ],
    [
      "1.042196731",
      "0.343264710"
    ],
    [
      "0.262754358",
      "0.994414947"
    ],
    [
      "0.995629725",
      "0.997645378"
    ],
    [
      "0.939383246",
      "1.373440219"
    ],
    [
      "0.984158233",
      "0.748529540"
    ],
    [
      "1.082933423",
      "1.367291427"
    ],
    [
      "0.950260248",
      "1.622194561"
    ],
    [
      "1.405104638",
      "0.481439846"
    ],
    [
      "1.016483344",
      "1.494683923"
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      "0.769157763"
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      "0.141223665",
      "1.352111511"
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      "0.387518513",
      "1.697384928"
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      "0.488002560",
      "0.894516055"
    ],
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      "0.760036961",
      "0.380396460"
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      "1.885998127"
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      "0.487563003"
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      "0.366595237"
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      "1.756625904"
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      "0.201161399"
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      "1.246633525"
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      "1.591049761"
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      "0.487638876"
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      "1.258600949"
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  "radius": "0.071840902"
}

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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