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 = 164

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 = 164
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[0.0708737244619971200258426261005, 0.0708737244619971200258426261015]见来源署名packomania.com ↗
本站纪录0.070873724Ἀθηνᾶ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 = 0.070873724

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 = 164 的当前纪录构型,0.070873724
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
    0.070873723→0.070873724
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": [
    [
      "0.558029061",
      "1.657394215"
    ],
    [
      "0.618820252",
      "1.086916483"
    ],
    [
      "1.368605354",
      "1.178287899"
    ],
    [
      "1.205083237",
      "0.444213729"
    ],
    [
      "1.141062238",
      "1.007573328"
    ],
    [
      "0.171204736",
      "1.419969100"
    ],
    [
      "1.548778189",
      "0.699199187"
    ],
    [
      "0.162258155",
      "0.598173872"
    ],
    [
      "0.798628154",
      "0.752747307"
    ],
    [
      "0.877754375",
      "1.118080920"
    ],
    [
      "0.814328347",
      "0.611872032"
    ],
    [
      "1.779822988",
      "0.863444371"
    ],
    [
      "1.189400471",
      "0.585090945"
    ],
    [
      "1.601496715",
      "0.291849354"
    ],
    [
      "0.605128730",
      "0.159697578"
    ],
    [
      "1.518054755",
      "0.981023537"
    ],
    [
      "0.551762930",
      "0.729555232"
    ],
    [
      "1.789157777",
      "1.145243088"
    ],
    [
      "1.450326468",
      "0.474289459"
    ],
    [
      "1.349938537",
      "1.318800849"
    ],
    [
      "1.316946631",
      "0.199980831"
    ],
    [
      "0.845142447",
      "0.330057375"
    ],
    [
      "0.991151388",
      "1.203130650"
    ],
    [
      "0.321852646",
      "0.364868673"
    ],
    [
      "0.221949514",
      "0.868669814"
    ],
    [
      "1.026931097",
      "0.920816113"
    ],
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      "1.067841347",
      "1.569808066"
    ],
    [
      "0.973407808",
      "1.343763167"
    ],
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      "1.403226333",
      "0.896915335"
    ],
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      "1.312907126",
      "1.599808880"
    ],
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      "1.073794048",
      "0.497650497"
    ],
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      "1.218259102",
      "1.372802370"
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      "0.727891261",
      "0.249484638"
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      "1.237584161",
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      "1.867417041",
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      "0.813113951"
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      "1.196166965"
    ],
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      "1.289009245"
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      "0.386500023"
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      "1.303551942",
      "0.669125348"
    ],
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      "0.466907829",
      "1.424924927"
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      "1.386151336",
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      "1.743225465",
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      "1.302048964",
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      "1.928489117"
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      "1.712836887"
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      "1.819381908"
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      "1.735574687"
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      "0.447804681"
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      "1.611577440",
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      "0.483864549",
      "1.284195367"
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      "0.749200613",
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      "0.855955129",
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      "0.921713794",
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      "1.565654486"
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      "0.371945074"
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      "1.585844542",
      "0.432729973"
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      "1.533077997",
      "0.840074463"
    ],
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      "0.779940838"
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      "0.730283506"
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      "0.335840614",
      "1.649744523"
    ],
    [
      "1.418926525",
      "0.756040059"
    ]
  ],
  "radius": "0.070873724"
}
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": 164
}

当前第一名的答案

{
  "centers": [
    [
      "0.558029061",
      "1.657394215"
    ],
    [
      "0.618820252",
      "1.086916483"
    ],
    [
      "1.368605354",
      "1.178287899"
    ],
    [
      "1.205083237",
      "0.444213729"
    ],
    [
      "1.141062238",
      "1.007573328"
    ],
    [
      "0.171204736",
      "1.419969100"
    ],
    [
      "1.548778189",
      "0.699199187"
    ],
    [
      "0.162258155",
      "0.598173872"
    ],
    [
      "0.798628154",
      "0.752747307"
    ],
    [
      "0.877754375",
      "1.118080920"
    ],
    [
      "0.814328347",
      "0.611872032"
    ],
    [
      "1.779822988",
      "0.863444371"
    ],
    [
      "1.189400471",
      "0.585090945"
    ],
    [
      "1.601496715",
      "0.291849354"
    ],
    [
      "0.605128730",
      "0.159697578"
    ],
    [
      "1.518054755",
      "0.981023537"
    ],
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      "0.551762930",
      "0.729555232"
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      "0.474289459"
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      "1.318800849"
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      "1.343763167"
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      "0.896915335"
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      "1.372802370"
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      "0.475416555"
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      "1.142521610"
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      "1.928489117"
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      "1.052969867",
      "1.712836887"
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      "1.819381908"
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      "1.062706716"
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      "0.836781710"
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      "1.876480908"
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      "0.645520829"
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      "1.509974657"
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      "1.541154370"
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      "0.262495519"
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  "radius": "0.070873724"
}

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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