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 = 167

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 = 167
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[0.0703282033595108826652134765445, 0.0703282033595108826652134765455]见来源署名packomania.com ↗
本站纪录0.070328203游侠纪录详情 ↓

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 = 0.070328203

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 = 167 的当前纪录构型,0.070328203
构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者游侠
解题方式AI · OpenAI: GPT-5.6 Sol
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. 游侠AI · OpenAI: GPT-5.6 Sol
    0.070328202→0.070328203
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": [
    [
      "0.432008249",
      "1.244864607"
    ],
    [
      "0.670296033",
      "0.735218945"
    ],
    [
      "0.468594116",
      "0.874525509"
    ],
    [
      "1.455977059",
      "0.782664363"
    ],
    [
      "1.157388375",
      "1.424401897"
    ],
    [
      "1.054608032",
      "1.927712562"
    ],
    [
      "1.098504413",
      "1.552139534"
    ],
    [
      "0.820475653",
      "1.809591930"
    ],
    [
      "1.267395033",
      "1.784678089"
    ],
    [
      "1.510075437",
      "0.652461983"
    ],
    [
      "0.675554420",
      "1.589456083"
    ],
    [
      "1.564479594",
      "0.261368148"
    ],
    [
      "0.350826192",
      "1.130000781"
    ],
    [
      "1.199696273",
      "1.907970841"
    ],
    [
      "0.788644802",
      "0.479335344"
    ],
    [
      "1.208842860",
      "0.440713911"
    ],
    [
      "1.882618978",
      "0.707983923"
    ],
    [
      "0.140054601",
      "0.647345781"
    ],
    [
      "0.847528765",
      "0.351597708"
    ],
    [
      "1.238570432",
      "1.539265723"
    ],
    [
      "0.589113976",
      "0.620355120"
    ],
    [
      "1.356338356",
      "1.283790450"
    ],
    [
      "1.378636451",
      "1.526391912"
    ],
    [
      "1.461542648",
      "1.642163394"
    ],
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      "1.289444072",
      "0.555986065"
    ],
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      "0.712140287",
      "1.219116985"
    ],
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      "0.874504401",
      "1.448844635"
    ],
    [
      "1.076206318",
      "1.309538072"
    ],
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      "1.216272337",
      "1.296664261"
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      "0.977984553",
      "1.806309923"
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      "1.629094848",
      "1.684492017"
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      "1.717652424",
      "1.372906351"
    ],
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      "1.068776841",
      "0.453587722"
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      "1.127660803",
      "0.325850086"
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      "1.362462073"
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      "1.028203986"
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      "1.577586432",
      "1.385780464"
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      "0.950428071",
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      "0.266552223",
      "1.016799787"
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      "1.506570600"
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      "0.767283876"
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      "0.844480974"
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      "1.452661451",
      "0.523926200"
    ],
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      "0.891544109",
      "0.837208960"
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      "0.530230014",
      "0.748092756"
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    [
      "1.539501507",
      "0.408576725"
    ],
    [
      "0.698349248",
      "1.879372773"
    ]
  ],
  "radius": "0.070328203"
}
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": 167
}

当前第一名的答案

{
  "centers": [
    [
      "0.432008249",
      "1.244864607"
    ],
    [
      "0.670296033",
      "0.735218945"
    ],
    [
      "0.468594116",
      "0.874525509"
    ],
    [
      "1.455977059",
      "0.782664363"
    ],
    [
      "1.157388375",
      "1.424401897"
    ],
    [
      "1.054608032",
      "1.927712562"
    ],
    [
      "1.098504413",
      "1.552139534"
    ],
    [
      "0.820475653",
      "1.809591930"
    ],
    [
      "1.267395033",
      "1.784678089"
    ],
    [
      "1.510075437",
      "0.652461983"
    ],
    [
      "0.675554420",
      "1.589456083"
    ],
    [
      "1.564479594",
      "0.261368148"
    ],
    [
      "0.350826192",
      "1.130000781"
    ],
    [
      "1.199696273",
      "1.907970841"
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      "0.479335344"
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      "0.440713911"
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      "0.707983923"
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      "0.351597708"
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      "0.620355120"
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      "1.526391912"
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      "1.296664261"
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      "1.684492017"
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      "0.748914129"
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  "radius": "0.070328203"
}

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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