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 = 125

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 = 125
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[0.0808523433287059465571607235435, 0.0808523433287059465571607235445]见来源署名packomania.com ↗
本站纪录0.080852343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 = 0.080852343

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 = 125 的当前纪录构型,0.080852343
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
    0.080852342→0.080852343
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": [
    [
      "0.961398464",
      "0.081663278"
    ],
    [
      "1.122931585",
      "0.089110193"
    ],
    [
      "0.801060099",
      "0.102639799"
    ],
    [
      "0.646879123",
      "0.151390513"
    ],
    [
      "1.388211973",
      "0.166859028"
    ],
    [
      "1.231967168",
      "0.208523958"
    ],
    [
      "0.503627590",
      "0.226406538"
    ],
    [
      "0.912642503",
      "0.235842595"
    ],
    [
      "1.074152337",
      "0.243778530"
    ],
    [
      "1.528209643",
      "0.247785278"
    ],
    [
      "0.754957109",
      "0.271671662"
    ],
    [
      "1.341405937",
      "0.327568325"
    ],
    [
      "0.616003916",
      "0.354378469"
    ],
    [
      "1.658390653",
      "0.358686022"
    ],
    [
      "1.183591107",
      "0.362822898"
    ],
    [
      "0.456522042",
      "0.381098079"
    ],
    [
      "0.865354296",
      "0.390478397"
    ],
    [
      "1.026660407",
      "0.401824961"
    ],
    [
      "1.484230305",
      "0.403394479"
    ],
    [
      "0.297040168",
      "0.407817689"
    ],
    [
      "0.726152604",
      "0.472766273"
    ],
    [
      "1.297519500",
      "0.484419092"
    ],
    [
      "1.765109938",
      "0.490648257"
    ],
    [
      "0.559402840",
      "0.505853628"
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    [
      "1.604721480",
      "0.511238266"
    ],
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      "1.138902537",
      "0.518229903"
    ],
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      "0.399920966",
      "0.532573238"
    ],
    [
      "0.204140657",
      "0.540173597"
    ],
    [
      "1.448032461",
      "0.560995627"
    ],
    [
      "0.987379473",
      "0.574702613"
    ],
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      "0.827478556",
      "0.598787926"
    ],
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      "0.668020258",
      "0.625647877"
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      "1.702747179",
      "0.639843740"
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      "0.879604550"
    ],
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      "0.719823487",
      "0.903829263"
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      "1.772002513",
      "0.925965502"
    ],
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      "0.560341613",
      "0.930548873"
    ],
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      "0.400266712",
      "0.953449350"
    ],
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      "1.617393207",
      "0.973340264"
    ],
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      "1.141095039",
      "0.978774611"
    ],
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      "0.240570452",
      "0.978856240"
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      "0.993080164"
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      "1.457396156",
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      "0.080861874",
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      "1.005194963"
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      "0.662192754",
      "1.056146447"
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      "0.502496493",
      "1.081553336"
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      "1.748846640",
      "1.093089282"
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  ],
  "radius": "0.080852343"
}
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": 125
}

当前第一名的答案

{
  "centers": [
    [
      "0.961398464",
      "0.081663278"
    ],
    [
      "1.122931585",
      "0.089110193"
    ],
    [
      "0.801060099",
      "0.102639799"
    ],
    [
      "0.646879123",
      "0.151390513"
    ],
    [
      "1.388211973",
      "0.166859028"
    ],
    [
      "1.231967168",
      "0.208523958"
    ],
    [
      "0.503627590",
      "0.226406538"
    ],
    [
      "0.912642503",
      "0.235842595"
    ],
    [
      "1.074152337",
      "0.243778530"
    ],
    [
      "1.528209643",
      "0.247785278"
    ],
    [
      "0.754957109",
      "0.271671662"
    ],
    [
      "1.341405937",
      "0.327568325"
    ],
    [
      "0.616003916",
      "0.354378469"
    ],
    [
      "1.658390653",
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    [
      "1.183591107",
      "0.362822898"
    ],
    [
      "0.456522042",
      "0.381098079"
    ],
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      "0.865354296",
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    ],
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      "1.026660407",
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    ],
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      "1.222715346",
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      "0.735870623",
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      "1.062718295",
      "1.622943527"
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    [
      "0.575687931",
      "1.628491803"
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    [
      "0.385802687",
      "1.683808507"
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    [
      "1.596037733",
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      "1.314233868",
      "1.732819860"
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    [
      "0.835130516",
      "1.734013185"
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    [
      "0.674947824",
      "1.756147117"
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    [
      "1.154259927",
      "1.756415278"
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    [
      "0.993035097",
      "1.768863566"
    ],
    [
      "0.515143076",
      "1.780862458"
    ],
    [
      "1.464194519",
      "1.793319520"
    ],
    [
      "1.254681129",
      "1.883159066"
    ],
    [
      "0.766967148",
      "1.889116474"
    ],
    [
      "1.095726257",
      "1.914067948"
    ],
    [
      "0.926388430",
      "1.916195258"
    ]
  ],
  "radius": "0.080852343"
}

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

DISCUSSION

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聊思路、贴方法、问为什么卡住。所有登录用户都可以发帖;发言公开署名,与纪录使用同一个名字,署名后的 #编号是账号注册序号,冒不了名。新发言经自动审核后公开。

还没有帖子。第一个聊聊这道题的思路?