极值竞技场
中/EN
P01 · 装箱与覆盖 · 经典问题 · 难

单位正方形内的等圆装箱 · n = 108

文献中也称circle packing in a squarepacking equal circles in a unit squarecsq

在边长 1 的正方形内放置 n 个半径相同、互不重叠的圆，使共同半径尽可能大。

子题n = 108
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[0.0492435207865, 0.0492435207875]见来源署名packomania.com ↗
本站纪录暂无选手纪录——

严格定义

  • 容器单位正方形：左下角是原点 (0, 0)，右上角是 (1, 1)
  • 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
放大来摆，然后提交 ↗

当前展示：起步布局

1y0
0x1
已验证构造r = 0.049243520

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 = 1–30 已证明最优，适合观察经典接触结构；n = 31–300 接入 Packomania 的公开前沿，除 n = 36 外最优性仍未知。大规模部分集中成一张表，供算法批量挑战。

查看来源
单位正方形内的等圆装箱 n = 108 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

容器是边长 1 的正方形，左下角是原点 (0, 0)，右上角是 (1, 1)。坐标和长度用同一个单位，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 "0.25"。

起步布局答案

{
  "centers": [
    [
      "0.049243521",
      "0.049243521"
    ],
    [
      "0.147730562",
      "0.049243521"
    ],
    [
      "0.246217604",
      "0.049243521"
    ],
    [
      "0.344704646",
      "0.049243521"
    ],
    [
      "0.521359758",
      "0.049243521"
    ],
    [
      "0.692186295",
      "0.049243521"
    ],
    [
      "0.864615847",
      "0.049243521"
    ],
    [
      "0.433032202",
      "0.092808876"
    ],
    [
      "0.778401071",
      "0.096853496"
    ],
    [
      "0.95075643",
      "0.09698871"
    ],
    [
      "0.606773027",
      "0.098276886"
    ],
    [
      "0.055678595",
      "0.147520106"
    ],
    [
      "0.154165636",
      "0.147520106"
    ],
    [
      "0.252652678",
      "0.147520106"
    ],
    [
      "0.35113972",
      "0.147520106"
    ],
    [
      "0.521602426",
      "0.147730562"
    ],
    [
      "0.692186986",
      "0.147730562"
    ],
    [
      "0.864616537",
      "0.147730562"
    ],
    [
      "0.778401761",
      "0.195340538"
    ],
    [
      "0.950756479",
      "0.195475801"
    ],
    [
      "0.436310146",
      "0.196974083"
    ],
    [
      "0.606894706",
      "0.196974083"
    ],
    [
      "0.049243521",
      "0.245796691"
    ],
    [
      "0.147730562",
      "0.245796691"
    ],
    [
      "0.246217604",
      "0.245796691"
    ],
    [
      "0.344704646",
      "0.245796691"
    ],
    [
      "0.521602426",
      "0.246217604"
    ],
    [
      "0.692186986",
      "0.246217604"
    ],
    [
      "0.864616537",
      "0.246217604"
    ],
    [
      "0.778401761",
      "0.29382758"
    ],
    [
      "0.950756479",
      "0.293962842"
    ],
    [
      "0.436310146",
      "0.295461125"
    ],
    [
      "0.606894706",
      "0.295461125"
    ],
    [
      "0.055212036",
      "0.344102714"
    ],
    [
      "0.153699078",
      "0.344102714"
    ],
    [
      "0.252186119",
      "0.344102714"
    ],
    [
      "0.350673161",
      "0.344102714"
    ],
    [
      "0.521602426",
      "0.344704646"
    ],
    [
      "0.692186986",
      "0.344704646"
    ],
    [
      "0.864616537",
      "0.344704646"
    ],
    [
      "0.778401761",
      "0.392314621"
    ],
    [
      "0.950756479",
      "0.392449884"
    ],
    [
      "0.606894706",
      "0.393948166"
    ],
    [
      "0.436310604",
      "0.39394896"
    ],
    [
      "0.049243521",
      "0.442408737"
    ],
    [
      "0.147761154",
      "0.44241059"
    ],
    [
      "0.246248196",
      "0.44241059"
    ],
    [
      "0.350673168",
      "0.442589756"
    ],
    [
      "0.692186986",
      "0.443191687"
    ],
    [
      "0.864616537",
      "0.443191687"
    ],
    [
      "0.521462861",
      "0.443434216"
    ],
    [
      "0.778401761",
      "0.490801663"
    ],
    [
      "0.950756479",
      "0.490936925"
    ],
    [
      "0.606894706",
      "0.492435208"
    ],
    [
      "0.423494281",
      "0.512132619"
    ],
    [
      "0.298317405",
      "0.526007808"
    ],
    [
      "0.098500734",
      "0.52769311"
    ],
    [
      "0.197004675",
      "0.52770287"
    ],
    [
      "0.692186986",
      "0.541678729"
    ],
    [
      "0.864616537",
      "0.541678729"
    ],
    [
      "0.521462861",
      "0.541921257"
    ],
    [
      "0.778401761",
      "0.589288704"
    ],
    [
      "0.950756479",
      "0.589423967"
    ],
    [
      "0.606894706",
      "0.590922249"
    ],
    [
      "0.351043288",
      "0.609192413"
    ],
    [
      "0.44953033",
      "0.609192413"
    ],
    [
      "0.249073884",
      "0.611300088"
    ],
    [
      "0.049243521",
      "0.612977484"
    ],
    [
      "0.147744255",
      "0.61298539"
    ],
    [
      "0.864616537",
      "0.64016577"
    ],
    [
      "0.688633141",
      "0.64632121"
    ],
    [
      "0.778401761",
      "0.687775746"
    ],
    [
      "0.950756479",
      "0.687911008"
    ],
    [
      "0.597103638",
      "0.688921394"
    ],
    [
      "0.301799768",
      "0.694484693"
    ],
    [
      "0.400286809",
      "0.694484693"
    ],
    [
      "0.498773851",
      "0.694484693"
    ],
    [
      "0.199813464",
      "0.696582608"
    ],
    [
      "0.098487042",
      "0.698269764"
    ],
    [
      "0.84698856",
      "0.768198927"
    ],
    [
      "0.645985158",
      "0.774421654"
    ],
    [
      "0.252194246",
      "0.779984953"
    ],
    [
      "0.350681287",
      "0.779984953"
    ],
    [
      "0.449168329",
      "0.779984953"
    ],
    [
      "0.54765537",
      "0.779984953"
    ],
    [
      "0.744304188",
      "0.780171919"
    ],
    [
      "0.150556251",
      "0.781866981"
    ],
    [
      "0.049243521",
      "0.783562044"
    ],
    [
      "0.950756479",
      "0.78639805"
    ],
    [
      "0.20262546",
      "0.865464199"
    ],
    [
      "0.301112501",
      "0.865464199"
    ],
    [
      "0.399599543",
      "0.865464199"
    ],
    [
      "0.498086584",
      "0.865464199"
    ],
    [
      "0.596573626",
      "0.865464199"
    ],
    [
      "0.695060668",
      "0.865464199"
    ],
    [
      "0.793547709",
      "0.865464199"
    ],
    [
      "0.892034751",
      "0.865464199"
    ],
    [
      "0.10131273",
      "0.867159261"
    ],
    [
      "0.948786736",
      "0.948786736"
    ],
    [
      "0.049243521",
      "0.950756479"
    ],
    [
      "0.153381939",
      "0.950756479"
    ],
    [
      "0.251868981",
      "0.950756479"
    ],
    [
      "0.350356022",
      "0.950756479"
    ],
    [
      "0.448843064",
      "0.950756479"
    ],
    [
      "0.547330105",
      "0.950756479"
    ],
    [
      "0.645817147",
      "0.950756479"
    ],
    [
      "0.744304188",
      "0.950756479"
    ],
    [
      "0.84279123",
      "0.950756479"
    ]
  ],
  "radius": "0.04924352"
}
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": 108
}

起步布局答案

{
  "centers": [
    [
      "0.049243521",
      "0.049243521"
    ],
    [
      "0.147730562",
      "0.049243521"
    ],
    [
      "0.246217604",
      "0.049243521"
    ],
    [
      "0.344704646",
      "0.049243521"
    ],
    [
      "0.521359758",
      "0.049243521"
    ],
    [
      "0.692186295",
      "0.049243521"
    ],
    [
      "0.864615847",
      "0.049243521"
    ],
    [
      "0.433032202",
      "0.092808876"
    ],
    [
      "0.778401071",
      "0.096853496"
    ],
    [
      "0.95075643",
      "0.09698871"
    ],
    [
      "0.606773027",
      "0.098276886"
    ],
    [
      "0.055678595",
      "0.147520106"
    ],
    [
      "0.154165636",
      "0.147520106"
    ],
    [
      "0.252652678",
      "0.147520106"
    ],
    [
      "0.35113972",
      "0.147520106"
    ],
    [
      "0.521602426",
      "0.147730562"
    ],
    [
      "0.692186986",
      "0.147730562"
    ],
    [
      "0.864616537",
      "0.147730562"
    ],
    [
      "0.778401761",
      "0.195340538"
    ],
    [
      "0.950756479",
      "0.195475801"
    ],
    [
      "0.436310146",
      "0.196974083"
    ],
    [
      "0.606894706",
      "0.196974083"
    ],
    [
      "0.049243521",
      "0.245796691"
    ],
    [
      "0.147730562",
      "0.245796691"
    ],
    [
      "0.246217604",
      "0.245796691"
    ],
    [
      "0.344704646",
      "0.245796691"
    ],
    [
      "0.521602426",
      "0.246217604"
    ],
    [
      "0.692186986",
      "0.246217604"
    ],
    [
      "0.864616537",
      "0.246217604"
    ],
    [
      "0.778401761",
      "0.29382758"
    ],
    [
      "0.950756479",
      "0.293962842"
    ],
    [
      "0.436310146",
      "0.295461125"
    ],
    [
      "0.606894706",
      "0.295461125"
    ],
    [
      "0.055212036",
      "0.344102714"
    ],
    [
      "0.153699078",
      "0.344102714"
    ],
    [
      "0.252186119",
      "0.344102714"
    ],
    [
      "0.350673161",
      "0.344102714"
    ],
    [
      "0.521602426",
      "0.344704646"
    ],
    [
      "0.692186986",
      "0.344704646"
    ],
    [
      "0.864616537",
      "0.344704646"
    ],
    [
      "0.778401761",
      "0.392314621"
    ],
    [
      "0.950756479",
      "0.392449884"
    ],
    [
      "0.606894706",
      "0.393948166"
    ],
    [
      "0.436310604",
      "0.39394896"
    ],
    [
      "0.049243521",
      "0.442408737"
    ],
    [
      "0.147761154",
      "0.44241059"
    ],
    [
      "0.246248196",
      "0.44241059"
    ],
    [
      "0.350673168",
      "0.442589756"
    ],
    [
      "0.692186986",
      "0.443191687"
    ],
    [
      "0.864616537",
      "0.443191687"
    ],
    [
      "0.521462861",
      "0.443434216"
    ],
    [
      "0.778401761",
      "0.490801663"
    ],
    [
      "0.950756479",
      "0.490936925"
    ],
    [
      "0.606894706",
      "0.492435208"
    ],
    [
      "0.423494281",
      "0.512132619"
    ],
    [
      "0.298317405",
      "0.526007808"
    ],
    [
      "0.098500734",
      "0.52769311"
    ],
    [
      "0.197004675",
      "0.52770287"
    ],
    [
      "0.692186986",
      "0.541678729"
    ],
    [
      "0.864616537",
      "0.541678729"
    ],
    [
      "0.521462861",
      "0.541921257"
    ],
    [
      "0.778401761",
      "0.589288704"
    ],
    [
      "0.950756479",
      "0.589423967"
    ],
    [
      "0.606894706",
      "0.590922249"
    ],
    [
      "0.351043288",
      "0.609192413"
    ],
    [
      "0.44953033",
      "0.609192413"
    ],
    [
      "0.249073884",
      "0.611300088"
    ],
    [
      "0.049243521",
      "0.612977484"
    ],
    [
      "0.147744255",
      "0.61298539"
    ],
    [
      "0.864616537",
      "0.64016577"
    ],
    [
      "0.688633141",
      "0.64632121"
    ],
    [
      "0.778401761",
      "0.687775746"
    ],
    [
      "0.950756479",
      "0.687911008"
    ],
    [
      "0.597103638",
      "0.688921394"
    ],
    [
      "0.301799768",
      "0.694484693"
    ],
    [
      "0.400286809",
      "0.694484693"
    ],
    [
      "0.498773851",
      "0.694484693"
    ],
    [
      "0.199813464",
      "0.696582608"
    ],
    [
      "0.098487042",
      "0.698269764"
    ],
    [
      "0.84698856",
      "0.768198927"
    ],
    [
      "0.645985158",
      "0.774421654"
    ],
    [
      "0.252194246",
      "0.779984953"
    ],
    [
      "0.350681287",
      "0.779984953"
    ],
    [
      "0.449168329",
      "0.779984953"
    ],
    [
      "0.54765537",
      "0.779984953"
    ],
    [
      "0.744304188",
      "0.780171919"
    ],
    [
      "0.150556251",
      "0.781866981"
    ],
    [
      "0.049243521",
      "0.783562044"
    ],
    [
      "0.950756479",
      "0.78639805"
    ],
    [
      "0.20262546",
      "0.865464199"
    ],
    [
      "0.301112501",
      "0.865464199"
    ],
    [
      "0.399599543",
      "0.865464199"
    ],
    [
      "0.498086584",
      "0.865464199"
    ],
    [
      "0.596573626",
      "0.865464199"
    ],
    [
      "0.695060668",
      "0.865464199"
    ],
    [
      "0.793547709",
      "0.865464199"
    ],
    [
      "0.892034751",
      "0.865464199"
    ],
    [
      "0.10131273",
      "0.867159261"
    ],
    [
      "0.948786736",
      "0.948786736"
    ],
    [
      "0.049243521",
      "0.950756479"
    ],
    [
      "0.153381939",
      "0.950756479"
    ],
    [
      "0.251868981",
      "0.950756479"
    ],
    [
      "0.350356022",
      "0.950756479"
    ],
    [
      "0.448843064",
      "0.950756479"
    ],
    [
      "0.547330105",
      "0.950756479"
    ],
    [
      "0.645817147",
      "0.950756479"
    ],
    [
      "0.744304188",
      "0.950756479"
    ],
    [
      "0.84279123",
      "0.950756479"
    ]
  ],
  "radius": "0.04924352"
}

提交 radius 与 centers。每个数写成十进制字符串，例如 "0.25"。 · 验证器 v1.0.0

DISCUSSION

讨论区

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。

啊？？！NUE_13#310 条回复空悲切/.lird#282 条回复