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 = 105

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 = 105
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[0.0880075646078596354662965694815, 0.0880075646078596354662965694825]见来源署名packomania.com ↗
本站纪录暂无选手纪录——

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:起步布局

2y0
0x2
已验证构造r = 0.088007564

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 = 105 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"。

起步布局答案

{
  "centers": [
    [
      "1.025115454",
      "0.088353459"
    ],
    [
      "0.849520476",
      "0.100507857"
    ],
    [
      "0.679530747",
      "0.146167663"
    ],
    [
      "1.325024995",
      "0.147891465"
    ],
    [
      "1.158085218",
      "0.203680679"
    ],
    [
      "0.521478258",
      "0.223632082"
    ],
    [
      "1.482661494",
      "0.226198937"
    ],
    [
      "0.981651289",
      "0.258917793"
    ],
    [
      "0.806056311",
      "0.271072191"
    ],
    [
      "1.289330348",
      "0.320966831"
    ],
    [
      "1.62236894",
      "0.333379617"
    ],
    [
      "0.647182635",
      "0.34683815"
    ],
    [
      "1.114621053",
      "0.374245013"
    ],
    [
      "0.477630861",
      "0.3940983"
    ],
    [
      "1.447288668",
      "0.398623091"
    ],
    [
      "0.302566044",
      "0.412363991"
    ],
    [
      "0.929370057",
      "0.426989202"
    ],
    [
      "1.740082469",
      "0.467075857"
    ],
    [
      "0.761690567",
      "0.480514098"
    ],
    [
      "1.245866183",
      "0.491531165"
    ],
    [
      "1.57284839",
      "0.521976575"
    ],
    [
      "0.592635634",
      "0.529521804"
    ],
    [
      "1.06233982",
      "0.542316422"
    ],
    [
      "0.202670669",
      "0.557285487"
    ],
    [
      "0.419309376",
      "0.560170363"
    ],
    [
      "1.40347901",
      "0.56988627"
    ],
    [
      "0.893306906",
      "0.599270294"
    ],
    [
      "1.828673386",
      "0.61917114"
    ],
    [
      "0.725627416",
      "0.652795191"
    ],
    [
      "1.19358495",
      "0.659602574"
    ],
    [
      "1.661439308",
      "0.674071857"
    ],
    [
      "0.309142777",
      "0.697446166"
    ],
    [
      "0.556572483",
      "0.701802897"
    ],
    [
      "1.026276669",
      "0.714597514"
    ],
    [
      "1.492069928",
      "0.721981552"
    ],
    [
      "0.124737284",
      "0.743821085"
    ],
    [
      "0.857356683",
      "0.771574987"
    ],
    [
      "1.32483585",
      "0.77688227"
    ],
    [
      "1.792601763",
      "0.791450459"
    ],
    [
      "0.430653725",
      "0.824789856"
    ],
    [
      "0.689677194",
      "0.825099884"
    ],
    [
      "1.157521799",
      "0.831883667"
    ],
    [
      "1.625367685",
      "0.846351176"
    ],
    [
      "0.259615381",
      "0.866349578"
    ],
    [
      "0.990326447",
      "0.886902207"
    ],
    [
      "1.455998305",
      "0.894260871"
    ],
    [
      "0.091622292",
      "0.918881929"
    ],
    [
      "1.908792278",
      "0.923666532"
    ],
    [
      "0.821406461",
      "0.94387968"
    ],
    [
      "0.560022791",
      "0.944142151"
    ],
    [
      "1.288766929",
      "0.949169819"
    ],
    [
      "1.741558199",
      "0.97856725"
    ],
    [
      "0.392624993",
      "0.998541628"
    ],
    [
      "1.121571577",
      "1.00418836"
    ],
    [
      "1.57218882",
      "1.026476945"
    ],
    [
      "0.221113186",
      "1.038102036"
    ],
    [
      "0.954376224",
      "1.0592069"
    ],
    [
      "0.690833018",
      "1.061913158"
    ],
    [
      "1.404957443",
      "1.081385893"
    ],
    [
      "1.906530014",
      "1.099667123"
    ],
    [
      "0.523435221",
      "1.116312635"
    ],
    [
      "1.237762091",
      "1.136404433"
    ],
    [
      "1.739295936",
      "1.15456784"
    ],
    [
      "0.103876873",
      "1.169391688"
    ],
    [
      "0.351389986",
      "1.169930968"
    ],
    [
      "0.83170632",
      "1.185434553"
    ],
    [
      "1.070566738",
      "1.191422974"
    ],
    [
      "1.539631522",
      "1.199454828"
    ],
    [
      "0.663026441",
      "1.235717991"
    ],
    [
      "1.372400146",
      "1.254363776"
    ],
    [
      "1.870500188",
      "1.271955188"
    ],
    [
      "0.495628644",
      "1.290117468"
    ],
    [
      "1.205204793",
      "1.309382317"
    ],
    [
      "1.670876652",
      "1.31674098"
    ],
    [
      "0.947896834",
      "1.317650627"
    ],
    [
      "0.153106957",
      "1.338381997"
    ],
    [
      "0.328929169",
      "1.34662066"
    ],
    [
      "0.779216955",
      "1.367934064"
    ],
    [
      "1.5036813",
      "1.371759521"
    ],
    [
      "1.336449923",
      "1.426668469"
    ],
    [
      "0.604942134",
      "1.428073571"
    ],
    [
      "1.080406551",
      "1.433506143"
    ],
    [
      "1.802044884",
      "1.434113126"
    ],
    [
      "0.438242659",
      "1.484576764"
    ],
    [
      "1.634849531",
      "1.489131667"
    ],
    [
      "0.909784563",
      "1.489637446"
    ],
    [
      "0.233883171",
      "1.49476783"
    ],
    [
      "0.740993505",
      "1.539748771"
    ],
    [
      "1.467618155",
      "1.544040615"
    ],
    [
      "1.211651681",
      "1.550792296"
    ],
    [
      "0.576252031",
      "1.601758682"
    ],
    [
      "1.044456328",
      "1.605810836"
    ],
    [
      "0.343196661",
      "1.632723934"
    ],
    [
      "1.624285168",
      "1.664829476"
    ],
    [
      "1.342976731",
      "1.668321913"
    ],
    [
      "0.759990439",
      "1.714735749"
    ],
    [
      "1.175701459",
      "1.723096989"
    ],
    [
      "0.933774541",
      "1.742671596"
    ],
    [
      "0.476975583",
      "1.747111546"
    ],
    [
      "1.484944413",
      "1.772372396"
    ],
    [
      "0.630236772",
      "1.833669813"
    ],
    [
      "1.327539843",
      "1.851145025"
    ],
    [
      "0.79927922",
      "1.889625758"
    ],
    [
      "1.15793466",
      "1.89821314"
    ],
    [
      "0.982446529",
      "1.911823491"
    ]
  ],
  "radius": "0.088007564"
}
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": 105
}

起步布局答案

{
  "centers": [
    [
      "1.025115454",
      "0.088353459"
    ],
    [
      "0.849520476",
      "0.100507857"
    ],
    [
      "0.679530747",
      "0.146167663"
    ],
    [
      "1.325024995",
      "0.147891465"
    ],
    [
      "1.158085218",
      "0.203680679"
    ],
    [
      "0.521478258",
      "0.223632082"
    ],
    [
      "1.482661494",
      "0.226198937"
    ],
    [
      "0.981651289",
      "0.258917793"
    ],
    [
      "0.806056311",
      "0.271072191"
    ],
    [
      "1.289330348",
      "0.320966831"
    ],
    [
      "1.62236894",
      "0.333379617"
    ],
    [
      "0.647182635",
      "0.34683815"
    ],
    [
      "1.114621053",
      "0.374245013"
    ],
    [
      "0.477630861",
      "0.3940983"
    ],
    [
      "1.447288668",
      "0.398623091"
    ],
    [
      "0.302566044",
      "0.412363991"
    ],
    [
      "0.929370057",
      "0.426989202"
    ],
    [
      "1.740082469",
      "0.467075857"
    ],
    [
      "0.761690567",
      "0.480514098"
    ],
    [
      "1.245866183",
      "0.491531165"
    ],
    [
      "1.57284839",
      "0.521976575"
    ],
    [
      "0.592635634",
      "0.529521804"
    ],
    [
      "1.06233982",
      "0.542316422"
    ],
    [
      "0.202670669",
      "0.557285487"
    ],
    [
      "0.419309376",
      "0.560170363"
    ],
    [
      "1.40347901",
      "0.56988627"
    ],
    [
      "0.893306906",
      "0.599270294"
    ],
    [
      "1.828673386",
      "0.61917114"
    ],
    [
      "0.725627416",
      "0.652795191"
    ],
    [
      "1.19358495",
      "0.659602574"
    ],
    [
      "1.661439308",
      "0.674071857"
    ],
    [
      "0.309142777",
      "0.697446166"
    ],
    [
      "0.556572483",
      "0.701802897"
    ],
    [
      "1.026276669",
      "0.714597514"
    ],
    [
      "1.492069928",
      "0.721981552"
    ],
    [
      "0.124737284",
      "0.743821085"
    ],
    [
      "0.857356683",
      "0.771574987"
    ],
    [
      "1.32483585",
      "0.77688227"
    ],
    [
      "1.792601763",
      "0.791450459"
    ],
    [
      "0.430653725",
      "0.824789856"
    ],
    [
      "0.689677194",
      "0.825099884"
    ],
    [
      "1.157521799",
      "0.831883667"
    ],
    [
      "1.625367685",
      "0.846351176"
    ],
    [
      "0.259615381",
      "0.866349578"
    ],
    [
      "0.990326447",
      "0.886902207"
    ],
    [
      "1.455998305",
      "0.894260871"
    ],
    [
      "0.091622292",
      "0.918881929"
    ],
    [
      "1.908792278",
      "0.923666532"
    ],
    [
      "0.821406461",
      "0.94387968"
    ],
    [
      "0.560022791",
      "0.944142151"
    ],
    [
      "1.288766929",
      "0.949169819"
    ],
    [
      "1.741558199",
      "0.97856725"
    ],
    [
      "0.392624993",
      "0.998541628"
    ],
    [
      "1.121571577",
      "1.00418836"
    ],
    [
      "1.57218882",
      "1.026476945"
    ],
    [
      "0.221113186",
      "1.038102036"
    ],
    [
      "0.954376224",
      "1.0592069"
    ],
    [
      "0.690833018",
      "1.061913158"
    ],
    [
      "1.404957443",
      "1.081385893"
    ],
    [
      "1.906530014",
      "1.099667123"
    ],
    [
      "0.523435221",
      "1.116312635"
    ],
    [
      "1.237762091",
      "1.136404433"
    ],
    [
      "1.739295936",
      "1.15456784"
    ],
    [
      "0.103876873",
      "1.169391688"
    ],
    [
      "0.351389986",
      "1.169930968"
    ],
    [
      "0.83170632",
      "1.185434553"
    ],
    [
      "1.070566738",
      "1.191422974"
    ],
    [
      "1.539631522",
      "1.199454828"
    ],
    [
      "0.663026441",
      "1.235717991"
    ],
    [
      "1.372400146",
      "1.254363776"
    ],
    [
      "1.870500188",
      "1.271955188"
    ],
    [
      "0.495628644",
      "1.290117468"
    ],
    [
      "1.205204793",
      "1.309382317"
    ],
    [
      "1.670876652",
      "1.31674098"
    ],
    [
      "0.947896834",
      "1.317650627"
    ],
    [
      "0.153106957",
      "1.338381997"
    ],
    [
      "0.328929169",
      "1.34662066"
    ],
    [
      "0.779216955",
      "1.367934064"
    ],
    [
      "1.5036813",
      "1.371759521"
    ],
    [
      "1.336449923",
      "1.426668469"
    ],
    [
      "0.604942134",
      "1.428073571"
    ],
    [
      "1.080406551",
      "1.433506143"
    ],
    [
      "1.802044884",
      "1.434113126"
    ],
    [
      "0.438242659",
      "1.484576764"
    ],
    [
      "1.634849531",
      "1.489131667"
    ],
    [
      "0.909784563",
      "1.489637446"
    ],
    [
      "0.233883171",
      "1.49476783"
    ],
    [
      "0.740993505",
      "1.539748771"
    ],
    [
      "1.467618155",
      "1.544040615"
    ],
    [
      "1.211651681",
      "1.550792296"
    ],
    [
      "0.576252031",
      "1.601758682"
    ],
    [
      "1.044456328",
      "1.605810836"
    ],
    [
      "0.343196661",
      "1.632723934"
    ],
    [
      "1.624285168",
      "1.664829476"
    ],
    [
      "1.342976731",
      "1.668321913"
    ],
    [
      "0.759990439",
      "1.714735749"
    ],
    [
      "1.175701459",
      "1.723096989"
    ],
    [
      "0.933774541",
      "1.742671596"
    ],
    [
      "0.476975583",
      "1.747111546"
    ],
    [
      "1.484944413",
      "1.772372396"
    ],
    [
      "0.630236772",
      "1.833669813"
    ],
    [
      "1.327539843",
      "1.851145025"
    ],
    [
      "0.79927922",
      "1.889625758"
    ],
    [
      "1.15793466",
      "1.89821314"
    ],
    [
      "0.982446529",
      "1.911823491"
    ]
  ],
  "radius": "0.088007564"
}

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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