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 = 203

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 = 203
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[0.0640564290300644841635121968415, 0.0640564290300644841635121968425]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 = 0.064056428

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 = 203 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": [
    [
      "1.52436183",
      "0.755324801"
    ],
    [
      "1.484239577",
      "0.490747802"
    ],
    [
      "0.888898603",
      "0.235325603"
    ],
    [
      "1.201877073",
      "0.859698708"
    ],
    [
      "1.933532622",
      "0.932864236"
    ],
    [
      "1.50148155",
      "0.357200583"
    ],
    [
      "1.706975776",
      "1.32270068"
    ],
    [
      "0.847920543",
      "0.709787064"
    ],
    [
      "1.195886689",
      "1.699483159"
    ],
    [
      "1.542110948",
      "0.882202196"
    ],
    [
      "0.572548208",
      "1.195601083"
    ],
    [
      "0.648194226",
      "0.259745907"
    ],
    [
      "1.164478408",
      "1.445189751"
    ],
    [
      "1.272665505",
      "0.528239621"
    ],
    [
      "1.129305557",
      "1.926968414"
    ],
    [
      "1.217581214",
      "0.986845412"
    ],
    [
      "1.313851035",
      "1.749456327"
    ],
    [
      "0.663309125",
      "0.126713322"
    ],
    [
      "1.337590537",
      "1.036549269"
    ],
    [
      "0.556844067",
      "1.068454379"
    ],
    [
      "0.486094186",
      "1.399964522"
    ],
    [
      "0.724029305",
      "1.499590491"
    ],
    [
      "1.013060803",
      "1.141192486"
    ],
    [
      "0.832216402",
      "0.58264036"
    ],
    [
      "1.552878082",
      "1.755157607"
    ],
    [
      "0.260640069",
      "0.426120867"
    ],
    [
      "1.036736813",
      "0.428293286"
    ],
    [
      "1.430530055",
      "1.696551594"
    ],
    [
      "0.745063515",
      "1.780996462"
    ],
    [
      "0.241321223",
      "1.303733429"
    ],
    [
      "1.001655328",
      "0.174507635"
    ],
    [
      "1.070944605",
      "0.066749106"
    ],
    [
      "1.617863323",
      "1.524065574"
    ],
    [
      "1.304137278",
      "0.782525171"
    ],
    [
      "1.471259024",
      "1.213669141"
    ],
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      "0.358614059",
      "1.355262984"
    ],
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      "0.484205453",
      "0.538424421"
    ],
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      "0.627695991",
      "0.736987434"
    ],
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      "0.509731645",
      "0.687014266"
    ],
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      "0.220099234",
      "0.753661898"
    ],
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      "1.186109441",
      "0.732559862"
    ],
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      "0.419749748",
      "1.734370488"
    ],
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      "0.504296509",
      "1.530314571"
    ],
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      "0.27488987",
      "1.591781774"
    ],
    [
      "0.871078229",
      "1.804087503"
    ],
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      "0.841993651",
      "1.549563658"
    ],
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      "0.863624684",
      "0.836933768"
    ],
    [
      "1.916927196",
      "1.187709576"
    ],
    [
      "0.879328824",
      "0.964080472"
    ],
    [
      "1.282442754",
      "1.495162919"
    ],
    [
      "0.58831584",
      "1.322739929"
    ],
    [
      "1.251034472",
      "1.24086951"
    ],
    [
      "1.58922337",
      "1.263642308"
    ],
    [
      "0.189057805",
      "1.052619284"
    ],
    [
      "0.98158903",
      "0.886906935"
    ],
    [
      "0.438879721",
      "1.018481212"
    ],
    [
      "1.500045942",
      "1.473746899"
    ],
    [
      "1.288369646",
      "0.655386325"
    ],
    [
      "0.525435786",
      "0.814160971"
    ],
    [
      "1.414825914",
      "1.56940489"
    ],
    [
      "1.406397484",
      "0.705351634"
    ],
    [
      "0.950180748",
      "0.632613527"
    ],
    [
      "0.151558441",
      "1.395142113"
    ],
    [
      "0.105546201",
      "1.275577159"
    ],
    [
      "1.128539179",
      "0.194713948"
    ],
    [
      "0.729956197",
      "0.659813897"
    ],
    [
      "1.115321008",
      "1.064018949"
    ],
    [
      "0.808540392",
      "1.295539559"
    ],
    [
      "0.690576046",
      "1.245566392"
    ],
    [
      "0.761364478",
      "0.914107305"
    ],
    [
      "1.021032672",
      "0.301146582"
    ],
    [
      "0.522009953",
      "1.657196951"
    ],
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      "1.368998818",
      "1.290842678"
    ],
    [
      "0.316613741",
      "0.980219814"
    ],
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      "0.525471286",
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    ],
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      "1.353294678",
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      "1.675256573",
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      "0.991280842"
    ],
    [
      "1.197724729",
      "0.085180182"
    ],
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      "1.819814415",
      "0.873863611"
    ],
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      "0.36001136",
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      "1.638136753",
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      "0.768693878",
      "1.906911157"
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      "1.799671915",
      "0.51367172"
    ],
    [
      "0.623036595",
      "1.578415468"
    ],
    [
      "0.076292804",
      "1.150848881"
    ],
    [
      "0.76429451",
      "0.205547682"
    ],
    [
      "1.697362781",
      "1.194948988"
    ],
    [
      "1.538660137",
      "0.234601003"
    ],
    [
      "1.148774267",
      "1.318043047"
    ],
    [
      "1.422101624",
      "0.832498338"
    ],
    [
      "1.390629851",
      "0.578212788"
    ],
    [
      "1.533814368",
      "1.620754132"
    ],
    [
      "0.541139927",
      "0.941307675"
    ],
    [
      "1.439850742",
      "0.959375732"
    ],
    [
      "1.256961365",
      "0.401092917"
    ],
    [
      "0.446127712",
      "0.412119261"
    ],
    [
      "1.910537293",
      "0.783407744"
    ],
    [
      "0.805945753",
      "0.084394562"
    ],
    [
      "1.099616867",
      "0.936872245"
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    [
      "0.792836251",
      "1.168392855"
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    [
      "0.402036304",
      "1.607488108"
    ],
    [
      "0.300909601",
      "0.85307311"
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    [
      "0.069930682",
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    [
      "1.384702959",
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    [
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      "0.714252056",
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      "1.422416954"
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      "0.132946652",
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      "0.643863512",
      "1.865518625"
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      "0.944253856",
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      "0.934476608",
      "0.505466823"
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    [
      "1.068145094",
      "0.682586694"
    ],
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      "1.395216584"
    ],
    [
      "1.298146895",
      "1.622309623"
    ],
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      "1.077922343",
      "1.649509992"
    ],
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      "0.25918573",
      "1.46463507"
    ],
    [
      "1.600958166",
      "0.437930424"
    ],
    [
      "1.842944997",
      "1.40673591"
    ],
    [
      "1.443934998",
      "1.823961216"
    ],
    [
      "1.602306147",
      "1.396573362"
    ],
    [
      "1.42909097",
      "0.168211982"
    ],
    [
      "1.68165864",
      "1.067802284"
    ],
    [
      "0.959957997",
      "1.599536825"
    ],
    [
      "1.632889618",
      "0.687246732"
    ],
    [
      "0.092941159",
      "0.769273697"
    ],
    [
      "0.706280187",
      "1.372713096"
    ],
    [
      "0.350535594",
      "1.227405083"
    ],
    [
      "1.180182548",
      "1.572336455"
    ],
    [
      "0.562856473",
      "0.355973091"
    ],
    [
      "1.36917832",
      "0.281524231"
    ],
    [
      "1.052440954",
      "0.55543999"
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    [
      "1.384762961",
      "0.410017961"
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    [
      "1.301197341",
      "0.16072039"
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    [
      "0.997356662",
      "1.014045781"
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    [
      "0.345935477",
      "0.330530084"
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    [
      "0.200484254",
      "1.182303423"
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    [
      "0.470326554",
      "1.272825676"
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    [
      "0.800808121",
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    [
      "1.138997018",
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    [
      "1.455554883",
      "1.086522437"
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    [
      "0.777068619",
      "1.041254009"
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    [
      "0.194642133",
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    ],
    [
      "0.854137159",
      "1.67709969"
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    [
      "0.627731491",
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    [
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    [
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    [
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      "1.008766163"
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      "1.725767437",
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    [
      "0.394877014",
      "0.630257609"
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      "0.266830372",
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    [
      "0.064346353",
      "1.023294239"
    ],
    [
      "1.872429359",
      "0.661093845"
    ],
    [
      "1.241257224",
      "0.273946212"
    ],
    [
      "0.643400132",
      "0.864134138"
    ],
    [
      "1.651059094",
      "1.672393058"
    ],
    [
      "1.209291632",
      "1.826892782"
    ],
    [
      "0.462199605",
      "0.276719704"
    ],
    [
      "0.304689058",
      "1.107776493"
    ],
    [
      "1.062218202",
      "1.522363288"
    ],
    [
      "1.319841419",
      "0.909671875"
    ],
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      "0.918772467",
      "0.378320119"
    ],
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      "1.573519229",
      "1.136495604"
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      "0.757749394"
    ],
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      "0.597819197",
      "0.479222888"
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      "0.965884889",
      "0.759760231"
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    [
      "1.82760127",
      "1.279545208"
    ],
    [
      "0.932798775",
      "0.066472087"
    ],
    [
      "0.745660338",
      "0.786960601"
    ],
    [
      "1.747932193",
      "0.630871999"
    ],
    [
      "1.091327286",
      "1.776919615"
    ],
    [
      "1.722319503",
      "1.449891381"
    ],
    [
      "1.514247953",
      "0.622428728"
    ],
    [
      "0.423175581",
      "0.891334507"
    ]
  ],
  "radius": "0.064056428"
}
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": 203
}

起步布局答案

{
  "centers": [
    [
      "1.52436183",
      "0.755324801"
    ],
    [
      "1.484239577",
      "0.490747802"
    ],
    [
      "0.888898603",
      "0.235325603"
    ],
    [
      "1.201877073",
      "0.859698708"
    ],
    [
      "1.933532622",
      "0.932864236"
    ],
    [
      "1.50148155",
      "0.357200583"
    ],
    [
      "1.706975776",
      "1.32270068"
    ],
    [
      "0.847920543",
      "0.709787064"
    ],
    [
      "1.195886689",
      "1.699483159"
    ],
    [
      "1.542110948",
      "0.882202196"
    ],
    [
      "0.572548208",
      "1.195601083"
    ],
    [
      "0.648194226",
      "0.259745907"
    ],
    [
      "1.164478408",
      "1.445189751"
    ],
    [
      "1.272665505",
      "0.528239621"
    ],
    [
      "1.129305557",
      "1.926968414"
    ],
    [
      "1.217581214",
      "0.986845412"
    ],
    [
      "1.313851035",
      "1.749456327"
    ],
    [
      "0.663309125",
      "0.126713322"
    ],
    [
      "1.337590537",
      "1.036549269"
    ],
    [
      "0.556844067",
      "1.068454379"
    ],
    [
      "0.486094186",
      "1.399964522"
    ],
    [
      "0.724029305",
      "1.499590491"
    ],
    [
      "1.013060803",
      "1.141192486"
    ],
    [
      "0.832216402",
      "0.58264036"
    ],
    [
      "1.552878082",
      "1.755157607"
    ],
    [
      "0.260640069",
      "0.426120867"
    ],
    [
      "1.036736813",
      "0.428293286"
    ],
    [
      "1.430530055",
      "1.696551594"
    ],
    [
      "0.745063515",
      "1.780996462"
    ],
    [
      "0.241321223",
      "1.303733429"
    ],
    [
      "1.001655328",
      "0.174507635"
    ],
    [
      "1.070944605",
      "0.066749106"
    ],
    [
      "1.617863323",
      "1.524065574"
    ],
    [
      "1.304137278",
      "0.782525171"
    ],
    [
      "1.471259024",
      "1.213669141"
    ],
    [
      "0.358614059",
      "1.355262984"
    ],
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  "radius": "0.064056428"
}

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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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?