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 = 236

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 = 236
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[0.0596146775924782962233350676605, 0.0596146775924782962233350676615]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 = 0.059614677

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 = 236 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": [
    [
      "1.412286838",
      "0.413560973"
    ],
    [
      "1.15803759",
      "0.072989388"
    ],
    [
      "1.257487826",
      "1.540464658"
    ],
    [
      "0.894512708",
      "0.065549902"
    ],
    [
      "0.951500561",
      "1.623195189"
    ],
    [
      "1.208105838",
      "0.70724596"
    ],
    [
      "0.289896391",
      "0.946196757"
    ],
    [
      "1.326915558",
      "0.71724044"
    ],
    [
      "1.089296118",
      "0.697251481"
    ],
    [
      "0.670333413",
      "0.119293184"
    ],
    [
      "1.366942632",
      "0.184780187"
    ],
    [
      "0.613749746",
      "0.662098057"
    ],
    [
      "1.459410817",
      "1.820528035"
    ],
    [
      "1.52747115",
      "1.722633039"
    ],
    [
      "0.426879693",
      "0.749998573"
    ],
    [
      "1.08792872",
      "1.422580702"
    ],
    [
      "1.29210972",
      "1.128895715"
    ],
    [
      "1.052673046",
      "1.839127018"
    ],
    [
      "1.127640581",
      "0.28400871"
    ],
    [
      "0.902426065",
      "0.785151997"
    ],
    [
      "1.021235785",
      "0.795146476"
    ],
    [
      "1.595826334",
      "0.272459256"
    ],
    [
      "0.324967733",
      "0.345281745"
    ],
    [
      "0.937047959",
      "0.373583054"
    ],
    [
      "1.038546732",
      "0.589362005"
    ],
    [
      "1.394975891",
      "0.619345444"
    ],
    [
      "0.629693294",
      "1.181642807"
    ],
    [
      "1.613820898",
      "1.57166692"
    ],
    [
      "1.410919439",
      "1.138890195"
    ],
    [
      "0.353910584",
      "1.635310009"
    ],
    [
      "0.512250973",
      "0.446319106"
    ],
    [
      "1.037179333",
      "1.314691227"
    ],
    [
      "0.107029345",
      "0.705164513"
    ],
    [
      "1.241360333",
      "1.02100624"
    ],
    [
      "0.205203802",
      "1.249554364"
    ],
    [
      "0.867312734",
      "1.201631767"
    ],
    [
      "1.461668826",
      "1.24677967"
    ],
    [
      "1.940277366",
      "1.01424882"
    ],
    [
      "1.485751425",
      "0.194785676"
    ],
    [
      "0.682117572",
      "0.559378566"
    ],
    [
      "0.728109062",
      "1.900222124"
    ],
    [
      "0.218088023",
      "0.661786993"
    ],
    [
      "0.463499824",
      "1.588343664"
    ],
    [
      "0.859381543",
      "1.929807059"
    ],
    [
      "0.78953046",
      "0.122068631"
    ],
    [
      "0.06161024",
      "0.938769186"
    ],
    [
      "1.91669881",
      "0.790267205"
    ],
    [
      "0.918062121",
      "1.309521242"
    ],
    [
      "0.968811507",
      "1.417410718"
    ],
    [
      "0.56300036",
      "0.554208581"
    ],
    [
      "1.700674178",
      "1.267721814"
    ],
    [
      "1.224049386",
      "1.226790711"
    ],
    [
      "1.122550613",
      "1.01101176"
    ],
    [
      "1.802159794",
      "0.757154477"
    ],
    [
      "1.761751472",
      "0.644981366"
    ],
    [
      "1.157356452",
      "0.599356485"
    ],
    [
      "0.800927292",
      "0.569373046"
    ],
    [
      "1.484068813",
      "0.314003158"
    ],
    [
      "0.919737012",
      "0.579367525"
    ],
    [
      "0.561632961",
      "1.279537803"
    ],
    [
      "0.087827292",
      "1.228616503"
    ],
    [
      "0.528194521",
      "0.965863857"
    ],
    [
      "0.442823241",
      "1.269543323"
    ],
    [
      "0.664499133",
      "0.769987532"
    ],
    [
      "0.174523614",
      "0.900478077"
    ],
    [
      "1.731669771",
      "1.590743849"
    ],
    [
      "0.731192068",
      "1.397421758"
    ],
    [
      "1.750481322",
      "0.433341073"
    ],
    [
      "1.07031028",
      "1.633189668"
    ],
    [
      "1.274798773",
      "1.334680186"
    ],
    [
      "0.697753628",
      "1.083747812"
    ],
    [
      "1.835408946",
      "1.070978473"
    ],
    [
      "0.815379894",
      "1.81898518"
    ],
    [
      "1.529729159",
      "1.148884675"
    ],
    [
      "1.258855225",
      "0.815135436"
    ],
    [
      "0.986429947",
      "1.206801752"
    ],
    [
      "0.241462926",
      "1.363136559"
    ],
    [
      "1.463036224",
      "0.521450449"
    ],
    [
      "1.512418213",
      "1.354669146"
    ],
    [
      "0.324013522",
      "1.259548844"
    ],
    [
      "0.425512295",
      "1.475327795"
    ],
    [
      "1.189427493",
      "1.638359653"
    ],
    [
      "0.783616346",
      "0.775157517"
    ],
    [
      "0.645820788",
      "1.701101225"
    ],
    [
      "1.597789492",
      "1.050989679"
    ],
    [
      "1.195700914",
      "0.186113714"
    ],
    [
      "1.342859106",
      "1.236785191"
    ],
    [
      "1.701549574",
      "0.542066981"
    ],
    [
      "1.648538879",
      "1.158879154"
    ],
    [
      "1.140045505",
      "0.805140956"
    ],
    [
      "0.273264135",
      "1.151659368"
    ],
    [
      "0.155887625",
      "1.130721507"
    ],
    [
      "1.280675661",
      "0.102478007"
    ],
    [
      "0.799252401",
      "1.299526763"
    ],
    [
      "0.304924975",
      "1.46407321"
    ],
    [
      "1.226526189",
      "1.864694465"
    ],
    [
      "0.900751174",
      "1.515305713"
    ],
    [
      "1.276166171",
      "0.609350965"
    ],
    [
      "0.884931174",
      "0.991022801"
    ],
    [
      "0.173648095",
      "0.551149152"
    ],
    [
      "1.879100396",
      "0.666073289"
    ],
    [
      "1.808346363",
      "1.480521292"
    ],
    [
      "0.494940026",
      "0.652103577"
    ],
    [
      "0.241910354",
      "0.444127816"
    ],
    [
      "1.174667398",
      "0.393572014"
    ],
    [
      "0.868987625",
      "0.47147805"
    ],
    [
      "1.019560894",
      "1.525300193"
    ],
    [
      "1.20673844",
      "1.432575182"
    ],
    [
      "1.784659559",
      "0.963088998"
    ],
    [
      "0.959714379",
      "0.165371649"
    ],
    [
      "0.544322015",
      "1.485322274"
    ],
    [
      "0.781941454",
      "1.505311234"
    ],
    [
      "0.834181787",
      "0.883133326"
    ],
    [
      "0.061404909",
      "1.057998364"
    ],
    [
      "0.663131734",
      "1.495316754"
    ],
    [
      "0.816563347",
      "1.093742291"
    ],
    [
      "0.7732006",
      "0.240174409"
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      "1.547040106",
      "0.943100204"
    ],
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      "0.477445135",
      "0.857974381"
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    [
      "1.749855799",
      "1.376624742"
    ],
    [
      "1.36526002",
      "0.303997669"
    ],
    [
      "1.580478546",
      "1.25677415"
    ],
    [
      "1.293477118",
      "0.403566493"
    ],
    [
      "1.32554816",
      "1.442569662"
    ],
    [
      "1.377664944",
      "0.825129915"
    ],
    [
      "0.444155016",
      "0.348448887"
    ],
    [
      "1.641452891",
      "1.68765016"
    ],
    [
      "1.716599212",
      "1.060984159"
    ],
    [
      "1.564187189",
      "1.46325966"
    ],
    [
      "0.52561008",
      "1.690117704"
    ],
    [
      "0.360799274",
      "0.458999553"
    ],
    [
      "0.890021141",
      "0.26401975"
    ],
    [
      "0.832690841",
      "1.613200709"
    ],
    [
      "1.071801227",
      "0.903122285"
    ],
    [
      "1.121052304",
      "1.741243776"
    ],
    [
      "1.1733",
      "1.118901236"
    ],
    [
      "1.2464503",
      "0.294003189"
    ],
    [
      "0.41602084",
      "1.737084049"
    ],
    [
      "1.68912403",
      "1.479227331"
    ],
    [
      "0.989633668",
      "1.940328184"
    ],
    [
      "0.409384802",
      "0.955869377"
    ],
    [
      "1.308237213",
      "1.648354133"
    ],
    [
      "0.561544881",
      "0.168085543"
    ],
    [
      "0.639745015",
      "1.820175673"
    ],
    [
      "0.851676679",
      "0.677262521"
    ],
    [
      "0.510883575",
      "1.171648328"
    ],
    [
      "0.680442681",
      "1.289532283"
    ],
    [
      "0.376130307",
      "0.642109097"
    ],
    [
      "0.357956724",
      "0.848301761"
    ],
    [
      "1.631227932",
      "1.364663626"
    ],
    [
      "1.155989053",
      "1.324685707"
    ],
    [
      "1.868122837",
      "1.361506976"
    ],
    [
      "1.393608493",
      "1.344674666"
    ],
    [
      "1.642863513",
      "0.382018112"
    ],
    [
      "1.478979773",
      "1.040995199"
    ],
    [
      "0.933510576",
      "1.835133898"
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    [
      "1.106607065",
      "0.491467009"
    ],
    [
      "0.392073855",
      "1.161653848"
    ],
    [
      "0.987797345",
      "0.48147253"
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    [
      "0.493572628",
      "1.377432799"
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    [
      "1.127872776",
      "1.931650743"
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    [
      "0.460134188",
      "1.063758852"
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    [
      "1.632595331",
      "0.639334404"
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      "1.055857678",
      "0.383577534"
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    [
      "0.715064574",
      "0.877963341"
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    [
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      "1.342198698"
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    [
      "1.88952798",
      "0.906359345"
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      "1.467753352"
    ],
    [
      "1.344226504",
      "0.511455969"
    ],
    [
      "0.595071401",
      "1.59321175"
    ],
    [
      "0.883440227",
      "1.721090184"
    ],
    [
      "0.647004241",
      "0.975858336"
    ],
    [
      "1.105239667",
      "1.216796231"
    ],
    [
      "0.545689413",
      "0.759993052"
    ],
    [
      "0.699428519",
      "0.353594095"
    ],
    [
      "1.564534997",
      "0.737229399"
    ],
    [
      "1.309420666",
      "0.923111244"
    ],
    [
      "0.197010716",
      "0.783388499"
    ],
    [
      "0.765813961",
      "0.985852816"
    ],
    [
      "0.952991507",
      "0.893127805"
    ],
    [
      "0.612382348",
      "1.387427279"
    ],
    [
      "0.194703154",
      "1.017987334"
    ],
    [
      "1.445725278",
      "0.72723492"
    ],
    [
      "0.631368186",
      "0.451489091"
    ],
    [
      "0.253195843",
      "1.571496374"
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      "0.578943908",
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      "0.81857853",
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      "1.530470178"
    ],
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      "1.582836394",
      "0.530984566"
    ],
    [
      "0.44419064",
      "0.544214102"
    ],
    [
      "1.665849826",
      "0.953094683"
    ],
    [
      "1.495120603",
      "1.560447343"
    ],
    [
      "1.190610946",
      "0.913116765"
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    [
      "1.767348599",
      "1.168873634"
    ],
    [
      "0.459804575",
      "0.230251043"
    ],
    [
      "1.078900712",
      "0.162168947"
    ],
    [
      "1.444357879",
      "1.452564142"
    ],
    [
      "1.358999936",
      "1.756237334"
    ],
    [
      "1.513785611",
      "0.629339924"
    ],
    [
      "1.345336559",
      "1.874681208"
    ],
    [
      "1.376297546",
      "1.550459137"
    ],
    [
      "1.820921389",
      "0.541303881"
    ],
    [
      "0.93568056",
      "1.098912276"
    ],
    [
      "0.565837183",
      "0.332915064"
    ],
    [
      "0.750177906",
      "0.46148357"
    ],
    [
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      "0.556894549"
    ],
    [
      "1.683344717",
      "0.747223879"
    ],
    [
      "0.764630508",
      "1.711095705"
    ],
    [
      "0.732866959",
      "0.667268041"
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    [
      "0.596254854",
      "0.867968861"
    ],
    [
      "1.615284384",
      "0.845118875"
    ],
    [
      "1.427060269",
      "1.658342338"
    ]
  ],
  "radius": "0.059614677"
}
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": 236
}

起步布局答案

{
  "centers": [
    [
      "1.412286838",
      "0.413560973"
    ],
    [
      "1.15803759",
      "0.072989388"
    ],
    [
      "1.257487826",
      "1.540464658"
    ],
    [
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  "radius": "0.059614677"
}

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?