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 = 183

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 = 183
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[0.0672690799445, 0.0672690799455]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 = 0.067269079

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 = 183 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": [
    [
      "1.069648",
      "0.06987306"
    ],
    [
      "0.886738765",
      "0.074215624"
    ],
    [
      "1.202736658",
      "0.089568884"
    ],
    [
      "0.754716995",
      "0.100114837"
    ],
    [
      "1.331607278",
      "0.128206686"
    ],
    [
      "0.608333711",
      "0.153486865"
    ],
    [
      "0.980568233",
      "0.170696229"
    ],
    [
      "1.463003581",
      "0.190299652"
    ],
    [
      "0.852388108",
      "0.211566415"
    ],
    [
      "0.490624072",
      "0.218640202"
    ],
    [
      "1.220057625",
      "0.222987401"
    ],
    [
      "0.719098426",
      "0.229852397"
    ],
    [
      "1.087663185",
      "0.253777746"
    ],
    [
      "1.348928245",
      "0.261625203"
    ],
    [
      "1.574675025",
      "0.265333011"
    ],
    [
      "0.601388787",
      "0.295005734"
    ],
    [
      "0.957735913",
      "0.303282811"
    ],
    [
      "1.460599689",
      "0.336658562"
    ],
    [
      "0.430713311",
      "0.339102716"
    ],
    [
      "0.829555788",
      "0.344152997"
    ],
    [
      "1.18540085",
      "0.352985187"
    ],
    [
      "1.674390067",
      "0.355651486"
    ],
    [
      "0.302705731",
      "0.380510152"
    ],
    [
      "0.701375664",
      "0.385023182"
    ],
    [
      "1.31427147",
      "0.391622989"
    ],
    [
      "1.057220595",
      "0.393854963"
    ],
    [
      "0.563357793",
      "0.424056732"
    ],
    [
      "1.560314731",
      "0.426977037"
    ],
    [
      "0.92904047",
      "0.434725148"
    ],
    [
      "1.760074084",
      "0.459375956"
    ],
    [
      "1.425942915",
      "0.466656348"
    ],
    [
      "0.438207483",
      "0.47343199"
    ],
    [
      "0.800860345",
      "0.475595334"
    ],
    [
      "1.156924109",
      "0.484475079"
    ],
    [
      "0.309507722",
      "0.514876255"
    ],
    [
      "0.671721414",
      "0.516252565"
    ],
    [
      "1.285794729",
      "0.523112881"
    ],
    [
      "1.028525151",
      "0.5252973"
    ],
    [
      "1.645998748",
      "0.530701507"
    ],
    [
      "0.180793469",
      "0.554031808"
    ],
    [
      "0.543542866",
      "0.557127709"
    ],
    [
      "0.900345026",
      "0.566167485"
    ],
    [
      "1.516031063",
      "0.566579529"
    ],
    [
      "1.829944377",
      "0.574348382"
    ],
    [
      "0.409512039",
      "0.604874327"
    ],
    [
      "0.771211173",
      "0.60682583"
    ],
    [
      "1.388640894",
      "0.609849164"
    ],
    [
      "1.128009832",
      "0.615869451"
    ],
    [
      "0.280812279",
      "0.646318592"
    ],
    [
      "0.643031049",
      "0.647696015"
    ],
    [
      "1.256880452",
      "0.654507253"
    ],
    [
      "0.999829708",
      "0.656739637"
    ],
    [
      "1.611996817",
      "0.660872107"
    ],
    [
      "0.125156028",
      "0.676526672"
    ],
    [
      "1.745316712",
      "0.678936501"
    ],
    [
      "0.514847423",
      "0.688570047"
    ],
    [
      "0.870695855",
      "0.697397981"
    ],
    [
      "1.88254726",
      "0.698176703"
    ],
    [
      "1.484606649",
      "0.704141743"
    ],
    [
      "0.380816596",
      "0.736316664"
    ],
    [
      "0.74251573",
      "0.738268167"
    ],
    [
      "1.356426524",
      "0.745011928"
    ],
    [
      "1.099314389",
      "0.747311789"
    ],
    [
      "0.22369264",
      "0.76812936"
    ],
    [
      "0.614335605",
      "0.779138352"
    ],
    [
      "1.228185009",
      "0.78594959"
    ],
    [
      "0.970180536",
      "0.787970133"
    ],
    [
      "1.594721514",
      "0.795746319"
    ],
    [
      "1.783811069",
      "0.807850041"
    ],
    [
      "0.0874715",
      "0.808282913"
    ],
    [
      "0.486151979",
      "0.820012384"
    ],
    [
      "1.916788304",
      "0.82828461"
    ],
    [
      "0.842000411",
      "0.828840319"
    ],
    [
      "1.455911205",
      "0.83558408"
    ],
    [
      "0.324353341",
      "0.858433074"
    ],
    [
      "0.713820286",
      "0.869710504"
    ],
    [
      "1.32773108",
      "0.876454265"
    ],
    [
      "1.069665217",
      "0.878542285"
    ],
    [
      "0.19635213",
      "0.899860196"
    ],
    [
      "1.687757323",
      "0.902052983"
    ],
    [
      "0.585640162",
      "0.91058069"
    ],
    [
      "1.198535838",
      "0.917180086"
    ],
    [
      "0.941485093",
      "0.91941247"
    ],
    [
      "1.555395886",
      "0.926156231"
    ],
    [
      "0.068997968",
      "0.943235695"
    ],
    [
      "0.456697767",
      "0.951286775"
    ],
    [
      "0.813304968",
      "0.960282656"
    ],
    [
      "1.931955106",
      "0.961965141"
    ],
    [
      "1.427215762",
      "0.967026417"
    ],
    [
      "1.798441288",
      "0.978535506"
    ],
    [
      "0.328756662",
      "0.992899156"
    ],
    [
      "0.685124843",
      "1.001152841"
    ],
    [
      "1.298081909",
      "1.007684761"
    ],
    [
      "1.040969774",
      "1.009984622"
    ],
    [
      "1.659061879",
      "1.03349532"
    ],
    [
      "0.200053288",
      "1.034347436"
    ],
    [
      "0.556944718",
      "1.042023027"
    ],
    [
      "1.169840394",
      "1.048622424"
    ],
    [
      "0.912789649",
      "1.050854807"
    ],
    [
      "1.526700443",
      "1.057598568"
    ],
    [
      "0.070516525",
      "1.077765285"
    ],
    [
      "0.428764593",
      "1.082893212"
    ],
    [
      "0.784609524",
      "1.091724993"
    ],
    [
      "1.927732113",
      "1.096437008"
    ],
    [
      "1.39756659",
      "1.098256913"
    ],
    [
      "1.794218295",
      "1.113007373"
    ],
    [
      "0.300061219",
      "1.124341493"
    ],
    [
      "0.656429399",
      "1.132595178"
    ],
    [
      "1.269386465",
      "1.139127098"
    ],
    [
      "1.01227433",
      "1.141426959"
    ],
    [
      "0.172120114",
      "1.165953873"
    ],
    [
      "1.67120303",
      "1.167484532"
    ],
    [
      "0.528249275",
      "1.173465364"
    ],
    [
      "1.141144951",
      "1.180064761"
    ],
    [
      "0.884094206",
      "1.182297144"
    ],
    [
      "1.497051271",
      "1.188829065"
    ],
    [
      "0.40006915",
      "1.214335549"
    ],
    [
      "0.755914081",
      "1.22316733"
    ],
    [
      "1.904207186",
      "1.228902453"
    ],
    [
      "1.368871147",
      "1.22969925"
    ],
    [
      "0.272128045",
      "1.25594793"
    ],
    [
      "1.772444426",
      "1.256088687"
    ],
    [
      "0.627733956",
      "1.264037515"
    ],
    [
      "1.240691022",
      "1.270569436"
    ],
    [
      "0.983578887",
      "1.272869296"
    ],
    [
      "1.596535953",
      "1.279401216"
    ],
    [
      "0.112356649",
      "1.286489531"
    ],
    [
      "0.499553831",
      "1.304907701"
    ],
    [
      "1.112449507",
      "1.311507098"
    ],
    [
      "0.855398762",
      "1.313739481"
    ],
    [
      "1.468355828",
      "1.320271402"
    ],
    [
      "0.727218637",
      "1.354609667"
    ],
    [
      "1.861869775",
      "1.356605468"
    ],
    [
      "0.360546434",
      "1.359221652"
    ],
    [
      "1.340175703",
      "1.361141587"
    ],
    [
      "1.697973906",
      "1.368167345"
    ],
    [
      "0.21236458",
      "1.376483587"
    ],
    [
      "0.599038512",
      "1.395479853"
    ],
    [
      "1.211995578",
      "1.402011773"
    ],
    [
      "0.954883443",
      "1.404311633"
    ],
    [
      "1.567840509",
      "1.410843553"
    ],
    [
      "0.470858388",
      "1.436350038"
    ],
    [
      "1.083754064",
      "1.442949435"
    ],
    [
      "0.826703319",
      "1.445181819"
    ],
    [
      "1.439660384",
      "1.451713739"
    ],
    [
      "1.801600729",
      "1.476889127"
    ],
    [
      "0.698523194",
      "1.486052004"
    ],
    [
      "1.31148026",
      "1.492583924"
    ],
    [
      "1.668985969",
      "1.499557207"
    ],
    [
      "0.353888284",
      "1.502821921"
    ],
    [
      "0.219588527",
      "1.510827664"
    ],
    [
      "1.183300135",
      "1.53345411"
    ],
    [
      "0.562998856",
      "1.535171172"
    ],
    [
      "0.926188",
      "1.53575397"
    ],
    [
      "1.539145066",
      "1.54228589"
    ],
    [
      "1.05505862",
      "1.574391772"
    ],
    [
      "0.798007875",
      "1.576624156"
    ],
    [
      "1.411411318",
      "1.584575315"
    ],
    [
      "0.444556782",
      "1.602218802"
    ],
    [
      "0.66982775",
      "1.617494341"
    ],
    [
      "1.282784816",
      "1.624026261"
    ],
    [
      "0.313142309",
      "1.631041585"
    ],
    [
      "1.68491556",
      "1.633148991"
    ],
    [
      "1.154604691",
      "1.664896447"
    ],
    [
      "0.897492556",
      "1.667196307"
    ],
    [
      "1.557250342",
      "1.675600239"
    ],
    [
      "0.552241274",
      "1.682869696"
    ],
    [
      "1.026363177",
      "1.705834109"
    ],
    [
      "1.380552057",
      "1.716449651"
    ],
    [
      "0.427782832",
      "1.736044222"
    ],
    [
      "0.668314533",
      "1.752023991"
    ],
    [
      "0.802450448",
      "1.762419803"
    ],
    [
      "1.25836515",
      "1.772760186"
    ],
    [
      "1.491287234",
      "1.79285801"
    ],
    [
      "1.125909248",
      "1.796338784"
    ],
    [
      "0.931321068",
      "1.801057605"
    ],
    [
      "0.550728057",
      "1.817399346"
    ],
    [
      "1.367176191",
      "1.856066872"
    ],
    [
      "0.699325614",
      "1.882939343"
    ],
    [
      "0.833458076",
      "1.893379603"
    ],
    [
      "1.229268815",
      "1.904114362"
    ],
    [
      "1.096812912",
      "1.927692961"
    ],
    [
      "0.962342768",
      "1.931970441"
    ]
  ],
  "radius": "0.067269079"
}
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": 183
}

起步布局答案

{
  "centers": [
    [
      "1.069648",
      "0.06987306"
    ],
    [
      "0.886738765",
      "0.074215624"
    ],
    [
      "1.202736658",
      "0.089568884"
    ],
    [
      "0.754716995",
      "0.100114837"
    ],
    [
      "1.331607278",
      "0.128206686"
    ],
    [
      "0.608333711",
      "0.153486865"
    ],
    [
      "0.980568233",
      "0.170696229"
    ],
    [
      "1.463003581",
      "0.190299652"
    ],
    [
      "0.852388108",
      "0.211566415"
    ],
    [
      "0.490624072",
      "0.218640202"
    ],
    [
      "1.220057625",
      "0.222987401"
    ],
    [
      "0.719098426",
      "0.229852397"
    ],
    [
      "1.087663185",
      "0.253777746"
    ],
    [
      "1.348928245",
      "0.261625203"
    ],
    [
      "1.574675025",
      "0.265333011"
    ],
    [
      "0.601388787",
      "0.295005734"
    ],
    [
      "0.957735913",
      "0.303282811"
    ],
    [
      "1.460599689",
      "0.336658562"
    ],
    [
      "0.430713311",
      "0.339102716"
    ],
    [
      "0.829555788",
      "0.344152997"
    ],
    [
      "1.18540085",
      "0.352985187"
    ],
    [
      "1.674390067",
      "0.355651486"
    ],
    [
      "0.302705731",
      "0.380510152"
    ],
    [
      "0.701375664",
      "0.385023182"
    ],
    [
      "1.31427147",
      "0.391622989"
    ],
    [
      "1.057220595",
      "0.393854963"
    ],
    [
      "0.563357793",
      "0.424056732"
    ],
    [
      "1.560314731",
      "0.426977037"
    ],
    [
      "0.92904047",
      "0.434725148"
    ],
    [
      "1.760074084",
      "0.459375956"
    ],
    [
      "1.425942915",
      "0.466656348"
    ],
    [
      "0.438207483",
      "0.47343199"
    ],
    [
      "0.800860345",
      "0.475595334"
    ],
    [
      "1.156924109",
      "0.484475079"
    ],
    [
      "0.309507722",
      "0.514876255"
    ],
    [
      "0.671721414",
      "0.516252565"
    ],
    [
      "1.285794729",
      "0.523112881"
    ],
    [
      "1.028525151",
      "0.5252973"
    ],
    [
      "1.645998748",
      "0.530701507"
    ],
    [
      "0.180793469",
      "0.554031808"
    ],
    [
      "0.543542866",
      "0.557127709"
    ],
    [
      "0.900345026",
      "0.566167485"
    ],
    [
      "1.516031063",
      "0.566579529"
    ],
    [
      "1.829944377",
      "0.574348382"
    ],
    [
      "0.409512039",
      "0.604874327"
    ],
    [
      "0.771211173",
      "0.60682583"
    ],
    [
      "1.388640894",
      "0.609849164"
    ],
    [
      "1.128009832",
      "0.615869451"
    ],
    [
      "0.280812279",
      "0.646318592"
    ],
    [
      "0.643031049",
      "0.647696015"
    ],
    [
      "1.256880452",
      "0.654507253"
    ],
    [
      "0.999829708",
      "0.656739637"
    ],
    [
      "1.611996817",
      "0.660872107"
    ],
    [
      "0.125156028",
      "0.676526672"
    ],
    [
      "1.745316712",
      "0.678936501"
    ],
    [
      "0.514847423",
      "0.688570047"
    ],
    [
      "0.870695855",
      "0.697397981"
    ],
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  "radius": "0.067269079"
}

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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