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 = 195

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 = 195
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[0.0654143218261515773009104633565, 0.0654143218261515773009104633575]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 = 0.065414321

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 = 195 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": [
    [
      "0.952110276",
      "0.066642103"
    ],
    [
      "1.089493636",
      "0.069980021"
    ],
    [
      "0.822243164",
      "0.08247458"
    ],
    [
      "1.239758287",
      "0.09669144"
    ],
    [
      "1.363549309",
      "0.13902248"
    ],
    [
      "0.594113441",
      "0.158152929"
    ],
    [
      "0.724458596",
      "0.169390126"
    ],
    [
      "1.01555547",
      "0.181057318"
    ],
    [
      "1.146192622",
      "0.188133227"
    ],
    [
      "0.885688358",
      "0.196889794"
    ],
    [
      "1.269983644",
      "0.230464267"
    ],
    [
      "1.438870314",
      "0.245993879"
    ],
    [
      "1.569084892",
      "0.258654753"
    ],
    [
      "0.416814103",
      "0.269695544"
    ],
    [
      "0.547213387",
      "0.280286131"
    ],
    [
      "0.677558542",
      "0.291523328"
    ],
    [
      "1.070587159",
      "0.299748713"
    ],
    [
      "0.807957105",
      "0.302122804"
    ],
    [
      "0.940720047",
      "0.315581189"
    ],
    [
      "1.195659066",
      "0.338130376"
    ],
    [
      "1.329177636",
      "0.347135623"
    ],
    [
      "0.320546692",
      "0.358288544"
    ],
    [
      "1.456877286",
      "0.375577377"
    ],
    [
      "1.587091864",
      "0.388238251"
    ],
    [
      "0.471589818",
      "0.388505287"
    ],
    [
      "0.602654267",
      "0.398786944"
    ],
    [
      "1.717306442",
      "0.400899125"
    ],
    [
      "0.733578408",
      "0.409751533"
    ],
    [
      "0.863881383",
      "0.421467703"
    ],
    [
      "0.994001126",
      "0.435068681"
    ],
    [
      "1.124215704",
      "0.447729555"
    ],
    [
      "1.254853058",
      "0.454801732"
    ],
    [
      "0.23763429",
      "0.459489669"
    ],
    [
      "0.375322408",
      "0.477098287"
    ],
    [
      "1.382552708",
      "0.483243486"
    ],
    [
      "1.512767286",
      "0.49590436"
    ],
    [
      "0.52827557",
      "0.506415674"
    ],
    [
      "1.642981863",
      "0.508565234"
    ],
    [
      "0.658674133",
      "0.51701515"
    ],
    [
      "1.773428775",
      "0.519143337"
    ],
    [
      "0.788977108",
      "0.52873132"
    ],
    [
      "0.91909685",
      "0.542332298"
    ],
    [
      "1.049311428",
      "0.554993172"
    ],
    [
      "1.179656583",
      "0.566230369"
    ],
    [
      "0.169563461",
      "0.571251187"
    ],
    [
      "0.299778039",
      "0.583912061"
    ],
    [
      "1.308147158",
      "0.590853653"
    ],
    [
      "0.43200816",
      "0.595008675"
    ],
    [
      "1.438361736",
      "0.603514527"
    ],
    [
      "1.568576314",
      "0.616175402"
    ],
    [
      "1.854609391",
      "0.621740145"
    ],
    [
      "0.584295436",
      "0.62464388"
    ],
    [
      "1.698972227",
      "0.626807414"
    ],
    [
      "0.714598411",
      "0.63636005"
    ],
    [
      "0.844718154",
      "0.649961027"
    ],
    [
      "0.974932731",
      "0.662621902"
    ],
    [
      "1.105281663",
      "0.673885084"
    ],
    [
      "0.117828566",
      "0.691416218"
    ],
    [
      "1.233893572",
      "0.698568736"
    ],
    [
      "0.248043144",
      "0.704077092"
    ],
    [
      "1.36410815",
      "0.71122961"
    ],
    [
      "0.378465537",
      "0.714379197"
    ],
    [
      "1.494322728",
      "0.723890484"
    ],
    [
      "1.780187636",
      "0.729375398"
    ],
    [
      "0.508223508",
      "0.731082575"
    ],
    [
      "1.624778797",
      "0.735117782"
    ],
    [
      "0.638438086",
      "0.743743449"
    ],
    [
      "0.768652664",
      "0.756404323"
    ],
    [
      "1.905265188",
      "0.767738662"
    ],
    [
      "0.898867242",
      "0.769065198"
    ],
    [
      "1.029082585",
      "0.781718196"
    ],
    [
      "1.157821645",
      "0.805007431"
    ],
    [
      "0.171893237",
      "0.810551205"
    ],
    [
      "1.288036223",
      "0.817668305"
    ],
    [
      "0.302107815",
      "0.82321208"
    ],
    [
      "1.4182508",
      "0.830329179"
    ],
    [
      "0.432151581",
      "0.83752127"
    ],
    [
      "1.548465378",
      "0.842990053"
    ],
    [
      "0.562366159",
      "0.850182145"
    ],
    [
      "1.678679956",
      "0.855650928"
    ],
    [
      "1.809502851",
      "0.856877367"
    ],
    [
      "0.692580736",
      "0.862843019"
    ],
    [
      "0.822795314",
      "0.875503893"
    ],
    [
      "0.953009892",
      "0.888164767"
    ],
    [
      "0.071407482",
      "0.894329168"
    ],
    [
      "1.930024302",
      "0.907776423"
    ],
    [
      "1.081750301",
      "0.911446543"
    ],
    [
      "1.211964879",
      "0.924107417"
    ],
    [
      "0.226035887",
      "0.929650775"
    ],
    [
      "1.342179456",
      "0.936768292"
    ],
    [
      "0.356079653",
      "0.943959966"
    ],
    [
      "1.472394034",
      "0.949429166"
    ],
    [
      "0.486294231",
      "0.95662084"
    ],
    [
      "1.602608612",
      "0.96209004"
    ],
    [
      "0.616508809",
      "0.969281714"
    ],
    [
      "1.738570514",
      "0.973055959"
    ],
    [
      "0.746723387",
      "0.981942588"
    ],
    [
      "0.876937964",
      "0.994603462"
    ],
    [
      "0.125550132",
      "1.013428737"
    ],
    [
      "1.005678373",
      "1.017885239"
    ],
    [
      "1.135892951",
      "1.030546113"
    ],
    [
      "1.903348821",
      "1.035856682"
    ],
    [
      "1.266107529",
      "1.043206987"
    ],
    [
      "0.278665647",
      "1.049426575"
    ],
    [
      "1.396322107",
      "1.055867861"
    ],
    [
      "0.410222304",
      "1.063059535"
    ],
    [
      "1.526536684",
      "1.068528735"
    ],
    [
      "0.540436881",
      "1.075720409"
    ],
    [
      "1.656751262",
      "1.081189609"
    ],
    [
      "0.670651459",
      "1.088381283"
    ],
    [
      "1.786814564",
      "1.095320124"
    ],
    [
      "0.800866037",
      "1.101042157"
    ],
    [
      "0.929606446",
      "1.124323934"
    ],
    [
      "0.075090123",
      "1.134134667"
    ],
    [
      "1.059821024",
      "1.136984808"
    ],
    [
      "1.190035601",
      "1.149645682"
    ],
    [
      "0.203938129",
      "1.156813408"
    ],
    [
      "1.320250179",
      "1.162306556"
    ],
    [
      "1.919788576",
      "1.165648317"
    ],
    [
      "0.334150376",
      "1.16949823"
    ],
    [
      "1.450464757",
      "1.17496743"
    ],
    [
      "0.464364954",
      "1.182159104"
    ],
    [
      "1.580680477",
      "1.187629121"
    ],
    [
      "0.594579532",
      "1.194819979"
    ],
    [
      "0.724794109",
      "1.207480853"
    ],
    [
      "1.721252962",
      "1.208535893"
    ],
    [
      "0.853534518",
      "1.230762629"
    ],
    [
      "0.983749096",
      "1.243423503"
    ],
    [
      "1.113963674",
      "1.256084377"
    ],
    [
      "0.102883276",
      "1.261977049"
    ],
    [
      "1.244178252",
      "1.268745251"
    ],
    [
      "0.258058874",
      "1.275922933"
    ],
    [
      "1.374392829",
      "1.281406126"
    ],
    [
      "0.388271221",
      "1.288607714"
    ],
    [
      "1.887644959",
      "1.292466777"
    ],
    [
      "1.504607407",
      "1.294067"
    ],
    [
      "0.518507604",
      "1.301258674"
    ],
    [
      "1.634820089",
      "1.306747361"
    ],
    [
      "0.648722182",
      "1.313919548"
    ],
    [
      "1.763090152",
      "1.332494691"
    ],
    [
      "0.777462591",
      "1.337201324"
    ],
    [
      "0.907677169",
      "1.349862198"
    ],
    [
      "1.037891746",
      "1.362523073"
    ],
    [
      "1.168106324",
      "1.375183947"
    ],
    [
      "0.14825633",
      "1.384685728"
    ],
    [
      "1.298320902",
      "1.387844821"
    ],
    [
      "1.42853548",
      "1.400505695"
    ],
    [
      "0.277449649",
      "1.405306595"
    ],
    [
      "0.442435677",
      "1.407697369"
    ],
    [
      "1.559960823",
      "1.414042395"
    ],
    [
      "0.572650254",
      "1.420358243"
    ],
    [
      "1.688230886",
      "1.439789725"
    ],
    [
      "0.701390663",
      "1.44364002"
    ],
    [
      "1.818580744",
      "1.450972233"
    ],
    [
      "0.831605241",
      "1.456300894"
    ],
    [
      "0.961819819",
      "1.468961768"
    ],
    [
      "1.092034397",
      "1.481622642"
    ],
    [
      "1.222248974",
      "1.494283516"
    ],
    [
      "1.352463552",
      "1.50694439"
    ],
    [
      "0.358471479",
      "1.508027562"
    ],
    [
      "1.48267813",
      "1.519605264"
    ],
    [
      "0.227742292",
      "1.526337434"
    ],
    [
      "0.486055472",
      "1.537800432"
    ],
    [
      "0.616857849",
      "1.543491596"
    ],
    [
      "1.610948193",
      "1.545352594"
    ],
    [
      "0.75548571",
      "1.562951427"
    ],
    [
      "1.741560756",
      "1.568108792"
    ],
    [
      "0.885747891",
      "1.575400463"
    ],
    [
      "1.015941297",
      "1.58828058"
    ],
    [
      "1.146177047",
      "1.600722211"
    ],
    [
      "1.276391625",
      "1.613383085"
    ],
    [
      "1.406606202",
      "1.62604396"
    ],
    [
      "0.308827331",
      "1.629071325"
    ],
    [
      "0.550127855",
      "1.656022667"
    ],
    [
      "1.65202026",
      "1.669567003"
    ],
    [
      "0.680200393",
      "1.670067915"
    ],
    [
      "0.81044779",
      "1.682386578"
    ],
    [
      "1.522360008",
      "1.687012726"
    ],
    [
      "0.424064558",
      "1.691010937"
    ],
    [
      "0.940640852",
      "1.695266817"
    ],
    [
      "1.070876596",
      "1.70770808"
    ],
    [
      "1.204899136",
      "1.722950225"
    ],
    [
      "1.335113714",
      "1.735611099"
    ],
    [
      "0.734655816",
      "1.789024801"
    ],
    [
      "0.50575302",
      "1.793202567"
    ],
    [
      "0.864458626",
      "1.80537609"
    ],
    [
      "1.470886024",
      "1.807289751"
    ],
    [
      "0.994984275",
      "1.814274911"
    ],
    [
      "1.125220018",
      "1.826716174"
    ],
    [
      "0.62136036",
      "1.854448602"
    ],
    [
      "1.353540406",
      "1.86513558"
    ],
    [
      "1.229266809",
      "1.906028211"
    ],
    [
      "0.78539659",
      "1.909612976"
    ],
    [
      "0.917688035",
      "1.9307363"
    ],
    [
      "1.049327697",
      "1.933283005"
    ]
  ],
  "radius": "0.065414321"
}
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": 195
}

起步布局答案

{
  "centers": [
    [
      "0.952110276",
      "0.066642103"
    ],
    [
      "1.089493636",
      "0.069980021"
    ],
    [
      "0.822243164",
      "0.08247458"
    ],
    [
      "1.239758287",
      "0.09669144"
    ],
    [
      "1.363549309",
      "0.13902248"
    ],
    [
      "0.594113441",
      "0.158152929"
    ],
    [
      "0.724458596",
      "0.169390126"
    ],
    [
      "1.01555547",
      "0.181057318"
    ],
    [
      "1.146192622",
      "0.188133227"
    ],
    [
      "0.885688358",
      "0.196889794"
    ],
    [
      "1.269983644",
      "0.230464267"
    ],
    [
      "1.438870314",
      "0.245993879"
    ],
    [
      "1.569084892",
      "0.258654753"
    ],
    [
      "0.416814103",
      "0.269695544"
    ],
    [
      "0.547213387",
      "0.280286131"
    ],
    [
      "0.677558542",
      "0.291523328"
    ],
    [
      "1.070587159",
      "0.299748713"
    ],
    [
      "0.807957105",
      "0.302122804"
    ],
    [
      "0.940720047",
      "0.315581189"
    ],
    [
      "1.195659066",
      "0.338130376"
    ],
    [
      "1.329177636",
      "0.347135623"
    ],
    [
      "0.320546692",
      "0.358288544"
    ],
    [
      "1.456877286",
      "0.375577377"
    ],
    [
      "1.587091864",
      "0.388238251"
    ],
    [
      "0.471589818",
      "0.388505287"
    ],
    [
      "0.602654267",
      "0.398786944"
    ],
    [
      "1.717306442",
      "0.400899125"
    ],
    [
      "0.733578408",
      "0.409751533"
    ],
    [
      "0.863881383",
      "0.421467703"
    ],
    [
      "0.994001126",
      "0.435068681"
    ],
    [
      "1.124215704",
      "0.447729555"
    ],
    [
      "1.254853058",
      "0.454801732"
    ],
    [
      "0.23763429",
      "0.459489669"
    ],
    [
      "0.375322408",
      "0.477098287"
    ],
    [
      "1.382552708",
      "0.483243486"
    ],
    [
      "1.512767286",
      "0.49590436"
    ],
    [
      "0.52827557",
      "0.506415674"
    ],
    [
      "1.642981863",
      "0.508565234"
    ],
    [
      "0.658674133",
      "0.51701515"
    ],
    [
      "1.773428775",
      "0.519143337"
    ],
    [
      "0.788977108",
      "0.52873132"
    ],
    [
      "0.91909685",
      "0.542332298"
    ],
    [
      "1.049311428",
      "0.554993172"
    ],
    [
      "1.179656583",
      "0.566230369"
    ],
    [
      "0.169563461",
      "0.571251187"
    ],
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  "radius": "0.065414321"
}

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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