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 = 227

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 = 227
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[0.0606488091136105627994790670185, 0.0606488091136105627994790670195]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 = 0.060648808

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 = 227 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": [
    [
      "0.873910134",
      "0.764572768"
    ],
    [
      "1.820276907",
      "0.881653446"
    ],
    [
      "0.212664468",
      "0.768678891"
    ],
    [
      "0.917695853",
      "1.318701063"
    ],
    [
      "0.715156613",
      "1.374532889"
    ],
    [
      "1.011980335",
      "0.474868773"
    ],
    [
      "1.284587941",
      "1.715930992"
    ],
    [
      "0.28030657",
      "1.055954719"
    ],
    [
      "1.612727317",
      "0.932241922"
    ],
    [
      "1.064898175",
      "0.271548732"
    ],
    [
      "0.174481419",
      "0.996674457"
    ],
    [
      "1.86955365",
      "1.355326765"
    ],
    [
      "0.991297014",
      "0.795124898"
    ],
    [
      "0.597769733",
      "1.343980759"
    ],
    [
      "0.800308973",
      "1.288148933"
    ],
    [
      "0.650687574",
      "1.140660719"
    ],
    [
      "0.198732427",
      "1.237120919"
    ],
    [
      "1.272117876",
      "1.277777743"
    ],
    [
      "1.758033415",
      "0.445233381"
    ],
    [
      "1.357270236",
      "1.191393788"
    ],
    [
      "1.032705029",
      "0.154454121"
    ],
    [
      "1.368821436",
      "1.628585998"
    ],
    [
      "1.599181745",
      "0.626584481"
    ],
    [
      "0.642639437",
      "1.764354852"
    ],
    [
      "1.117914645",
      "0.068087833"
    ],
    [
      "0.586218534",
      "0.90678855"
    ],
    [
      "0.174152851",
      "0.55238728"
    ],
    [
      "0.692095589",
      "0.499989943"
    ],
    [
      "1.352589775",
      "0.129332951"
    ],
    [
      "1.076449374",
      "0.708740942"
    ],
    [
      "0.431170007",
      "1.747538025"
    ],
    [
      "0.532239981",
      "1.81460495"
    ],
    [
      "0.404639704",
      "0.273414301"
    ],
    [
      "1.655080529",
      "0.38109435"
    ],
    [
      "0.814121556",
      "1.592761436"
    ],
    [
      "1.055766054",
      "1.028997067"
    ],
    [
      "1.186965516",
      "1.364161699"
    ],
    [
      "1.935816181",
      "0.918582951"
    ],
    [
      "1.331906455",
      "0.449589076"
    ],
    [
      "0.731284829",
      "0.140031639"
    ],
    [
      "0.639136374",
      "0.703468509"
    ],
    [
      "1.474657116",
      "1.221945917"
    ],
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      "0.618453054",
      "1.023724634"
    ],
    [
      "1.304352396",
      "1.394713828"
    ],
    [
      "0.830165789",
      "0.210285947"
    ],
    [
      "0.565535214",
      "1.227044675"
    ],
    [
      "1.877978486",
      "0.666086975"
    ],
    [
      "1.495340437",
      "0.901689792"
    ],
    [
      "1.152710354",
      "1.926855009"
    ],
    [
      "0.963735941",
      "1.740277585"
    ],
    [
      "1.161601734",
      "0.622356987"
    ],
    [
      "0.39778739",
      "1.08614361"
    ],
    [
      "1.669925307",
      "1.481815103"
    ],
    [
      "0.083264877",
      "1.199967758"
    ],
    [
      "1.544864326",
      "1.461200066"
    ],
    [
      "0.157293262",
      "0.876600815"
    ],
    [
      "1.246754095",
      "0.535973031"
    ],
    [
      "1.343457654",
      "0.886781286"
    ],
    [
      "0.622136315",
      "0.266086768"
    ],
    [
      "0.938379174",
      "0.998444938"
    ],
    [
      "1.267437415",
      "0.215716906"
    ],
    [
      "0.862400309",
      "0.327222032"
    ],
    [
      "1.527574957",
      "1.018625877"
    ],
    [
      "0.853226814",
      "1.084828893"
    ],
    [
      "0.533300694",
      "1.11010859"
    ],
    [
      "0.703605414",
      "0.937340679"
    ],
    [
      "0.087386918",
      "0.777473547"
    ],
    [
      "1.311223134",
      "0.769845201"
    ],
    [
      "1.416612299",
      "1.740072076"
    ],
    [
      "0.382906144",
      "0.965762296"
    ],
    [
      "0.959062494",
      "0.678188813"
    ],
    [
      "1.730114197",
      "0.962794051"
    ],
    [
      "0.747391133",
      "1.491468974"
    ],
    [
      "1.129367215",
      "0.505420902"
    ],
    [
      "0.894593455",
      "0.444316643"
    ],
    [
      "0.906144654",
      "0.881508853"
    ],
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      "0.525033262",
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    ],
    [
      "1.134047676",
      "1.567481739"
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      "1.384824295",
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      "1.105009833"
    ],
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      "1.501764659",
      "1.65368812"
    ],
    [
      "0.339584895",
      "1.668006399"
    ],
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      "0.885461334",
      "1.201764978"
    ],
    [
      "0.269874164",
      "0.921752255"
    ],
    [
      "1.258305294",
      "0.973165241"
    ],
    [
      "0.397827849",
      "1.521082797"
    ],
    [
      "1.384515968",
      "1.857046166"
    ],
    [
      "1.097132695",
      "0.388484817"
    ],
    [
      "0.740612116",
      "0.292098243"
    ],
    [
      "1.239883356",
      "1.160841659"
    ],
    [
      "1.173152934",
      "1.059549196"
    ],
    [
      "1.396375494",
      "0.683461246"
    ],
    [
      "1.270871462",
      "1.899449449"
    ],
    [
      "0.929513276",
      "1.936702878"
    ],
    [
      "0.063474879",
      "1.072810418"
    ],
    [
      "0.674873957",
      "1.881290937"
    ],
    [
      "1.699946937",
      "0.844628723"
    ],
    [
      "0.501066174",
      "0.993172505"
    ],
    [
      "0.454253859",
      "1.628457179"
    ],
    [
      "0.724172823",
      "0.61697019"
    ],
    [
      "1.931487216",
      "1.121293966"
    ],
    [
      "0.467723509",
      "0.525688244"
    ],
    [
      "1.219200036",
      "1.481097784"
    ],
    [
      "0.259011767",
      "1.577336209"
    ],
    [
      "1.453973796",
      "1.542202042"
    ],
    [
      "0.820992294",
      "0.967892808"
    ],
    [
      "1.325035716",
      "1.074457703"
    ],
    [
      "0.788757774",
      "0.850956724"
    ],
    [
      "1.069578636",
      "1.333609569"
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    [
      "1.048895316",
      "1.653865694"
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      "1.364140975",
      "0.566525161"
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      "0.238717329",
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      "0.407806548",
      "0.63115421"
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      "0.682922093",
      "1.257596804"
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      "1.53692961"
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      "1.801983471",
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      "1.198282261",
      "1.802269615"
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      "0.756523254",
      "0.734020639"
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      "0.777247949",
      "0.413605987"
    ],
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      "1.580492797",
      "0.815305837"
    ],
    [
      "0.30813383",
      "1.174457013"
    ],
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      "1.417058815",
      "0.363205121"
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      "0.833954875",
      "0.075440821"
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      "1.31342863"
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      "0.76111905",
      "1.795997936"
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      "1.421739276",
      "1.425265958"
    ],
    [
      "0.931508436",
      "1.623313565"
    ],
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      "0.503230955",
      "0.202754005"
    ],
    [
      "1.002848213",
      "1.232317107"
    ],
    [
      "0.354281305",
      "0.834639991"
    ],
    [
      "0.794993108",
      "1.916305133"
    ],
    [
      "1.677196356",
      "1.166114091"
    ],
    [
      "1.336586916",
      "1.511649913"
    ],
    [
      "0.671370894",
      "0.820404594"
    ],
    [
      "1.867067418",
      "1.018516528"
    ],
    [
      "0.315975691",
      "0.356189931"
    ],
    [
      "0.659955316",
      "0.382694009"
    ],
    [
      "0.468562814",
      "0.876097107"
    ],
    [
      "1.908091343",
      "1.240313903"
    ],
    [
      "1.787312187",
      "1.512367232"
    ],
    [
      "1.823200636",
      "0.54753854"
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      "0.448148334",
      "1.196492545"
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      "0.984426276",
      "1.419993525"
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      "1.338882002"
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      "1.101813156",
      "1.450545654"
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    [
      "1.714724733",
      "1.609548371"
    ],
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      "1.193836254",
      "0.739293072"
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      "1.237041282",
      "0.091059771"
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      "0.553984014",
      "0.789852465"
    ],
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      "0.610105501",
      "0.145387258"
    ],
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      "1.691753907",
      "0.546821466"
    ],
    [
      "0.362995973",
      "1.282876501"
    ],
    [
      "1.154730996",
      "1.247225614"
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      "0.832543493",
      "1.405085018"
    ],
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      "1.592043996",
      "1.252498047"
    ],
    [
      "0.630004253",
      "1.460916844"
    ],
    [
      "0.061900516",
      "0.951523017"
    ],
    [
      "0.846356075",
      "1.70969752"
    ],
    [
      "0.878475915",
      "1.826665157"
    ],
    [
      "0.426512328",
      "0.406137545"
    ],
    [
      "1.577068637",
      "0.507319555"
    ],
    [
      "1.278988614",
      "0.652909116"
    ],
    [
      "1.534445695",
      "0.39375725"
    ],
    [
      "1.533267561",
      "1.773308583"
    ],
    [
      "1.410188077",
      "0.988073748"
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      "1.665112016",
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      "0.284689489",
      "0.602334893"
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    [
      "1.917522745",
      "0.798672725"
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    [
      "0.735839934",
      "1.054276764"
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    [
      "1.389504756",
      "1.308329873"
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    [
      "0.257712551",
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    [
      "1.756921128",
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      "1.182285055",
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    [
      "1.628168364",
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      "1.794941021",
      "1.196591867"
    ],
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      "0.289947071",
      "1.460049726"
    ],
    [
      "1.044214855",
      "0.591804857"
    ],
    [
      "0.809441095",
      "0.530700599"
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    [
      "1.214519575",
      "0.419036947"
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      "0.544491237",
      "1.54739962"
    ],
    [
      "0.572432512",
      "1.665435592"
    ],
    [
      "1.108683894",
      "0.825677027"
    ],
    [
      "1.559809476",
      "1.135561962"
    ],
    [
      "1.597337853",
      "1.578996241"
    ],
    [
      "0.661878117",
      "1.577951749"
    ],
    [
      "0.899273916",
      "1.50637748"
    ],
    [
      "1.624278516",
      "1.369434131"
    ],
    [
      "0.322654187",
      "0.717538165"
    ],
    [
      "0.947552669",
      "0.240838076"
    ],
    [
      "1.762348716",
      "1.079730136"
    ],
    [
      "0.841675615",
      "0.647636683"
    ],
    [
      "1.039028118",
      "1.884552982"
    ],
    [
      "1.251434556",
      "1.598033868"
    ],
    [
      "1.226070774",
      "0.856229156"
    ],
    [
      "1.023531534",
      "0.912060982"
    ],
    [
      "0.176055781",
      "1.117961858"
    ],
    [
      "0.439880902",
      "0.74869919"
    ],
    [
      "0.120433182",
      "1.329761841"
    ]
  ],
  "radius": "0.060648808"
}
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": 227
}

起步布局答案

{
  "centers": [
    [
      "0.873910134",
      "0.764572768"
    ],
    [
      "1.820276907",
      "0.881653446"
    ],
    [
      "0.212664468",
      "0.768678891"
    ],
    [
      "0.917695853",
      "1.318701063"
    ],
    [
      "0.715156613",
      "1.374532889"
    ],
    [
      "1.011980335",
      "0.474868773"
    ],
    [
      "1.284587941",
      "1.715930992"
    ],
    [
      "0.28030657",
      "1.055954719"
    ],
    [
      "1.612727317",
      "0.932241922"
    ],
    [
      "1.064898175",
      "0.271548732"
    ],
    [
      "0.174481419",
      "0.996674457"
    ],
    [
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  "radius": "0.060648808"
}

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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