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 = 156

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 = 156
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[0.0729056248105, 0.0729056248115]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 = 0.072905624

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 = 156 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": [
    [
      "0.927515096",
      "0.075743586"
    ],
    [
      "1.073326331",
      "0.07580996"
    ],
    [
      "0.783496868",
      "0.098539865"
    ],
    [
      "1.217323745",
      "0.098737347"
    ],
    [
      "1.355945382",
      "0.143958608"
    ],
    [
      "0.655895597",
      "0.169100734"
    ],
    [
      "0.515768391",
      "0.209415578"
    ],
    [
      "1.485762258",
      "0.210355137"
    ],
    [
      "0.981344996",
      "0.211254701"
    ],
    [
      "1.127141684",
      "0.213315339"
    ],
    [
      "0.783872883",
      "0.249349801"
    ],
    [
      "1.280178347",
      "0.268539011"
    ],
    [
      "0.397801214",
      "0.295116603"
    ],
    [
      "0.651536967",
      "0.314846824"
    ],
    [
      "1.583827151",
      "0.318263419"
    ],
    [
      "0.903328379",
      "0.334438824"
    ],
    [
      "1.424414948",
      "0.343100031"
    ],
    [
      "1.16258563",
      "0.354753125"
    ],
    [
      "0.519309286",
      "0.37632414"
    ],
    [
      "0.77177695",
      "0.397328744"
    ],
    [
      "1.70553457",
      "0.398564258"
    ],
    [
      "1.296045601",
      "0.413484348"
    ],
    [
      "1.024973309",
      "0.416397138"
    ],
    [
      "0.388434109",
      "0.440626664"
    ],
    [
      "1.526178489",
      "0.452194567"
    ],
    [
      "0.640225521",
      "0.460218664"
    ],
    [
      "0.244334482",
      "0.462902611"
    ],
    [
      "0.893024619",
      "0.47988556"
    ],
    [
      "1.144816031",
      "0.499477559"
    ],
    [
      "1.791107903",
      "0.516624095"
    ],
    [
      "1.396607443",
      "0.519069558"
    ],
    [
      "0.508674092",
      "0.523108584"
    ],
    [
      "1.646418472",
      "0.534676486"
    ],
    [
      "0.76147319",
      "0.54277548"
    ],
    [
      "1.013264602",
      "0.562367479"
    ],
    [
      "1.265056014",
      "0.581959478"
    ],
    [
      "0.377122663",
      "0.585998504"
    ],
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      "1.516847425",
      "0.601551478"
    ],
    [
      "0.629921761",
      "0.6056654"
    ],
    [
      "0.233023036",
      "0.608274451"
    ],
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      "0.881713173",
      "0.625257399"
    ],
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      "1.133504585",
      "0.644849399"
    ],
    [
      "1.85711218",
      "0.646640848"
    ],
    [
      "1.385295996",
      "0.664441398"
    ],
    [
      "0.498370332",
      "0.66855532"
    ],
    [
      "1.639559204",
      "0.680326309"
    ],
    [
      "0.750161744",
      "0.688147319"
    ],
    [
      "0.121881368",
      "0.702659037"
    ],
    [
      "1.001953156",
      "0.707739319"
    ],
    [
      "1.253744567",
      "0.727331318"
    ],
    [
      "0.366818903",
      "0.73144524"
    ],
    [
      "1.505535979",
      "0.746923317"
    ],
    [
      "0.618610315",
      "0.751037239"
    ],
    [
      "1.757366662",
      "0.766246758"
    ],
    [
      "0.870401727",
      "0.770629239"
    ],
    [
      "1.901914701",
      "0.785398387"
    ],
    [
      "1.122193138",
      "0.790221238"
    ],
    [
      "0.235267474",
      "0.79433516"
    ],
    [
      "1.37398455",
      "0.809813237"
    ],
    [
      "0.487058886",
      "0.813927159"
    ],
    [
      "1.625874755",
      "0.829261034"
    ],
    [
      "0.738850298",
      "0.833519159"
    ],
    [
      "0.086121784",
      "0.84401735"
    ],
    [
      "0.990641709",
      "0.853111158"
    ],
    [
      "1.242433121",
      "0.872703157"
    ],
    [
      "0.355507457",
      "0.876817079"
    ],
    [
      "1.494224533",
      "0.892295157"
    ],
    [
      "0.607298869",
      "0.896409079"
    ],
    [
      "1.757945158",
      "0.912325857"
    ],
    [
      "0.85909028",
      "0.916001078"
    ],
    [
      "1.924407218",
      "0.929464369"
    ],
    [
      "1.110881692",
      "0.935593077"
    ],
    [
      "0.19950789",
      "0.935693473"
    ],
    [
      "1.362673104",
      "0.955185077"
    ],
    [
      "0.47574744",
      "0.959298999"
    ],
    [
      "1.614563309",
      "0.974632873"
    ],
    [
      "0.727538851",
      "0.978890998"
    ],
    [
      "0.979330263",
      "0.998482997"
    ],
    [
      "0.073077831",
      "1.011502779"
    ],
    [
      "1.231121675",
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    ],
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      "0.344196011",
      "1.022188919"
    ],
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      "1.483069156",
      "1.037679221"
    ],
    [
      "0.595987422",
      "1.041780918"
    ],
    [
      "1.734703809",
      "1.057259629"
    ],
    [
      "0.847778834",
      "1.061372918"
    ],
    [
      "1.880090554",
      "1.068377845"
    ],
    [
      "1.099570246",
      "1.080964917"
    ],
    [
      "0.210800659",
      "1.081066765"
    ],
    [
      "1.351361658",
      "1.100556916"
    ],
    [
      "0.464435993",
      "1.104670838"
    ],
    [
      "1.603308628",
      "1.120475347"
    ],
    [
      "0.716227405",
      "1.124262838"
    ],
    [
      "0.968018817",
      "1.143854837"
    ],
    [
      "0.086286015",
      "1.15694182"
    ],
    [
      "1.219810229",
      "1.163446836"
    ],
    [
      "0.328488045",
      "1.167151605"
    ],
    [
      "1.471601641",
      "1.183038836"
    ],
    [
      "0.584675976",
      "1.187152758"
    ],
    [
      "1.747557554",
      "1.205548808"
    ],
    [
      "0.836467388",
      "1.206744757"
    ],
    [
      "1.902419406",
      "1.212469284"
    ],
    [
      "1.0882588",
      "1.226336756"
    ],
    [
      "1.340050212",
      "1.245928756"
    ],
    [
      "0.453124547",
      "1.250042678"
    ],
    [
      "1.591997182",
      "1.265847186"
    ],
    [
      "0.181329676",
      "1.267520403"
    ],
    [
      "0.704915959",
      "1.269634677"
    ],
    [
      "0.956707371",
      "1.289226676"
    ],
    [
      "1.208498783",
      "1.308818676"
    ],
    [
      "0.319960567",
      "1.312713285"
    ],
    [
      "1.460290195",
      "1.328410675"
    ],
    [
      "0.57336453",
      "1.332524597"
    ],
    [
      "1.712162453",
      "1.348437913"
    ],
    [
      "1.857944972",
      "1.351332329"
    ],
    [
      "0.825155942",
      "1.352116596"
    ],
    [
      "1.076947354",
      "1.371708596"
    ],
    [
      "1.328738766",
      "1.391300595"
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    [
      "0.440145674",
      "1.395275144"
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      "0.169921561",
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    [
      "1.582327123",
      "1.41479835"
    ],
    [
      "0.693604513",
      "1.415006516"
    ],
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      "0.945395925",
      "1.434598516"
    ],
    [
      "1.197187337",
      "1.454190515"
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      "0.308552453",
      "1.45807757"
    ],
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      "1.448978749",
      "1.473782514"
    ],
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      "0.560385657",
      "1.477757064"
    ],
    [
      "1.792248182",
      "1.481504724"
    ],
    [
      "0.813844496",
      "1.497488436"
    ],
    [
      "1.065635908",
      "1.517080435"
    ],
    [
      "1.31742732",
      "1.536672435"
    ],
    [
      "0.428250617",
      "1.541488621"
    ],
    [
      "1.566734721",
      "1.559773512"
    ],
    [
      "0.68062564",
      "1.560238983"
    ],
    [
      "0.935803011",
      "1.583875314"
    ],
    [
      "1.706954129",
      "1.599766488"
    ],
    [
      "0.296107541",
      "1.603356766"
    ],
    [
      "1.435183292",
      "1.622663432"
    ],
    [
      "0.549074211",
      "1.623128903"
    ],
    [
      "1.201569912",
      "1.625204811"
    ],
    [
      "0.800865622",
      "1.642720902"
    ],
    [
      "1.060690761",
      "1.662807804"
    ],
    [
      "1.604172669",
      "1.703192269"
    ],
    [
      "0.399414055",
      "1.7062581"
    ],
    [
      "0.671502079",
      "1.710555891"
    ],
    [
      "1.317241364",
      "1.714691474"
    ],
    [
      "1.471581189",
      "1.763858735"
    ],
    [
      "1.163695484",
      "1.766011232"
    ],
    [
      "0.531562009",
      "1.767884709"
    ],
    [
      "0.85945645",
      "1.776242569"
    ],
    [
      "1.003775138",
      "1.797052105"
    ],
    [
      "1.357934483",
      "1.855211603"
    ],
    [
      "0.646825894",
      "1.857188445"
    ],
    [
      "1.219418348",
      "1.900754999"
    ],
    [
      "0.785593102",
      "1.901961009"
    ],
    [
      "1.075474622",
      "1.924017079"
    ],
    [
      "0.929663936",
      "1.924422424"
    ]
  ],
  "radius": "0.072905624"
}
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": 156
}

起步布局答案

{
  "centers": [
    [
      "0.927515096",
      "0.075743586"
    ],
    [
      "1.073326331",
      "0.07580996"
    ],
    [
      "0.783496868",
      "0.098539865"
    ],
    [
      "1.217323745",
      "0.098737347"
    ],
    [
      "1.355945382",
      "0.143958608"
    ],
    [
      "0.655895597",
      "0.169100734"
    ],
    [
      "0.515768391",
      "0.209415578"
    ],
    [
      "1.485762258",
      "0.210355137"
    ],
    [
      "0.981344996",
      "0.211254701"
    ],
    [
      "1.127141684",
      "0.213315339"
    ],
    [
      "0.783872883",
      "0.249349801"
    ],
    [
      "1.280178347",
      "0.268539011"
    ],
    [
      "0.397801214",
      "0.295116603"
    ],
    [
      "0.651536967",
      "0.314846824"
    ],
    [
      "1.583827151",
      "0.318263419"
    ],
    [
      "0.903328379",
      "0.334438824"
    ],
    [
      "1.424414948",
      "0.343100031"
    ],
    [
      "1.16258563",
      "0.354753125"
    ],
    [
      "0.519309286",
      "0.37632414"
    ],
    [
      "0.77177695",
      "0.397328744"
    ],
    [
      "1.70553457",
      "0.398564258"
    ],
    [
      "1.296045601",
      "0.413484348"
    ],
    [
      "1.024973309",
      "0.416397138"
    ],
    [
      "0.388434109",
      "0.440626664"
    ],
    [
      "1.526178489",
      "0.452194567"
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    ],
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      "0.751037239"
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    [
      "1.757366662",
      "0.766246758"
    ],
    [
      "0.870401727",
      "0.770629239"
    ],
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      "0.785398387"
    ],
    [
      "1.122193138",
      "0.790221238"
    ],
    [
      "0.235267474",
      "0.79433516"
    ],
    [
      "1.37398455",
      "0.809813237"
    ],
    [
      "0.487058886",
      "0.813927159"
    ],
    [
      "1.625874755",
      "0.829261034"
    ],
    [
      "0.738850298",
      "0.833519159"
    ],
    [
      "0.086121784",
      "0.84401735"
    ],
    [
      "0.990641709",
      "0.853111158"
    ],
    [
      "1.242433121",
      "0.872703157"
    ],
    [
      "0.355507457",
      "0.876817079"
    ],
    [
      "1.494224533",
      "0.892295157"
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    [
      "0.607298869",
      "0.896409079"
    ],
    [
      "1.757945158",
      "0.912325857"
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    [
      "0.85909028",
      "0.916001078"
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    [
      "1.924407218",
      "0.929464369"
    ],
    [
      "1.110881692",
      "0.935593077"
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    [
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      "0.935693473"
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    [
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      "0.955185077"
    ],
    [
      "0.47574744",
      "0.959298999"
    ],
    [
      "1.614563309",
      "0.974632873"
    ],
    [
      "0.727538851",
      "0.978890998"
    ],
    [
      "0.979330263",
      "0.998482997"
    ],
    [
      "0.073077831",
      "1.011502779"
    ],
    [
      "1.231121675",
      "1.018074997"
    ],
    [
      "0.344196011",
      "1.022188919"
    ],
    [
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      "1.037679221"
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    [
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      "1.041780918"
    ],
    [
      "1.734703809",
      "1.057259629"
    ],
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  "radius": "0.072905624"
}

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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还没有帖子。第一个聊聊这道题的思路?