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 = 132

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 = 132
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[0.0788181266638514058250950945295, 0.0788181266638514058250950945305]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 = 0.078818126

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 = 132 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": [
    [
      "1.022519482",
      "0.079093427"
    ],
    [
      "0.865178518",
      "0.088737572"
    ],
    [
      "1.179200998",
      "0.096416553"
    ],
    [
      "0.711785586",
      "0.125066577"
    ],
    [
      "1.330634901",
      "0.140199671"
    ],
    [
      "1.472386691",
      "0.209160663"
    ],
    [
      "0.525135882",
      "0.210645762"
    ],
    [
      "0.979732712",
      "0.230811852"
    ],
    [
      "0.822174369",
      "0.241833872"
    ],
    [
      "1.196936764",
      "0.253051897"
    ],
    [
      "0.668315682",
      "0.276590687"
    ],
    [
      "1.600305395",
      "0.301280116"
    ],
    [
      "0.397506656",
      "0.303165863"
    ],
    [
      "1.346224548",
      "0.303671518"
    ],
    [
      "1.076751733",
      "0.355055556"
    ],
    [
      "0.920644018",
      "0.376954641"
    ],
    [
      "0.76488144",
      "0.401187006"
    ],
    [
      "1.486737868",
      "0.411644147"
    ],
    [
      "1.710645118",
      "0.41386046"
    ],
    [
      "0.59661095",
      "0.418014116"
    ],
    [
      "0.279511125",
      "0.427069689"
    ],
    [
      "0.439914039",
      "0.45499077"
    ],
    [
      "1.336495483",
      "0.461007254"
    ],
    [
      "1.177074194",
      "0.479636835"
    ],
    [
      "1.019856287",
      "0.502066089"
    ],
    [
      "0.863748573",
      "0.523965173"
    ],
    [
      "1.599129045",
      "0.525275692"
    ],
    [
      "1.800174738",
      "0.543604961"
    ],
    [
      "0.705365907",
      "0.547156488"
    ],
    [
      "0.18916651",
      "0.562821437"
    ],
    [
      "0.549194861",
      "0.568599266"
    ],
    [
      "1.448542183",
      "0.571888836"
    ],
    [
      "0.345706547",
      "0.581379601"
    ],
    [
      "1.272682846",
      "0.605150016"
    ],
    [
      "1.1165118",
      "0.626592793"
    ],
    [
      "0.960404086",
      "0.648491878"
    ],
    [
      "1.688658666",
      "0.655020193"
    ],
    [
      "0.80423304",
      "0.669934655"
    ],
    [
      "1.866272525",
      "0.686714258"
    ],
    [
      "0.645850374",
      "0.69312597"
    ],
    [
      "0.454987368",
      "0.694988097"
    ],
    [
      "1.538071803",
      "0.701633337"
    ],
    [
      "0.126501126",
      "0.707466651"
    ],
    [
      "1.381425672",
      "0.719273573"
    ],
    [
      "0.283041162",
      "0.726024815"
    ],
    [
      "1.213167313",
      "0.751119498"
    ],
    [
      "1.057059599",
      "0.773018583"
    ],
    [
      "0.900888553",
      "0.79446136"
    ],
    [
      "1.754756453",
      "0.79812949"
    ],
    [
      "0.744717507",
      "0.815904138"
    ],
    [
      "0.555588186",
      "0.822361901"
    ],
    [
      "1.90700291",
      "0.838997626"
    ],
    [
      "0.399235326",
      "0.842436048"
    ],
    [
      "1.60416959",
      "0.844742634"
    ],
    [
      "0.089414754",
      "0.860678235"
    ],
    [
      "1.447523459",
      "0.86238287"
    ],
    [
      "0.245954791",
      "0.879236399"
    ],
    [
      "1.291928527",
      "0.887669341"
    ],
    [
      "1.135820813",
      "0.909568425"
    ],
    [
      "0.979649767",
      "0.931011203"
    ],
    [
      "0.665842481",
      "0.952388271"
    ],
    [
      "0.823478721",
      "0.95245398"
    ],
    [
      "1.71906764",
      "0.954316418"
    ],
    [
      "0.509577294",
      "0.973133869"
    ],
    [
      "1.544755086",
      "0.990753268"
    ],
    [
      "0.353406248",
      "0.994576646"
    ],
    [
      "1.92117317",
      "0.99599569"
    ],
    [
      "1.38858404",
      "1.012196045"
    ],
    [
      "0.078993412",
      "1.017969633"
    ],
    [
      "1.232476326",
      "1.03409513"
    ],
    [
      "1.07630528",
      "1.055537907"
    ],
    [
      "0.920134234",
      "1.076980685"
    ],
    [
      "0.230521531",
      "1.093311314"
    ],
    [
      "1.797582803",
      "1.093845623"
    ],
    [
      "0.606326948",
      "1.098357753"
    ],
    [
      "0.763963188",
      "1.098423462"
    ],
    [
      "1.6414106",
      "1.115279972"
    ],
    [
      "0.450061761",
      "1.119103351"
    ],
    [
      "1.485239553",
      "1.13672275"
    ],
    [
      "1.329131839",
      "1.158621834"
    ],
    [
      "0.095542269",
      "1.17473482"
    ],
    [
      "1.172960793",
      "1.180064612"
    ],
    [
      "1.016789747",
      "1.201507389"
    ],
    [
      "1.895085705",
      "1.217709953"
    ],
    [
      "0.860618701",
      "1.222950167"
    ],
    [
      "1.739064597",
      "1.240217779"
    ],
    [
      "0.546811415",
      "1.244327235"
    ],
    [
      "0.704447655",
      "1.244392944"
    ],
    [
      "0.233546724",
      "1.250918537"
    ],
    [
      "1.581895067",
      "1.261249454"
    ],
    [
      "0.390546228",
      "1.265072833"
    ],
    [
      "1.425787352",
      "1.283148539"
    ],
    [
      "1.269616306",
      "1.304591316"
    ],
    [
      "1.11344526",
      "1.326034094"
    ],
    [
      "0.957274214",
      "1.347476872"
    ],
    [
      "0.801103168",
      "1.368919649"
    ],
    [
      "1.8418184",
      "1.371305843"
    ],
    [
      "1.67855058",
      "1.385776159"
    ],
    [
      "0.487295882",
      "1.390296717"
    ],
    [
      "0.644932122",
      "1.390362427"
    ],
    [
      "0.167312995",
      "1.393964968"
    ],
    [
      "1.522442865",
      "1.407675244"
    ],
    [
      "0.324312499",
      "1.408119264"
    ],
    [
      "1.366271819",
      "1.429118021"
    ],
    [
      "1.210100773",
      "1.450560799"
    ],
    [
      "1.053929727",
      "1.472003576"
    ],
    [
      "0.897758681",
      "1.493446354"
    ],
    [
      "0.741587635",
      "1.514889131"
    ],
    [
      "1.759782725",
      "1.52087067"
    ],
    [
      "0.420440472",
      "1.533053652"
    ],
    [
      "0.583923661",
      "1.537422198"
    ],
    [
      "1.462927332",
      "1.553644726"
    ],
    [
      "0.264240871",
      "1.554287423"
    ],
    [
      "1.306756286",
      "1.575087503"
    ],
    [
      "1.616896354",
      "1.587448996"
    ],
    [
      "1.15058524",
      "1.596530281"
    ],
    [
      "0.994414194",
      "1.617973058"
    ],
    [
      "0.838243148",
      "1.639415836"
    ],
    [
      "0.682072102",
      "1.660858613"
    ],
    [
      "0.386245636",
      "1.686936405"
    ],
    [
      "1.403411799",
      "1.699614208"
    ],
    [
      "1.247240753",
      "1.721056985"
    ],
    [
      "0.538806139",
      "1.726616139"
    ],
    [
      "1.557380821",
      "1.733418478"
    ],
    [
      "0.941670421",
      "1.766523665"
    ],
    [
      "1.099084105",
      "1.774897454"
    ],
    [
      "0.785499375",
      "1.787966442"
    ],
    [
      "0.639650079",
      "1.847775901"
    ],
    [
      "1.346395623",
      "1.853241451"
    ],
    [
      "1.195563235",
      "1.900003153"
    ],
    [
      "0.880907786",
      "1.913451196"
    ],
    [
      "1.038391624",
      "1.920381512"
    ]
  ],
  "radius": "0.078818126"
}
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": 132
}

起步布局答案

{
  "centers": [
    [
      "1.022519482",
      "0.079093427"
    ],
    [
      "0.865178518",
      "0.088737572"
    ],
    [
      "1.179200998",
      "0.096416553"
    ],
    [
      "0.711785586",
      "0.125066577"
    ],
    [
      "1.330634901",
      "0.140199671"
    ],
    [
      "1.472386691",
      "0.209160663"
    ],
    [
      "0.525135882",
      "0.210645762"
    ],
    [
      "0.979732712",
      "0.230811852"
    ],
    [
      "0.822174369",
      "0.241833872"
    ],
    [
      "1.196936764",
      "0.253051897"
    ],
    [
      "0.668315682",
      "0.276590687"
    ],
    [
      "1.600305395",
      "0.301280116"
    ],
    [
      "0.397506656",
      "0.303165863"
    ],
    [
      "1.346224548",
      "0.303671518"
    ],
    [
      "1.076751733",
      "0.355055556"
    ],
    [
      "0.920644018",
      "0.376954641"
    ],
    [
      "0.76488144",
      "0.401187006"
    ],
    [
      "1.486737868",
      "0.411644147"
    ],
    [
      "1.710645118",
      "0.41386046"
    ],
    [
      "0.59661095",
      "0.418014116"
    ],
    [
      "0.279511125",
      "0.427069689"
    ],
    [
      "0.439914039",
      "0.45499077"
    ],
    [
      "1.336495483",
      "0.461007254"
    ],
    [
      "1.177074194",
      "0.479636835"
    ],
    [
      "1.019856287",
      "0.502066089"
    ],
    [
      "0.863748573",
      "0.523965173"
    ],
    [
      "1.599129045",
      "0.525275692"
    ],
    [
      "1.800174738",
      "0.543604961"
    ],
    [
      "0.705365907",
      "0.547156488"
    ],
    [
      "0.18916651",
      "0.562821437"
    ],
    [
      "0.549194861",
      "0.568599266"
    ],
    [
      "1.448542183",
      "0.571888836"
    ],
    [
      "0.345706547",
      "0.581379601"
    ],
    [
      "1.272682846",
      "0.605150016"
    ],
    [
      "1.1165118",
      "0.626592793"
    ],
    [
      "0.960404086",
      "0.648491878"
    ],
    [
      "1.688658666",
      "0.655020193"
    ],
    [
      "0.80423304",
      "0.669934655"
    ],
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      "1.866272525",
      "0.686714258"
    ],
    [
      "0.645850374",
      "0.69312597"
    ],
    [
      "0.454987368",
      "0.694988097"
    ],
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      "1.538071803",
      "0.701633337"
    ],
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      "0.126501126",
      "0.707466651"
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      "1.381425672",
      "0.719273573"
    ],
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      "0.283041162",
      "0.726024815"
    ],
    [
      "1.213167313",
      "0.751119498"
    ],
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      "1.057059599",
      "0.773018583"
    ],
    [
      "0.900888553",
      "0.79446136"
    ],
    [
      "1.754756453",
      "0.79812949"
    ],
    [
      "0.744717507",
      "0.815904138"
    ],
    [
      "0.555588186",
      "0.822361901"
    ],
    [
      "1.90700291",
      "0.838997626"
    ],
    [
      "0.399235326",
      "0.842436048"
    ],
    [
      "1.60416959",
      "0.844742634"
    ],
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      "0.089414754",
      "0.860678235"
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      "0.86238287"
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      "0.879236399"
    ],
    [
      "1.291928527",
      "0.887669341"
    ],
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      "0.909568425"
    ],
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      "0.931011203"
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      "0.952388271"
    ],
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      "0.95245398"
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      "0.954316418"
    ],
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      "0.973133869"
    ],
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      "0.990753268"
    ],
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      "0.994576646"
    ],
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      "0.99599569"
    ],
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      "1.012196045"
    ],
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      "1.017969633"
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      "1.232476326",
      "1.03409513"
    ],
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      "1.055537907"
    ],
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      "1.076980685"
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      "1.093845623"
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      "1.098357753"
    ],
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      "1.119103351"
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      "1.13672275"
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      "1.158621834"
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      "1.17473482"
    ],
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      "1.172960793",
      "1.180064612"
    ],
    [
      "1.016789747",
      "1.201507389"
    ],
    [
      "1.895085705",
      "1.217709953"
    ],
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      "0.860618701",
      "1.222950167"
    ],
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      "1.739064597",
      "1.240217779"
    ],
    [
      "0.546811415",
      "1.244327235"
    ],
    [
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      "1.244392944"
    ],
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      "1.250918537"
    ],
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      "1.581895067",
      "1.261249454"
    ],
    [
      "0.390546228",
      "1.265072833"
    ],
    [
      "1.425787352",
      "1.283148539"
    ],
    [
      "1.269616306",
      "1.304591316"
    ],
    [
      "1.11344526",
      "1.326034094"
    ],
    [
      "0.957274214",
      "1.347476872"
    ],
    [
      "0.801103168",
      "1.368919649"
    ],
    [
      "1.8418184",
      "1.371305843"
    ],
    [
      "1.67855058",
      "1.385776159"
    ],
    [
      "0.487295882",
      "1.390296717"
    ],
    [
      "0.644932122",
      "1.390362427"
    ],
    [
      "0.167312995",
      "1.393964968"
    ],
    [
      "1.522442865",
      "1.407675244"
    ],
    [
      "0.324312499",
      "1.408119264"
    ],
    [
      "1.366271819",
      "1.429118021"
    ],
    [
      "1.210100773",
      "1.450560799"
    ],
    [
      "1.053929727",
      "1.472003576"
    ],
    [
      "0.897758681",
      "1.493446354"
    ],
    [
      "0.741587635",
      "1.514889131"
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    [
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      "0.880907786",
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    [
      "1.038391624",
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  ],
  "radius": "0.078818126"
}

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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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?