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 = 143

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 = 143
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[0.0757724959074619115989327676275, 0.0757724959074619115989327676285]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 = 0.075772495

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 = 143 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": [
    [
      "1.074341975",
      "0.078767266"
    ],
    [
      "0.910307291",
      "0.080161222"
    ],
    [
      "1.22388803",
      "0.103300145"
    ],
    [
      "0.761165738",
      "0.107164811"
    ],
    [
      "1.367414641",
      "0.151941653"
    ],
    [
      "0.618471602",
      "0.15819684"
    ],
    [
      "0.998212271",
      "0.209802195"
    ],
    [
      "1.501062961",
      "0.223384015"
    ],
    [
      "0.8481901",
      "0.23123191"
    ],
    [
      "0.486035219",
      "0.231861547"
    ],
    [
      "1.25674041",
      "0.255465075"
    ],
    [
      "0.705495963",
      "0.282263939"
    ],
    [
      "1.114303879",
      "0.307211759"
    ],
    [
      "1.621239727",
      "0.315706437"
    ],
    [
      "1.39038873",
      "0.326907438"
    ],
    [
      "0.57305958",
      "0.355928646"
    ],
    [
      "0.971898919",
      "0.359045259"
    ],
    [
      "0.42210214",
      "0.369260404"
    ],
    [
      "0.821876747",
      "0.380474974"
    ],
    [
      "1.23030268",
      "0.404731822"
    ],
    [
      "1.510565496",
      "0.419229859"
    ],
    [
      "0.279647887",
      "0.42095828"
    ],
    [
      "1.724713872",
      "0.426426741"
    ],
    [
      "0.689058381",
      "0.453448709"
    ],
    [
      "1.087897719",
      "0.456565322"
    ],
    [
      "1.363973292",
      "0.476132466"
    ],
    [
      "0.546604128",
      "0.505146586"
    ],
    [
      "0.938805684",
      "0.506932792"
    ],
    [
      "0.395646688",
      "0.518478344"
    ],
    [
      "0.194706087",
      "0.546460373"
    ],
    [
      "1.634336648",
      "0.548108498"
    ],
    [
      "1.808703395",
      "0.552568108"
    ],
    [
      "1.20389652",
      "0.554085386"
    ],
    [
      "0.796846192",
      "0.559974163"
    ],
    [
      "1.484150058",
      "0.568454888"
    ],
    [
      "1.054804485",
      "0.604452855"
    ],
    [
      "0.654391939",
      "0.61167204"
    ],
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      "1.337629007",
      "0.625370073"
    ],
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      "0.310704888",
      "0.643980436"
    ],
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      "0.508244285",
      "0.651756295"
    ],
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      "0.912844993",
      "0.657494227"
    ],
    [
      "1.721175531",
      "0.676280507"
    ],
    [
      "0.13141538",
      "0.684156309"
    ],
    [
      "1.570550902",
      "0.692957011"
    ],
    [
      "1.871784408",
      "0.693098672"
    ],
    [
      "1.170803286",
      "0.701972919"
    ],
    [
      "0.77039074",
      "0.709192103"
    ],
    [
      "1.428761579",
      "0.746451611"
    ],
    [
      "1.028843793",
      "0.75501429"
    ],
    [
      "0.627936487",
      "0.76088998"
    ],
    [
      "0.404956356",
      "0.762650337"
    ],
    [
      "0.247414181",
      "0.781676372"
    ],
    [
      "1.286802086",
      "0.799492982"
    ],
    [
      "0.886389541",
      "0.806712167"
    ],
    [
      "1.759609163",
      "0.82287089"
    ],
    [
      "0.090762866",
      "0.834358678"
    ],
    [
      "1.91021804",
      "0.839689056"
    ],
    [
      "1.545983923",
      "0.842497467"
    ],
    [
      "1.144842594",
      "0.852534354"
    ],
    [
      "0.743935288",
      "0.858410043"
    ],
    [
      "0.517256711",
      "0.864407474"
    ],
    [
      "1.403717008",
      "0.895912823"
    ],
    [
      "1.002388341",
      "0.90423223"
    ],
    [
      "0.367202803",
      "0.909417342"
    ],
    [
      "0.217083415",
      "0.930155089"
    ],
    [
      "1.662590798",
      "0.939289623"
    ],
    [
      "1.261757515",
      "0.948954195"
    ],
    [
      "0.859934089",
      "0.955930107"
    ],
    [
      "0.633255511",
      "0.961927537"
    ],
    [
      "0.075890912",
      "0.985205626"
    ],
    [
      "1.924179594",
      "0.990589551"
    ],
    [
      "1.520631929",
      "0.992332664"
    ],
    [
      "1.119798023",
      "1.001995566"
    ],
    [
      "0.490801259",
      "1.013625414"
    ],
    [
      "1.779506853",
      "1.035708088"
    ],
    [
      "1.378672437",
      "1.045374036"
    ],
    [
      "0.97734377",
      "1.053693443"
    ],
    [
      "0.749254312",
      "1.059447601"
    ],
    [
      "0.346374145",
      "1.059524142"
    ],
    [
      "0.194162328",
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    ],
    [
      "1.637547985",
      "1.08875113"
    ],
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      "1.236712945",
      "1.098415407"
    ],
    [
      "0.606800059",
      "1.111145477"
    ],
    [
      "1.495589116",
      "1.141794171"
    ],
    [
      "1.913279546",
      "1.141834234"
    ],
    [
      "1.094753452",
      "1.151456779"
    ],
    [
      "0.865814868",
      "1.15629553"
    ],
    [
      "0.462449711",
      "1.162531536"
    ],
    [
      "1.768577423",
      "1.186858451"
    ],
    [
      "1.353629624",
      "1.194835543"
    ],
    [
      "0.096816389",
      "1.196101616"
    ],
    [
      "0.287650424",
      "1.199228863"
    ],
    [
      "0.723665424",
      "1.208825715"
    ],
    [
      "1.612505172",
      "1.238212637"
    ],
    [
      "1.211670132",
      "1.247876914"
    ],
    [
      "0.982382064",
      "1.253135467"
    ],
    [
      "0.581207639",
      "1.260513856"
    ],
    [
      "1.470546303",
      "1.291255678"
    ],
    [
      "1.876457011",
      "1.29329096"
    ],
    [
      "0.840232621",
      "1.305665652"
    ],
    [
      "0.439058196",
      "1.313044041"
    ],
    [
      "1.73034396",
      "1.333501169"
    ],
    [
      "0.141004275",
      "1.341061319"
    ],
    [
      "1.328586811",
      "1.34429705"
    ],
    [
      "1.099297696",
      "1.349554445"
    ],
    [
      "0.292267024",
      "1.35070352"
    ],
    [
      "0.698083177",
      "1.358195836"
    ],
    [
      "1.587462359",
      "1.387674144"
    ],
    [
      "0.957338595",
      "1.402596864"
    ],
    [
      "0.555625392",
      "1.409883977"
    ],
    [
      "1.445092572",
      "1.443504951"
    ],
    [
      "1.216213329",
      "1.445973424"
    ],
    [
      "0.815189152",
      "1.455127049"
    ],
    [
      "0.395811155",
      "1.469801506"
    ],
    [
      "1.795649887",
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      "0.208287077",
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      "1.567319613",
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    ],
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    ],
    [
      "0.932368674",
      "1.553119103"
    ],
    [
      "0.530581923",
      "1.559345374"
    ],
    [
      "1.708106501",
      "1.59395426"
    ],
    [
      "1.192887374",
      "1.595712478"
    ],
    [
      "0.296855834",
      "1.599820608"
    ],
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      "0.789606905",
      "1.60449717"
    ],
    [
      "1.457766012",
      "1.642583134"
    ],
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      "1.050928273",
      "1.648754897"
    ],
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      "0.634754925",
      "1.66961741"
    ],
    [
      "0.429815883",
      "1.672535876"
    ],
    [
      "1.600975895",
      "1.702122591"
    ],
    [
      "0.915819587",
      "1.717395464"
    ],
    [
      "1.335599635",
      "1.732256213"
    ],
    [
      "1.184777506",
      "1.747040316"
    ],
    [
      "0.768918364",
      "1.754623346"
    ],
    [
      "0.523305193",
      "1.791807136"
    ],
    [
      "1.474341988",
      "1.793218859"
    ],
    [
      "1.042818404",
      "1.800082735"
    ],
    [
      "0.65910843",
      "1.859063104"
    ],
    [
      "1.291204299",
      "1.877152516"
    ],
    [
      "0.804076861",
      "1.903222344"
    ],
    [
      "1.143947688",
      "1.912948817"
    ],
    [
      "0.954312872",
      "1.923097593"
    ]
  ],
  "radius": "0.075772495"
}
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": 143
}

起步布局答案

{
  "centers": [
    [
      "1.074341975",
      "0.078767266"
    ],
    [
      "0.910307291",
      "0.080161222"
    ],
    [
      "1.22388803",
      "0.103300145"
    ],
    [
      "0.761165738",
      "0.107164811"
    ],
    [
      "1.367414641",
      "0.151941653"
    ],
    [
      "0.618471602",
      "0.15819684"
    ],
    [
      "0.998212271",
      "0.209802195"
    ],
    [
      "1.501062961",
      "0.223384015"
    ],
    [
      "0.8481901",
      "0.23123191"
    ],
    [
      "0.486035219",
      "0.231861547"
    ],
    [
      "1.25674041",
      "0.255465075"
    ],
    [
      "0.705495963",
      "0.282263939"
    ],
    [
      "1.114303879",
      "0.307211759"
    ],
    [
      "1.621239727",
      "0.315706437"
    ],
    [
      "1.39038873",
      "0.326907438"
    ],
    [
      "0.57305958",
      "0.355928646"
    ],
    [
      "0.971898919",
      "0.359045259"
    ],
    [
      "0.42210214",
      "0.369260404"
    ],
    [
      "0.821876747",
      "0.380474974"
    ],
    [
      "1.23030268",
      "0.404731822"
    ],
    [
      "1.510565496",
      "0.419229859"
    ],
    [
      "0.279647887",
      "0.42095828"
    ],
    [
      "1.724713872",
      "0.426426741"
    ],
    [
      "0.689058381",
      "0.453448709"
    ],
    [
      "1.087897719",
      "0.456565322"
    ],
    [
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      "0.476132466"
    ],
    [
      "0.546604128",
      "0.505146586"
    ],
    [
      "0.938805684",
      "0.506932792"
    ],
    [
      "0.395646688",
      "0.518478344"
    ],
    [
      "0.194706087",
      "0.546460373"
    ],
    [
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      "0.548108498"
    ],
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      "0.552568108"
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      "0.559974163"
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      "0.568454888"
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      "0.604452855"
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      "0.61167204"
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      "0.625370073"
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      "1.759609163",
      "0.82287089"
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      "0.834358678"
    ],
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      "0.839689056"
    ],
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      "0.842497467"
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      "0.852534354"
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      "0.858410043"
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      "0.864407474"
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      "0.895912823"
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      "0.90423223"
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      "0.909417342"
    ],
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      "0.930155089"
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      "0.939289623"
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      "0.948954195"
    ],
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      "0.955930107"
    ],
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      "0.961927537"
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      "0.985205626"
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    [
      "1.924179594",
      "0.990589551"
    ],
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      "0.992332664"
    ],
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      "1.001995566"
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      "1.013625414"
    ],
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      "1.035708088"
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      "1.045374036"
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      "1.059524142"
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      "1.079956652"
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      "1.08875113"
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      "1.098415407"
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    [
      "0.606800059",
      "1.111145477"
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    [
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      "1.141794171"
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      "1.141834234"
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      "1.151456779"
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      "0.865814868",
      "1.15629553"
    ],
    [
      "0.462449711",
      "1.162531536"
    ],
    [
      "1.768577423",
      "1.186858451"
    ],
    [
      "1.353629624",
      "1.194835543"
    ],
    [
      "0.096816389",
      "1.196101616"
    ],
    [
      "0.287650424",
      "1.199228863"
    ],
    [
      "0.723665424",
      "1.208825715"
    ],
    [
      "1.612505172",
      "1.238212637"
    ],
    [
      "1.211670132",
      "1.247876914"
    ],
    [
      "0.982382064",
      "1.253135467"
    ],
    [
      "0.581207639",
      "1.260513856"
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  "radius": "0.075772495"
}

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?