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 = 193

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 = 193
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[0.0657921811390669068000205567405, 0.0657921811390669068000205567415]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 = 0.065792180

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 = 193 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": [
    [
      "0.994368975",
      "0.065809551"
    ],
    [
      "1.125892275",
      "0.074313561"
    ],
    [
      "0.86310449",
      "0.075876703"
    ],
    [
      "1.254703857",
      "0.101184004"
    ],
    [
      "0.734621679",
      "0.104277612"
    ],
    [
      "1.448601436",
      "0.180548354"
    ],
    [
      "0.542273743",
      "0.185610092"
    ],
    [
      "1.061097873",
      "0.188839234"
    ],
    [
      "0.929515037",
      "0.189472846"
    ],
    [
      "1.321488799",
      "0.214560439"
    ],
    [
      "1.189909455",
      "0.215709678"
    ],
    [
      "0.669631546",
      "0.218692328"
    ],
    [
      "0.801215909",
      "0.218692328"
    ],
    [
      "1.514647696",
      "0.294356684"
    ],
    [
      "0.47696665",
      "0.299844185"
    ],
    [
      "0.995678792",
      "0.303212909"
    ],
    [
      "1.643193629",
      "0.322470514"
    ],
    [
      "1.387535059",
      "0.328368769"
    ],
    [
      "1.255955716",
      "0.329518008"
    ],
    [
      "0.348899727",
      "0.330065165"
    ],
    [
      "1.124376017",
      "0.330625838"
    ],
    [
      "0.604324453",
      "0.332926421"
    ],
    [
      "0.735908816",
      "0.332926421"
    ],
    [
      "0.867493178",
      "0.332926421"
    ],
    [
      "1.732007384",
      "0.420876838"
    ],
    [
      "0.261231463",
      "0.428191206"
    ],
    [
      "1.458036584",
      "0.439472231"
    ],
    [
      "1.589582848",
      "0.442638432"
    ],
    [
      "1.326567401",
      "0.444976612"
    ],
    [
      "0.537082509",
      "0.446032412"
    ],
    [
      "1.194988057",
      "0.446125851"
    ],
    [
      "1.063408359",
      "0.447233681"
    ],
    [
      "0.668656144",
      "0.447712648"
    ],
    [
      "0.800240506",
      "0.447712648"
    ],
    [
      "0.931824868",
      "0.447712648"
    ],
    [
      "0.405528122",
      "0.448840875"
    ],
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      "1.806298784",
      "0.530798539"
    ],
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      "0.188219688",
      "0.537661408"
    ],
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      "1.656509842",
      "0.555931071"
    ],
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      "1.524932042",
      "0.557245228"
    ],
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      "1.393352343",
      "0.558353046"
    ],
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      "1.261772999",
      "0.559502285"
    ],
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      "1.130193656",
      "0.560651524"
    ],
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      "0.998613957",
      "0.561759354"
    ],
    [
      "0.47227738",
      "0.562238322"
    ],
    [
      "0.603861742",
      "0.562238322"
    ],
    [
      "0.735446104",
      "0.562238322"
    ],
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      "0.867030467",
      "0.562238322"
    ],
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      "0.340697681",
      "0.563346152"
    ],
    [
      "1.748045279",
      "0.650459945"
    ],
    [
      "0.246772101",
      "0.655500532"
    ],
    [
      "1.460101591",
      "0.671750499"
    ],
    [
      "1.328522248",
      "0.672899738"
    ],
    [
      "1.597196194",
      "0.673718805"
    ],
    [
      "1.19694309",
      "0.674070088"
    ],
    [
      "1.065363215",
      "0.675156801"
    ],
    [
      "0.933783517",
      "0.676264631"
    ],
    [
      "0.407446939",
      "0.676743598"
    ],
    [
      "0.539031302",
      "0.676743598"
    ],
    [
      "0.670615664",
      "0.676743598"
    ],
    [
      "0.802200026",
      "0.676743598"
    ],
    [
      "1.876805244",
      "0.677576655"
    ],
    [
      "0.119334633",
      "0.688274545"
    ],
    [
      "1.788941502",
      "0.7755277"
    ],
    [
      "0.211436495",
      "0.782251624"
    ],
    [
      "1.526755094",
      "0.785204256"
    ],
    [
      "1.263746222",
      "0.787435806"
    ],
    [
      "1.000625565",
      "0.789714564"
    ],
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      "1.658227318",
      "0.790635531"
    ],
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      "1.395288395",
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      "1.132205309",
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    ],
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    ],
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    ],
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      "0.540085833",
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    ],
    [
      "0.101860198",
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    [
      "1.000630229",
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    [
      "1.132214591",
      "1.018733195"
    ],
    [
      "1.263798954",
      "1.018733195"
    ],
    [
      "1.395383316",
      "1.018733195"
    ],
    [
      "1.526967678",
      "1.018733195"
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    [
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      "0.474293652",
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    [
      "0.605878014",
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    ],
    [
      "0.737462376",
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      "0.869046738",
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      "1.329591139",
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      "1.461179821",
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      "1.361346097"
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      "0.40629617",
      "1.362380372"
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    [
      "1.731362558",
      "1.366430277"
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    [
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      "0.408526392",
      "1.591550015"
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    [
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      "1.000629961",
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      "1.70339764"
    ],
    [
      "1.593367393",
      "1.721567312"
    ],
    [
      "0.408543389",
      "1.723134376"
    ],
    [
      "1.486100138",
      "1.797778731"
    ],
    [
      "0.516011865",
      "1.799061783"
    ],
    [
      "0.93483778",
      "1.816900624"
    ],
    [
      "1.066422142",
      "1.816900624"
    ],
    [
      "0.761966046",
      "1.832614777"
    ],
    [
      "1.24006743",
      "1.832824487"
    ],
    [
      "1.369189103",
      "1.858162953"
    ],
    [
      "0.633082221",
      "1.859136539"
    ],
    [
      "1.144244817",
      "1.923004703"
    ],
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      "0.85735523",
      "1.923253334"
    ],
    [
      "1.000629961",
      "1.93282979"
    ]
  ],
  "radius": "0.06579218"
}
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": 193
}

起步布局答案

{
  "centers": [
    [
      "0.994368975",
      "0.065809551"
    ],
    [
      "1.125892275",
      "0.074313561"
    ],
    [
      "0.86310449",
      "0.075876703"
    ],
    [
      "1.254703857",
      "0.101184004"
    ],
    [
      "0.734621679",
      "0.104277612"
    ],
    [
      "1.448601436",
      "0.180548354"
    ],
    [
      "0.542273743",
      "0.185610092"
    ],
    [
      "1.061097873",
      "0.188839234"
    ],
    [
      "0.929515037",
      "0.189472846"
    ],
    [
      "1.321488799",
      "0.214560439"
    ],
    [
      "1.189909455",
      "0.215709678"
    ],
    [
      "0.669631546",
      "0.218692328"
    ],
    [
      "0.801215909",
      "0.218692328"
    ],
    [
      "1.514647696",
      "0.294356684"
    ],
    [
      "0.47696665",
      "0.299844185"
    ],
    [
      "0.995678792",
      "0.303212909"
    ],
    [
      "1.643193629",
      "0.322470514"
    ],
    [
      "1.387535059",
      "0.328368769"
    ],
    [
      "1.255955716",
      "0.329518008"
    ],
    [
      "0.348899727",
      "0.330065165"
    ],
    [
      "1.124376017",
      "0.330625838"
    ],
    [
      "0.604324453",
      "0.332926421"
    ],
    [
      "0.735908816",
      "0.332926421"
    ],
    [
      "0.867493178",
      "0.332926421"
    ],
    [
      "1.732007384",
      "0.420876838"
    ],
    [
      "0.261231463",
      "0.428191206"
    ],
    [
      "1.458036584",
      "0.439472231"
    ],
    [
      "1.589582848",
      "0.442638432"
    ],
    [
      "1.326567401",
      "0.444976612"
    ],
    [
      "0.537082509",
      "0.446032412"
    ],
    [
      "1.194988057",
      "0.446125851"
    ],
    [
      "1.063408359",
      "0.447233681"
    ],
    [
      "0.668656144",
      "0.447712648"
    ],
    [
      "0.800240506",
      "0.447712648"
    ],
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      "0.931824868",
      "0.447712648"
    ],
    [
      "0.405528122",
      "0.448840875"
    ],
    [
      "1.806298784",
      "0.530798539"
    ],
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      "0.188219688",
      "0.537661408"
    ],
    [
      "1.656509842",
      "0.555931071"
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      "1.524932042",
      "0.557245228"
    ],
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      "1.393352343",
      "0.558353046"
    ],
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      "1.261772999",
      "0.559502285"
    ],
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      "1.130193656",
      "0.560651524"
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      "0.561759354"
    ],
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      "0.562238322"
    ],
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      "0.603861742",
      "0.562238322"
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  "radius": "0.06579218"
}

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。

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