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 = 174

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 = 174
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[0.0689995466815819658768691394285, 0.0689995466815819658768691394295]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 = 0.068999546

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 = 174 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": [
    [
      "0.912041747",
      "0.073163882"
    ],
    [
      "1.087958253",
      "0.073163882"
    ],
    [
      "1.224343288",
      "0.096433769"
    ],
    [
      "0.775656712",
      "0.096433769"
    ],
    [
      "1",
      "0.179498709"
    ],
    [
      "0.537893662",
      "0.191779995"
    ],
    [
      "0.675892756",
      "0.191779995"
    ],
    [
      "1.324107244",
      "0.191779995"
    ],
    [
      "1.462106338",
      "0.191779995"
    ],
    [
      "0.863932212",
      "0.202505377"
    ],
    [
      "1.136067788",
      "0.202505377"
    ],
    [
      "0.373486263",
      "0.311347416"
    ],
    [
      "1.626513737",
      "0.311347416"
    ],
    [
      "1.212966903",
      "0.318947649"
    ],
    [
      "0.788357938",
      "0.320016501"
    ],
    [
      "0.931000453",
      "0.323110521"
    ],
    [
      "1.068999547",
      "0.323110521"
    ],
    [
      "0.510588969",
      "0.327050847"
    ],
    [
      "0.648588063",
      "0.327050847"
    ],
    [
      "1.351411937",
      "0.327050847"
    ],
    [
      "1.489411031",
      "0.327050847"
    ],
    [
      "0.278572927",
      "0.411523303"
    ],
    [
      "1.721427073",
      "0.411523303"
    ],
    [
      "0.724001813",
      "0.442621241"
    ],
    [
      "0.862000907",
      "0.442621241"
    ],
    [
      "1",
      "0.442621241"
    ],
    [
      "1.137999093",
      "0.442621241"
    ],
    [
      "1.275998187",
      "0.442621241"
    ],
    [
      "0.441589423",
      "0.446561568"
    ],
    [
      "0.579588516",
      "0.446561568"
    ],
    [
      "1.420411484",
      "0.446561568"
    ],
    [
      "1.558410577",
      "0.446561568"
    ],
    [
      "0.195311645",
      "0.532963314"
    ],
    [
      "1.805293697",
      "0.534006672"
    ],
    [
      "1.663027517",
      "0.536556266"
    ],
    [
      "0.336972483",
      "0.536556266"
    ],
    [
      "0.655002267",
      "0.562131962"
    ],
    [
      "0.79300136",
      "0.562131962"
    ],
    [
      "0.931000453",
      "0.562131962"
    ],
    [
      "1.068999547",
      "0.562131962"
    ],
    [
      "1.20699864",
      "0.562131962"
    ],
    [
      "1.344997733",
      "0.562131962"
    ],
    [
      "0.510588969",
      "0.566072288"
    ],
    [
      "1.489411031",
      "0.566072288"
    ],
    [
      "0.267950359",
      "0.656053948"
    ],
    [
      "1.732049641",
      "0.656053948"
    ],
    [
      "0.405972029",
      "0.656066986"
    ],
    [
      "1.594027971",
      "0.656066986"
    ],
    [
      "0.130421847",
      "0.667440711"
    ],
    [
      "1.869578153",
      "0.667440711"
    ],
    [
      "0.724001813",
      "0.681642683"
    ],
    [
      "0.862000907",
      "0.681642683"
    ],
    [
      "1",
      "0.681642683"
    ],
    [
      "0.58600272",
      "0.681642683"
    ],
    [
      "1.137999093",
      "0.681642683"
    ],
    [
      "1.275998187",
      "0.681642683"
    ],
    [
      "1.41399728",
      "0.681642683"
    ],
    [
      "0.24134496",
      "0.791464076"
    ],
    [
      "1.75865504",
      "0.791464076"
    ],
    [
      "0.379344053",
      "0.791472675"
    ],
    [
      "1.620655947",
      "0.791472675"
    ],
    [
      "1.909183926",
      "0.799634253"
    ],
    [
      "0.090816074",
      "0.799634253"
    ],
    [
      "0.517003173",
      "0.801153403"
    ],
    [
      "0.655002267",
      "0.801153403"
    ],
    [
      "0.79300136",
      "0.801153403"
    ],
    [
      "0.931000453",
      "0.801153403"
    ],
    [
      "1.068999547",
      "0.801153403"
    ],
    [
      "1.20699864",
      "0.801153403"
    ],
    [
      "1.344997733",
      "0.801153403"
    ],
    [
      "1.482996827",
      "0.801153403"
    ],
    [
      "0.172345414",
      "0.910974796"
    ],
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      "1.827654586",
      "0.910974796"
    ],
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      "0.310344507",
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    ],
    [
      "1.689655493",
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    ],
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      "0.448003627",
      "0.920664124"
    ],
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      "0.58600272",
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    [
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    ],
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      "1.030494116"
    ],
    [
      "1.75865504",
      "1.030494116"
    ],
    [
      "0.37900408",
      "1.040174844"
    ],
    [
      "0.517003173",
      "1.040174844"
    ],
    [
      "0.655002267",
      "1.040174844"
    ],
    [
      "0.79300136",
      "1.040174844"
    ],
    [
      "0.931000453",
      "1.040174844"
    ],
    [
      "1.068999547",
      "1.040174844"
    ],
    [
      "1.20699864",
      "1.040174844"
    ],
    [
      "1.344997733",
      "1.040174844"
    ],
    [
      "1.482996827",
      "1.040174844"
    ],
    [
      "1.62099592",
      "1.040174844"
    ],
    [
      "1.862002773",
      "1.121943534"
    ],
    [
      "0.137997227",
      "1.121943534"
    ],
    [
      "1.698209236",
      "1.154550775"
    ],
    [
      "0.301790764",
      "1.154550775"
    ],
    [
      "0.448003627",
      "1.159685565"
    ],
    [
      "0.58600272",
      "1.159685565"
    ],
    [
      "0.724001813",
      "1.159685565"
    ],
    [
      "0.862000907",
      "1.159685565"
    ],
    [
      "1",
      "1.159685565"
    ],
    [
      "1.137999093",
      "1.159685565"
    ],
    [
      "1.275998187",
      "1.159685565"
    ],
    [
      "1.41399728",
      "1.159685565"
    ],
    [
      "1.551996373",
      "1.159685565"
    ],
    [
      "0.104860687",
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    ],
    [
      "1.895139313",
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    ],
    [
      "1.758992953",
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    ],
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      "0.241007047",
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    ],
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      "1.518217726"
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      "0.79300136",
      "1.518217726"
    ],
    [
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      "1.518217726"
    ],
    [
      "1.483489908",
      "1.525724732"
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    [
      "0.515178331",
      "1.527635291"
    ],
    [
      "1.761402438",
      "1.535750101"
    ],
    [
      "0.238597562",
      "1.535750101"
    ],
    [
      "0.593376591",
      "1.641692525"
    ],
    [
      "0.731375685",
      "1.641692525"
    ],
    [
      "0.869374778",
      "1.641692525"
    ],
    [
      "1.130625222",
      "1.641692525"
    ],
    [
      "1.268624315",
      "1.641692525"
    ],
    [
      "1.406623409",
      "1.641692525"
    ],
    [
      "0.329845894",
      "1.646262577"
    ],
    [
      "1.670154106",
      "1.646262577"
    ],
    [
      "0.462321174",
      "1.68491552"
    ],
    [
      "1.537678826",
      "1.68491552"
    ],
    [
      "1",
      "1.686198718"
    ],
    [
      "0.662376138",
      "1.761203246"
    ],
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      "0.800375231",
      "1.761203246"
    ],
    [
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      "1.761203246"
    ],
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      "1.337623862",
      "1.761203246"
    ],
    [
      "0.531320721",
      "1.804426241"
    ],
    [
      "1.468679279",
      "1.804426241"
    ],
    [
      "0.931000453",
      "1.805709438"
    ],
    [
      "1.068999547",
      "1.805709438"
    ],
    [
      "1.281695516",
      "1.887360964"
    ],
    [
      "0.718304484",
      "1.887360964"
    ],
    [
      "0.852567896",
      "1.91925275"
    ],
    [
      "1.147432104",
      "1.91925275"
    ],
    [
      "1.000442989",
      "1.93036464"
    ]
  ],
  "radius": "0.068999546"
}
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": 174
}

起步布局答案

{
  "centers": [
    [
      "0.912041747",
      "0.073163882"
    ],
    [
      "1.087958253",
      "0.073163882"
    ],
    [
      "1.224343288",
      "0.096433769"
    ],
    [
      "0.775656712",
      "0.096433769"
    ],
    [
      "1",
      "0.179498709"
    ],
    [
      "0.537893662",
      "0.191779995"
    ],
    [
      "0.675892756",
      "0.191779995"
    ],
    [
      "1.324107244",
      "0.191779995"
    ],
    [
      "1.462106338",
      "0.191779995"
    ],
    [
      "0.863932212",
      "0.202505377"
    ],
    [
      "1.136067788",
      "0.202505377"
    ],
    [
      "0.373486263",
      "0.311347416"
    ],
    [
      "1.626513737",
      "0.311347416"
    ],
    [
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      "0.318947649"
    ],
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      "0.320016501"
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      "0.323110521"
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      "0.327050847"
    ],
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    ],
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    ],
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    ],
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      "0.446561568"
    ],
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    ],
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    ],
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      "0.566072288"
    ],
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      "0.656053948"
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      "0.681642683"
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      "0.862000907",
      "0.681642683"
    ],
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      "1",
      "0.681642683"
    ],
    [
      "0.58600272",
      "0.681642683"
    ],
    [
      "1.137999093",
      "0.681642683"
    ],
    [
      "1.275998187",
      "0.681642683"
    ],
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      "0.681642683"
    ],
    [
      "0.24134496",
      "0.791464076"
    ],
    [
      "1.75865504",
      "0.791464076"
    ],
    [
      "0.379344053",
      "0.791472675"
    ],
    [
      "1.620655947",
      "0.791472675"
    ],
    [
      "1.909183926",
      "0.799634253"
    ],
    [
      "0.090816074",
      "0.799634253"
    ],
    [
      "0.517003173",
      "0.801153403"
    ],
    [
      "0.655002267",
      "0.801153403"
    ],
    [
      "0.79300136",
      "0.801153403"
    ],
    [
      "0.931000453",
      "0.801153403"
    ],
    [
      "1.068999547",
      "0.801153403"
    ],
    [
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  "radius": "0.068999546"
}

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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