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 = 138

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 = 138
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[0.0771527290782822552371317541205, 0.0771527290782822552371317541215]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 = 0.077152728

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 = 138 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": [
    [
      "1.076020559",
      "1.236311548"
    ],
    [
      "0.501045637",
      "0.655154654"
    ],
    [
      "0.690580525",
      "1.494478725"
    ],
    [
      "0.215171704",
      "1.134661306"
    ],
    [
      "1.708799959",
      "0.658150888"
    ],
    [
      "0.21500034",
      "0.514796312"
    ],
    [
      "0.339213096",
      "1.644210954"
    ],
    [
      "1.821366613",
      "1.4207184"
    ],
    [
      "1.495725981",
      "0.978858138"
    ],
    [
      "1.182890689",
      "0.842505711"
    ],
    [
      "1.378795836",
      "1.685422676"
    ],
    [
      "1.502296792",
      "1.592914902"
    ],
    [
      "0.606588423",
      "1.778012201"
    ],
    [
      "1.31120215",
      "0.756794898"
    ],
    [
      "1.879985116",
      "1.277980719"
    ],
    [
      "0.670436503",
      "1.186525942"
    ],
    [
      "0.798747964",
      "1.100815129"
    ],
    [
      "1.142392982",
      "0.226614004"
    ],
    [
      "0.510760075",
      "0.810387044"
    ],
    [
      "1.704324729",
      "0.40369994"
    ],
    [
      "1.862503263",
      "0.671769582"
    ],
    [
      "0.830371262",
      "0.248544975"
    ],
    [
      "0.079200175",
      "1.06107874"
    ],
    [
      "1.247250432",
      "1.766082338"
    ],
    [
      "1.332495106",
      "1.064668218"
    ],
    [
      "0.552197053",
      "1.426213146"
    ],
    [
      "1.382876451",
      "1.839674168"
    ],
    [
      "1.301130139",
      "0.602818507"
    ],
    [
      "0.957157879",
      "1.784815593"
    ],
    [
      "1.453092915",
      "0.830559129"
    ],
    [
      "0.493739966",
      "0.495834648"
    ],
    [
      "0.542222923",
      "0.198696541"
    ],
    [
      "0.540110925",
      "1.616077857"
    ],
    [
      "0.722315031",
      "1.880078487"
    ],
    [
      "1.152674656",
      "0.380576537"
    ],
    [
      "0.403741571",
      "1.203971177"
    ],
    [
      "1.243176247",
      "0.109768346"
    ],
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      "1.064799613",
      "1.082414619"
    ],
    [
      "1.044507217",
      "0.774240133"
    ],
    [
      "0.111320554",
      "1.248788924"
    ],
    [
      "0.410464563",
      "0.290003483"
    ],
    [
      "0.455788908",
      "1.745306228"
    ],
    [
      "0.378456543",
      "0.907619179"
    ],
    [
      "0.412468543",
      "1.35880461"
    ],
    [
      "1.612866326",
      "1.193547377"
    ],
    [
      "0.806637048",
      "1.750850116"
    ],
    [
      "0.081874609",
      "0.906764535"
    ],
    [
      "0.767740274",
      "0.637708976"
    ],
    [
      "0.270780219",
      "1.505910138"
    ],
    [
      "0.52198102",
      "0.964283973"
    ],
    [
      "1.4294416",
      "0.517107694"
    ],
    [
      "1.439513611",
      "0.671084085"
    ],
    [
      "0.937131436",
      "1.169080708"
    ],
    [
      "1.768760475",
      "0.959779392"
    ],
    [
      "0.269047793",
      "1.279255709"
    ],
    [
      "0.217491445",
      "0.980373285"
    ],
    [
      "0.237270916",
      "0.66748618"
    ],
    [
      "0.906123746",
      "0.705974554"
    ],
    [
      "0.649649198",
      "0.877617885"
    ],
    [
      "1.922463779",
      "0.973398085"
    ],
    [
      "0.532053031",
      "1.118260364"
    ],
    [
      "1.517431175",
      "1.76414139"
    ],
    [
      "1.09722904",
      "1.719375361"
    ],
    [
      "0.3654288",
      "0.581545903"
    ],
    [
      "0.78803267",
      "0.945883463"
    ],
    [
      "1.56224172",
      "0.268331573"
    ],
    [
      "0.872835862",
      "1.914043964"
    ],
    [
      "0.165196644",
      "1.393383327"
    ],
    [
      "0.961522755",
      "0.167246388"
    ],
    [
      "1.577759371",
      "0.739628116"
    ],
    [
      "0.926416141",
      "1.014149041"
    ],
    [
      "1.739784511",
      "1.551693722"
    ],
    [
      "1.026911923",
      "1.922454787"
    ],
    [
      "0.629356802",
      "0.569443398"
    ],
    [
      "1.914001176",
      "1.127471311"
    ],
    [
      "1.79383485",
      "0.529390483"
    ],
    [
      "0.705109497",
      "0.338654156"
    ],
    [
      "0.947203447",
      "1.323057099"
    ],
    [
      "1.432409907",
      "0.184939552"
    ],
    [
      "1.089834216",
      "0.081535576"
    ],
    [
      "1.293975923",
      "0.255472043"
    ],
    [
      "0.916195757",
      "0.859950946"
    ],
    [
      "0.957275458",
      "1.47703349"
    ],
    [
      "1.034435206",
      "0.620263741"
    ],
    [
      "1.362986853",
      "1.526560437"
    ],
    [
      "1.204183645",
      "1.150379031"
    ],
    [
      "0.639310998",
      "0.723659137"
    ],
    [
      "1.412753286",
      "0.353971346"
    ],
    [
      "1.471682708",
      "1.131278923"
    ],
    [
      "1.054579228",
      "0.928216524"
    ],
    [
      "1.620392437",
      "0.887927125"
    ],
    [
      "0.424902477",
      "1.513427051"
    ],
    [
      "0.757668263",
      "0.483732585"
    ],
    [
      "0.818891986",
      "1.408767912"
    ],
    [
      "1.352914842",
      "1.372584046"
    ],
    [
      "1.162746667",
      "0.534552929"
    ],
    [
      "0.660364492",
      "1.032549551"
    ],
    [
      "1.344402345",
      "1.218513569"
    ],
    [
      "0.551447022",
      "0.352726053"
    ],
    [
      "0.69470286",
      "0.175031308"
    ],
    [
      "0.350087224",
      "1.059294346"
    ],
    [
      "1.214521845",
      "1.304337779"
    ],
    [
      "1.492224781",
      "1.438938511"
    ],
    [
      "0.896051735",
      "0.551998163"
    ],
    [
      "0.82564912",
      "1.571285116"
    ],
    [
      "1.096186067",
      "1.544262924"
    ],
    [
      "1.726917039",
      "1.297482979"
    ],
    [
      "0.808819975",
      "1.254791521"
    ],
    [
      "1.488348453",
      "1.28468175"
    ],
    [
      "0.968517785",
      "1.630928858"
    ],
    [
      "0.69091044",
      "1.64878383"
    ],
    [
      "1.180235586",
      "1.905075808"
    ],
    [
      "1.637719887",
      "1.04125662"
    ],
    [
      "0.245860764",
      "0.828698119"
    ],
    [
      "0.542125042",
      "1.272236755"
    ],
    [
      "0.343158224",
      "0.428856035"
    ],
    [
      "1.759886198",
      "1.136157438"
    ],
    [
      "0.777812285",
      "0.791685367"
    ],
    [
      "0.110243928",
      "0.755089368"
    ],
    [
      "1.646209795",
      "1.428999195"
    ],
    [
      "1.224593856",
      "1.45831417"
    ],
    [
      "1.321274161",
      "0.910771289"
    ],
    [
      "1.551136757",
      "0.422236924"
    ],
    [
      "1.905136329",
      "0.820068591"
    ],
    [
      "0.825854354",
      "0.093732722"
    ],
    [
      "1.172818678",
      "0.68852932"
    ],
    [
      "1.637519665",
      "1.667244904"
    ],
    [
      "1.1929627",
      "0.996482102"
    ],
    [
      "1.751433025",
      "0.806449897"
    ],
    [
      "1.014081522",
      "0.312324817"
    ],
    [
      "0.372887752",
      "0.74109493"
    ],
    [
      "1.291058128",
      "0.448842116"
    ],
    [
      "1.573731421",
      "0.581282097"
    ],
    [
      "1.024363195",
      "0.46628735"
    ],
    [
      "0.882930028",
      "0.393623404"
    ],
    [
      "1.234687353",
      "1.612289154"
    ],
    [
      "0.680508514",
      "1.340502334"
    ],
    [
      "1.08609257",
      "1.39028794"
    ]
  ],
  "radius": "0.077152728"
}
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": 138
}

起步布局答案

{
  "centers": [
    [
      "1.076020559",
      "1.236311548"
    ],
    [
      "0.501045637",
      "0.655154654"
    ],
    [
      "0.690580525",
      "1.494478725"
    ],
    [
      "0.215171704",
      "1.134661306"
    ],
    [
      "1.708799959",
      "0.658150888"
    ],
    [
      "0.21500034",
      "0.514796312"
    ],
    [
      "0.339213096",
      "1.644210954"
    ],
    [
      "1.821366613",
      "1.4207184"
    ],
    [
      "1.495725981",
      "0.978858138"
    ],
    [
      "1.182890689",
      "0.842505711"
    ],
    [
      "1.378795836",
      "1.685422676"
    ],
    [
      "1.502296792",
      "1.592914902"
    ],
    [
      "0.606588423",
      "1.778012201"
    ],
    [
      "1.31120215",
      "0.756794898"
    ],
    [
      "1.879985116",
      "1.277980719"
    ],
    [
      "0.670436503",
      "1.186525942"
    ],
    [
      "0.798747964",
      "1.100815129"
    ],
    [
      "1.142392982",
      "0.226614004"
    ],
    [
      "0.510760075",
      "0.810387044"
    ],
    [
      "1.704324729",
      "0.40369994"
    ],
    [
      "1.862503263",
      "0.671769582"
    ],
    [
      "0.830371262",
      "0.248544975"
    ],
    [
      "0.079200175",
      "1.06107874"
    ],
    [
      "1.247250432",
      "1.766082338"
    ],
    [
      "1.332495106",
      "1.064668218"
    ],
    [
      "0.552197053",
      "1.426213146"
    ],
    [
      "1.382876451",
      "1.839674168"
    ],
    [
      "1.301130139",
      "0.602818507"
    ],
    [
      "0.957157879",
      "1.784815593"
    ],
    [
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      "0.830559129"
    ],
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      "0.495834648"
    ],
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      "0.542222923",
      "0.198696541"
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      "1.616077857"
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      "1.880078487"
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      "0.380576537"
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      "1.203971177"
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      "0.973398085"
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      "1.014149041"
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    ],
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      "1.922454787"
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      "0.569443398"
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      "1.127471311"
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      "0.529390483"
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      "0.338654156"
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      "1.323057099"
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      "0.184939552"
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      "0.081535576"
    ],
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      "0.255472043"
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      "1.150379031"
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      "0.723659137"
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      "1.131278923"
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      "0.928216524"
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      "0.887927125"
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      "1.513427051"
    ],
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      "0.483732585"
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      "1.408767912"
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    [
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      "1.372584046"
    ],
    [
      "1.162746667",
      "0.534552929"
    ],
    [
      "0.660364492",
      "1.032549551"
    ],
    [
      "1.344402345",
      "1.218513569"
    ],
    [
      "0.551447022",
      "0.352726053"
    ],
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      "0.69470286",
      "0.175031308"
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      "1.059294346"
    ],
    [
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      "1.304337779"
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  "radius": "0.077152728"
}

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?