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 = 79

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 = 79
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[0.1009584646544562289893940324775, 0.1009584646544562289893940324785]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 = 0.100958464

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 = 79 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",
      "0.100958465"
    ],
    [
      "1.238286729",
      "0.13311182"
    ],
    [
      "0.746684529",
      "0.137383657"
    ],
    [
      "1.425741071",
      "0.208153915"
    ],
    [
      "0.52711171",
      "0.235374374"
    ],
    [
      "0.895573337",
      "0.273774894"
    ],
    [
      "1.097450086",
      "0.277802823"
    ],
    [
      "0.703010657",
      "0.334520765"
    ],
    [
      "1.620389931",
      "0.349314196"
    ],
    [
      "0.368396178",
      "0.360193549"
    ],
    [
      "1.427827252",
      "0.410060067"
    ],
    [
      "1.226303808",
      "0.434651274"
    ],
    [
      "1.025265015",
      "0.466375784"
    ],
    [
      "0.825403746",
      "0.495114595"
    ],
    [
      "0.623436239",
      "0.521169862"
    ],
    [
      "1.576716059",
      "0.546451304"
    ],
    [
      "0.221407191",
      "0.550479232"
    ],
    [
      "1.778592809",
      "0.550479232"
    ],
    [
      "0.423283941",
      "0.554507161"
    ],
    [
      "1.34994546",
      "0.596352506"
    ],
    [
      "1.15008419",
      "0.625091316"
    ],
    [
      "0.95022292",
      "0.653830126"
    ],
    [
      "0.750361651",
      "0.682568937"
    ],
    [
      "0.550500381",
      "0.711307747"
    ],
    [
      "1.674166145",
      "0.723295662"
    ],
    [
      "0.318857278",
      "0.72732359"
    ],
    [
      "1.474764634",
      "0.755068037"
    ],
    [
      "1.869890551",
      "0.772918271"
    ],
    [
      "1.274903365",
      "0.783806848"
    ],
    [
      "0.126294598",
      "0.788069461"
    ],
    [
      "1.075042095",
      "0.812545658"
    ],
    [
      "0.875180825",
      "0.841284468"
    ],
    [
      "0.675319556",
      "0.870023279"
    ],
    [
      "0.475458286",
      "0.898762089"
    ],
    [
      "1.602668765",
      "0.913299798"
    ],
    [
      "0.275183405",
      "0.924460698"
    ],
    [
      "1.399722539",
      "0.942522379"
    ],
    [
      "1.19986127",
      "0.97126119"
    ],
    [
      "1.898629361",
      "0.972779541"
    ],
    [
      "1",
      "1"
    ],
    [
      "0.101370639",
      "1.027220459"
    ],
    [
      "0.80013873",
      "1.02873881"
    ],
    [
      "0.600277461",
      "1.057477621"
    ],
    [
      "1.724816595",
      "1.075539302"
    ],
    [
      "0.397322403",
      "1.086686004"
    ],
    [
      "1.524541714",
      "1.101237911"
    ],
    [
      "1.324680444",
      "1.129976721"
    ],
    [
      "1.124819175",
      "1.158715532"
    ],
    [
      "0.924957905",
      "1.187454342"
    ],
    [
      "1.873705402",
      "1.211930539"
    ],
    [
      "0.725096635",
      "1.216193152"
    ],
    [
      "0.130109449",
      "1.227081729"
    ],
    [
      "0.525235366",
      "1.244931963"
    ],
    [
      "1.681142722",
      "1.27267641"
    ],
    [
      "0.325833855",
      "1.276704338"
    ],
    [
      "1.449499619",
      "1.288692253"
    ],
    [
      "1.249638349",
      "1.317431063"
    ],
    [
      "1.04977708",
      "1.346169874"
    ],
    [
      "0.84991581",
      "1.374908684"
    ],
    [
      "0.65005454",
      "1.403647494"
    ],
    [
      "1.576716059",
      "1.445492839"
    ],
    [
      "0.221407191",
      "1.449520768"
    ],
    [
      "1.778592809",
      "1.449520768"
    ],
    [
      "0.423283941",
      "1.453548696"
    ],
    [
      "1.376563474",
      "1.478831083"
    ],
    [
      "1.174596254",
      "1.504885405"
    ],
    [
      "0.974734985",
      "1.533624216"
    ],
    [
      "0.773702797",
      "1.565356859"
    ],
    [
      "0.572172748",
      "1.589939933"
    ],
    [
      "1.631603822",
      "1.639806451"
    ],
    [
      "0.379610069",
      "1.650685804"
    ],
    [
      "1.296989343",
      "1.665479235"
    ],
    [
      "0.902549914",
      "1.722197177"
    ],
    [
      "1.104426663",
      "1.726225106"
    ],
    [
      "1.47288829",
      "1.764625626"
    ],
    [
      "0.574258929",
      "1.791846085"
    ],
    [
      "1.253315471",
      "1.862616343"
    ],
    [
      "0.761713271",
      "1.86688818"
    ],
    [
      "1",
      "1.899041535"
    ]
  ],
  "radius": "0.100958464"
}
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": 79
}

起步布局答案

{
  "centers": [
    [
      "1",
      "0.100958465"
    ],
    [
      "1.238286729",
      "0.13311182"
    ],
    [
      "0.746684529",
      "0.137383657"
    ],
    [
      "1.425741071",
      "0.208153915"
    ],
    [
      "0.52711171",
      "0.235374374"
    ],
    [
      "0.895573337",
      "0.273774894"
    ],
    [
      "1.097450086",
      "0.277802823"
    ],
    [
      "0.703010657",
      "0.334520765"
    ],
    [
      "1.620389931",
      "0.349314196"
    ],
    [
      "0.368396178",
      "0.360193549"
    ],
    [
      "1.427827252",
      "0.410060067"
    ],
    [
      "1.226303808",
      "0.434651274"
    ],
    [
      "1.025265015",
      "0.466375784"
    ],
    [
      "0.825403746",
      "0.495114595"
    ],
    [
      "0.623436239",
      "0.521169862"
    ],
    [
      "1.576716059",
      "0.546451304"
    ],
    [
      "0.221407191",
      "0.550479232"
    ],
    [
      "1.778592809",
      "0.550479232"
    ],
    [
      "0.423283941",
      "0.554507161"
    ],
    [
      "1.34994546",
      "0.596352506"
    ],
    [
      "1.15008419",
      "0.625091316"
    ],
    [
      "0.95022292",
      "0.653830126"
    ],
    [
      "0.750361651",
      "0.682568937"
    ],
    [
      "0.550500381",
      "0.711307747"
    ],
    [
      "1.674166145",
      "0.723295662"
    ],
    [
      "0.318857278",
      "0.72732359"
    ],
    [
      "1.474764634",
      "0.755068037"
    ],
    [
      "1.869890551",
      "0.772918271"
    ],
    [
      "1.274903365",
      "0.783806848"
    ],
    [
      "0.126294598",
      "0.788069461"
    ],
    [
      "1.075042095",
      "0.812545658"
    ],
    [
      "0.875180825",
      "0.841284468"
    ],
    [
      "0.675319556",
      "0.870023279"
    ],
    [
      "0.475458286",
      "0.898762089"
    ],
    [
      "1.602668765",
      "0.913299798"
    ],
    [
      "0.275183405",
      "0.924460698"
    ],
    [
      "1.399722539",
      "0.942522379"
    ],
    [
      "1.19986127",
      "0.97126119"
    ],
    [
      "1.898629361",
      "0.972779541"
    ],
    [
      "1",
      "1"
    ],
    [
      "0.101370639",
      "1.027220459"
    ],
    [
      "0.80013873",
      "1.02873881"
    ],
    [
      "0.600277461",
      "1.057477621"
    ],
    [
      "1.724816595",
      "1.075539302"
    ],
    [
      "0.397322403",
      "1.086686004"
    ],
    [
      "1.524541714",
      "1.101237911"
    ],
    [
      "1.324680444",
      "1.129976721"
    ],
    [
      "1.124819175",
      "1.158715532"
    ],
    [
      "0.924957905",
      "1.187454342"
    ],
    [
      "1.873705402",
      "1.211930539"
    ],
    [
      "0.725096635",
      "1.216193152"
    ],
    [
      "0.130109449",
      "1.227081729"
    ],
    [
      "0.525235366",
      "1.244931963"
    ],
    [
      "1.681142722",
      "1.27267641"
    ],
    [
      "0.325833855",
      "1.276704338"
    ],
    [
      "1.449499619",
      "1.288692253"
    ],
    [
      "1.249638349",
      "1.317431063"
    ],
    [
      "1.04977708",
      "1.346169874"
    ],
    [
      "0.84991581",
      "1.374908684"
    ],
    [
      "0.65005454",
      "1.403647494"
    ],
    [
      "1.576716059",
      "1.445492839"
    ],
    [
      "0.221407191",
      "1.449520768"
    ],
    [
      "1.778592809",
      "1.449520768"
    ],
    [
      "0.423283941",
      "1.453548696"
    ],
    [
      "1.376563474",
      "1.478831083"
    ],
    [
      "1.174596254",
      "1.504885405"
    ],
    [
      "0.974734985",
      "1.533624216"
    ],
    [
      "0.773702797",
      "1.565356859"
    ],
    [
      "0.572172748",
      "1.589939933"
    ],
    [
      "1.631603822",
      "1.639806451"
    ],
    [
      "0.379610069",
      "1.650685804"
    ],
    [
      "1.296989343",
      "1.665479235"
    ],
    [
      "0.902549914",
      "1.722197177"
    ],
    [
      "1.104426663",
      "1.726225106"
    ],
    [
      "1.47288829",
      "1.764625626"
    ],
    [
      "0.574258929",
      "1.791846085"
    ],
    [
      "1.253315471",
      "1.862616343"
    ],
    [
      "0.761713271",
      "1.86688818"
    ],
    [
      "1",
      "1.899041535"
    ]
  ],
  "radius": "0.100958464"
}

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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