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 = 155

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 = 155
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[0.0731330487065521015780453975165, 0.0731330487065521015780453975175]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 = 0.073133048

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 = 155 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": [
    [
      "0.951888567",
      "0.074382565"
    ],
    [
      "1.101393584",
      "0.078750395"
    ],
    [
      "0.766623863",
      "0.102995026"
    ],
    [
      "1.283181859",
      "0.117452336"
    ],
    [
      "0.626182444",
      "0.151859222"
    ],
    [
      "1.422373476",
      "0.174964854"
    ],
    [
      "0.876394725",
      "0.199660061"
    ],
    [
      "1.02263514",
      "0.202400899"
    ],
    [
      "1.168770748",
      "0.208577902"
    ],
    [
      "0.497411773",
      "0.221226978"
    ],
    [
      "0.737794917",
      "0.246391899"
    ],
    [
      "1.546904688",
      "0.251683485"
    ],
    [
      "1.304893359",
      "0.262098048"
    ],
    [
      "0.378923481",
      "0.311998327"
    ],
    [
      "0.94568394",
      "0.328473011"
    ],
    [
      "1.091879876",
      "0.33300284"
    ],
    [
      "1.659858595",
      "0.349101406"
    ],
    [
      "0.514667465",
      "0.366471642"
    ],
    [
      "0.660883908",
      "0.370815041"
    ],
    [
      "0.807084132",
      "0.375204848"
    ],
    [
      "1.228002487",
      "0.386522985"
    ],
    [
      "1.375032465",
      "0.390450208"
    ],
    [
      "1.521205246",
      "0.395674144"
    ],
    [
      "0.278423845",
      "0.418269652"
    ],
    [
      "1.015068058",
      "0.457476595"
    ],
    [
      "1.754038464",
      "0.461011744"
    ],
    [
      "0.414167829",
      "0.472742967"
    ],
    [
      "0.58399785",
      "0.495262437"
    ],
    [
      "0.730198632",
      "0.499633105"
    ],
    [
      "0.876399413",
      "0.504003773"
    ],
    [
      "1.615385114",
      "0.507584481"
    ],
    [
      "1.151190669",
      "0.510996741"
    ],
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      "1.297394013",
      "0.515280849"
    ],
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      "1.443594794",
      "0.519651517"
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      "0.195893657",
      "0.539027837"
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      "1.829440474",
      "0.586344533"
    ],
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    ],
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      "0.337385787",
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    ],
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      "0.645614655",
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      "1.537774663",
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      "0.791815436",
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    ],
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      "0.641028655"
    ],
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      "1.230420343",
      "0.645312764"
    ],
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      "1.376621124",
      "0.649683432"
    ],
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      "0.13338817",
      "0.671265636"
    ],
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      "1.884186886",
      "0.721978605"
    ],
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      "0.422914894",
      "0.734663145"
    ],
    [
      "0.264188349",
      "0.736809564"
    ],
    [
      "0.568151546",
      "0.75198614"
    ],
    [
      "0.714929936",
      "0.756715576"
    ],
    [
      "0.861130718",
      "0.761086244"
    ],
    [
      "1.007331499",
      "0.765456912"
    ],
    [
      "1.592491314",
      "0.767207935"
    ],
    [
      "1.153532281",
      "0.76982758"
    ],
    [
      "1.745754591",
      "0.771628219"
    ],
    [
      "1.299735624",
      "0.774111688"
    ],
    [
      "1.44663176",
      "0.778105712"
    ],
    [
      "0.092463962",
      "0.811689923"
    ],
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      "0.346029394",
      "0.859091402"
    ],
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      "1.916914347",
      "0.864536259"
    ],
    [
      "0.491266046",
      "0.876414397"
    ],
    [
      "0.637466828",
      "0.880785065"
    ],
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      "0.784245217",
      "0.885514501"
    ],
    [
      "0.930445999",
      "0.889885169"
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      "1.07664678",
      "0.894255837"
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      "1.661806596",
      "0.89600686"
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      "1.222923806",
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      "1.369746259",
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      "1.78338916",
      "1.095119382"
    ],
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      "0.079770655",
      "1.110726232"
    ],
    [
      "0.337492484",
      "1.125357471"
    ],
    [
      "0.483695827",
      "1.129641579"
    ],
    [
      "0.629896609",
      "1.134012247"
    ],
    [
      "0.77609739",
      "1.138382915"
    ],
    [
      "0.92287578",
      "1.143112351"
    ],
    [
      "1.069152805",
      "1.147441958"
    ],
    [
      "1.216048941",
      "1.151435982"
    ],
    [
      "1.362249722",
      "1.15580665"
    ],
    [
      "1.508450504",
      "1.160177318"
    ],
    [
      "1.912152226",
      "1.164501253"
    ],
    [
      "1.654651285",
      "1.164547986"
    ],
    [
      "0.255473085",
      "1.251759846"
    ],
    [
      "0.108647812",
      "1.254113404"
    ],
    [
      "0.406807765",
      "1.254156396"
    ],
    [
      "0.553011108",
      "1.258440504"
    ],
    [
      "0.69921189",
      "1.262811172"
    ],
    [
      "0.845412672",
      "1.26718184"
    ],
    [
      "0.992267305",
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    ],
    [
      "1.139163441",
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    ],
    [
      "1.285364222",
      "1.280234906"
    ],
    [
      "1.431565004",
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    ],
    [
      "1.577765785",
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      "1.537832756"
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      "1.542203424"
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      "1.562210155"
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      "1.63241963"
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      "1.640512109"
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      "0.761030634",
      "1.644882777"
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      "0.907929332",
      "1.648790241"
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      "1.054130113",
      "1.653160909"
    ],
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      "1.200462272",
      "1.66051581"
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      "1.346865733",
      "1.662391787"
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    [
      "1.493066515",
      "1.666762455"
    ],
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      "1.639267296",
      "1.671133123"
    ],
    [
      "0.831117055",
      "1.773263714"
    ],
    [
      "0.4983246",
      "1.779361366"
    ],
    [
      "0.644525381",
      "1.783732034"
    ],
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      "1.119545795",
      "1.783983537"
    ],
    [
      "0.974446839",
      "1.802424373"
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      "1.308215186",
      "1.803458815"
    ],
    [
      "1.454415967",
      "1.807829483"
    ],
    [
      "0.744124373",
      "1.890847916"
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    [
      "1.202446704",
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    ],
    [
      "1.055262159",
      "1.925178852"
    ]
  ],
  "radius": "0.073133048"
}
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": 155
}

起步布局答案

{
  "centers": [
    [
      "0.951888567",
      "0.074382565"
    ],
    [
      "1.101393584",
      "0.078750395"
    ],
    [
      "0.766623863",
      "0.102995026"
    ],
    [
      "1.283181859",
      "0.117452336"
    ],
    [
      "0.626182444",
      "0.151859222"
    ],
    [
      "1.422373476",
      "0.174964854"
    ],
    [
      "0.876394725",
      "0.199660061"
    ],
    [
      "1.02263514",
      "0.202400899"
    ],
    [
      "1.168770748",
      "0.208577902"
    ],
    [
      "0.497411773",
      "0.221226978"
    ],
    [
      "0.737794917",
      "0.246391899"
    ],
    [
      "1.546904688",
      "0.251683485"
    ],
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      "0.262098048"
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    [
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      "0.311998327"
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      "0.94568394",
      "0.328473011"
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    [
      "1.091879876",
      "0.33300284"
    ],
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    ],
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      "0.366471642"
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  "radius": "0.073133048"
}

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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