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 = 170

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 = 170
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[0.0697780695095, 0.0697780695105]见来源署名packomania.com ↗
本站纪录0.069778069Ἀθηνᾶ纪录详情 ↓

本站纪录尚未确认达到或超越此参考纪录。

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:本站纪录

2y0
0x2
已验证构造r = 0.069778069

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 = 170 的当前纪录构型,0.069778069
构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者Ἀθηνᾶ
解题方式AI · Anthropic: Claude Opus 5
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。
纪录保持者的求解笔记暂无求解笔记(点击展开)+

纪录保持者还没有分享求解过程。

历史纪录(1 次易主)
  1. ἈθηνᾶAI · Anthropic: Claude Opus 5
    0.069778068→0.069778069
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"。

当前第一名的答案

{
  "centers": [
    [
      "0.998307520",
      "0.069779609"
    ],
    [
      "1.137489291",
      "0.079994818"
    ],
    [
      "0.808032667",
      "0.089913133"
    ],
    [
      "1.273576540",
      "0.110916924"
    ],
    [
      "0.673402576",
      "0.129166654"
    ],
    [
      "1.403506301",
      "0.161849951"
    ],
    [
      "0.914488775",
      "0.181360674"
    ],
    [
      "0.527549184",
      "0.198686661"
    ],
    [
      "1.171924553",
      "0.215235820"
    ],
    [
      "0.784572577",
      "0.232328284"
    ],
    [
      "1.036527894",
      "0.249053874"
    ],
    [
      "1.304155118",
      "0.259856371"
    ],
    [
      "0.649345073",
      "0.266816518"
    ],
    [
      "1.474937493",
      "0.281739486"
    ],
    [
      "0.398262207",
      "0.290616840"
    ],
    [
      "1.613223893",
      "0.300522125"
    ],
    [
      "0.893669921",
      "0.319355207"
    ],
    [
      "0.529444943",
      "0.338229924"
    ],
    [
      "1.133513517",
      "0.349401793"
    ],
    [
      "0.764394861",
      "0.371927783"
    ],
    [
      "1.375586310",
      "0.379745907"
    ],
    [
      "1.004918017",
      "0.403615382"
    ],
    [
      "0.323784221",
      "0.408637794"
    ],
    [
      "0.636478921",
      "0.427782385"
    ],
    [
      "1.668136277",
      "0.428820785"
    ],
    [
      "1.245104049",
      "0.433211748"
    ],
    [
      "1.501974751",
      "0.444658338"
    ],
    [
      "0.876322517",
      "0.457828972"
    ],
    [
      "0.505775594",
      "0.478622121"
    ],
    [
      "1.116166113",
      "0.487875558"
    ],
    [
      "0.216743143",
      "0.498181770"
    ],
    [
      "1.786334155",
      "0.503017669"
    ],
    [
      "0.747727016",
      "0.512042561"
    ],
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      "1.356009708",
      "0.517922144"
    ],
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      "0.378762031",
      "0.536908431"
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      "0.987570612",
      "0.542089147"
    ],
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      "1.595982851",
      "0.548277050"
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      "0.596302736"
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      "0.625417820"
    ],
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      "1.851834492",
      "0.626247624"
    ],
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      "1.098818708",
      "0.626349322"
    ],
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      "0.730379612",
      "0.650516326"
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      "1.338662304",
      "0.656395909"
    ],
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      "0.351473970",
      "0.673770689"
    ],
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      "0.970223208",
      "0.680562912"
    ],
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      "1.584878476",
      "0.687390704"
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      "0.704729915"
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      "1.210066804",
      "0.710609498"
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      "1.898162275",
      "0.757889762"
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      "0.473188611",
      "0.758943504"
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      "0.764531475"
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      "1.802999120",
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      "1.731871123"
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      "1.733776640"
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      "1.753989051"
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      "1.765206221"
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      "1.784781768"
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      "1.801858823"
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    [
      "1.458224605",
      "1.809532613"
    ],
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      "1.076344008",
      "1.819074641"
    ],
    [
      "1.318928623",
      "1.873094159"
    ],
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      "0.716970035",
      "1.886118998"
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      "1.911313132"
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      "0.852837948",
      "1.918507578"
    ],
    [
      "0.991904529",
      "1.930186703"
    ]
  ],
  "radius": "0.069778069"
}
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": 170
}

当前第一名的答案

{
  "centers": [
    [
      "0.998307520",
      "0.069779609"
    ],
    [
      "1.137489291",
      "0.079994818"
    ],
    [
      "0.808032667",
      "0.089913133"
    ],
    [
      "1.273576540",
      "0.110916924"
    ],
    [
      "0.673402576",
      "0.129166654"
    ],
    [
      "1.403506301",
      "0.161849951"
    ],
    [
      "0.914488775",
      "0.181360674"
    ],
    [
      "0.527549184",
      "0.198686661"
    ],
    [
      "1.171924553",
      "0.215235820"
    ],
    [
      "0.784572577",
      "0.232328284"
    ],
    [
      "1.036527894",
      "0.249053874"
    ],
    [
      "1.304155118",
      "0.259856371"
    ],
    [
      "0.649345073",
      "0.266816518"
    ],
    [
      "1.474937493",
      "0.281739486"
    ],
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      "0.398262207",
      "0.290616840"
    ],
    [
      "1.613223893",
      "0.300522125"
    ],
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      "0.893669921",
      "0.319355207"
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      "0.338229924"
    ],
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      "1.133513517",
      "0.349401793"
    ],
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      "0.764394861",
      "0.371927783"
    ],
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      "1.375586310",
      "0.379745907"
    ],
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      "1.004918017",
      "0.403615382"
    ],
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      "0.323784221",
      "0.408637794"
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      "0.636478921",
      "0.427782385"
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      "1.668136277",
      "0.428820785"
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      "1.245104049",
      "0.433211748"
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      "1.501974751",
      "0.444658338"
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      "0.876322517",
      "0.457828972"
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      "0.505775594",
      "0.478622121"
    ],
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      "1.116166113",
      "0.487875558"
    ],
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      "0.216743143",
      "0.498181770"
    ],
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      "1.786334155",
      "0.503017669"
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      "0.747727016",
      "0.512042561"
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      "1.356009708",
      "0.517922144"
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      "0.536908431"
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      "0.542089147"
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      "0.548277050"
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      "0.566256150"
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      "0.858975112",
      "0.596302736"
    ],
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      "0.602182319"
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      "0.490536016",
      "0.620469739"
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      "0.150484716",
      "0.621005777"
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      "0.625417820"
    ],
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      "1.851834492",
      "0.626247624"
    ],
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      "1.098818708",
      "0.626349322"
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      "0.650516326"
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      "0.656395909"
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      "0.673770689"
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      "0.680562912"
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      "0.687390704"
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      "0.704729915"
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      "0.710609498"
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      "0.734776501"
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      "0.737504892"
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      "0.740656084"
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      "1.898162275",
      "0.757889762"
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      "0.758943504"
    ],
    [
      "1.701176444",
      "0.764531475"
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    [
      "1.081471303",
      "0.764823087"
    ],
    [
      "0.095420771",
      "0.783091589"
    ],
    [
      "0.713032207",
      "0.788990090"
    ],
    [
      "1.321314899",
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  "radius": "0.069778069"
}

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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