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 = 172

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 = 172
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[0.0693636354061843568401796427055, 0.0693636354061843568401796427065]见来源署名packomania.com ↗
本站纪录0.069363635Ἀθηνᾶ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 = 0.069363635

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 = 172 的当前纪录构型,0.069363635
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
    0.069363634→0.069363635
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": [
    [
      "0.976489585",
      "0.069660652"
    ],
    [
      "1.129947442",
      "0.078480762"
    ],
    [
      "0.798062227",
      "0.091536914"
    ],
    [
      "1.303867856",
      "0.120370323"
    ],
    [
      "0.665260567",
      "0.131648945"
    ],
    [
      "1.431250863",
      "0.175314159"
    ],
    [
      "0.900135608",
      "0.185485200"
    ],
    [
      "1.046591438",
      "0.189372747"
    ],
    [
      "0.539897154",
      "0.191056606"
    ],
    [
      "1.192077455",
      "0.202517520"
    ],
    [
      "1.549051044",
      "0.248583342"
    ],
    [
      "0.424757687",
      "0.268439801"
    ],
    [
      "0.653567329",
      "0.270582370"
    ],
    [
      "0.792277946",
      "0.272731864"
    ],
    [
      "1.288559737",
      "0.302199138"
    ],
    [
      "0.970237462",
      "0.305197295"
    ],
    [
      "1.108721451",
      "0.313409504"
    ],
    [
      "1.426774730",
      "0.314110120"
    ],
    [
      "1.654650760",
      "0.338549756"
    ],
    [
      "0.322400681",
      "0.362078997"
    ],
    [
      "0.478712598",
      "0.396244822"
    ],
    [
      "0.617196587",
      "0.404457031"
    ],
    [
      "0.755907204",
      "0.406606524"
    ],
    [
      "0.893883486",
      "0.421021843"
    ],
    [
      "1.032367475",
      "0.429234052"
    ],
    [
      "1.170851464",
      "0.437446261"
    ],
    [
      "1.309565989",
      "0.439326790"
    ],
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      "1.745703480",
      "0.443214257"
    ],
    [
      "1.447780981",
      "0.451237772"
    ],
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      "1.586276292",
      "0.459256789"
    ],
    [
      "0.234904620",
      "0.470811401"
    ],
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      "0.376355592",
      "0.489884017"
    ],
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      "0.540561531",
      "0.520421974"
    ],
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      "0.679045520",
      "0.528634183"
    ],
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      "0.817529509",
      "0.536846392"
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      "0.956013499",
      "0.545058601"
    ],
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      "1.094497488",
      "0.553270810"
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      "1.232981477",
      "0.561483019"
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      "1.677329012",
      "0.563921290"
    ],
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      "0.564538151"
    ],
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      "1.510083965",
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      "0.160556189",
      "0.598231246"
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      "0.605117483"
    ],
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      "0.438204525",
      "0.614061170"
    ],
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      "0.602691544",
      "0.644458731"
    ],
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      "0.741175533",
      "0.652670940"
    ],
    [
      "0.879659523",
      "0.660883149"
    ],
    [
      "1.018143512",
      "0.669095358"
    ],
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      "1.156627501",
      "0.677307567"
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      "1.601136685",
      "0.679852239"
    ],
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      "1.295111490",
      "0.685519776"
    ],
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      "1.742262513",
      "0.686513688"
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      "1.434062148",
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      "0.734920962"
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      "0.388614417",
      "0.743622263"
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      "0.526337568",
      "0.760283279"
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      "0.664821557",
      "0.768495488"
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      "0.803305546",
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      "0.809556533"
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      "1.772951313",
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      "1.915492538",
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      "1.314569614",
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      "1.619751710"
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      "0.967035348",
      "1.679065093"
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      "1.684323730"
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      "1.705812454"
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      "0.454709154",
      "1.754149810"
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      "0.693564853",
      "1.758426909"
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      "0.832048842",
      "1.766639118"
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      "1.092941197",
      "1.775313770"
    ],
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      "1.231425186",
      "1.783525979"
    ],
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      "1.468599707",
      "1.804051216"
    ],
    [
      "0.960932698",
      "1.817963646"
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      "0.572873847",
      "1.826829664"
    ],
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      "1.343869124",
      "1.864776311"
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      "0.732862897",
      "1.891471711"
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      "1.181066503",
      "1.912846497"
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      "0.868350403",
      "1.921277606"
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    [
      "1.043342111",
      "1.929588649"
    ]
  ],
  "radius": "0.069363635"
}
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": 172
}

当前第一名的答案

{
  "centers": [
    [
      "0.976489585",
      "0.069660652"
    ],
    [
      "1.129947442",
      "0.078480762"
    ],
    [
      "0.798062227",
      "0.091536914"
    ],
    [
      "1.303867856",
      "0.120370323"
    ],
    [
      "0.665260567",
      "0.131648945"
    ],
    [
      "1.431250863",
      "0.175314159"
    ],
    [
      "0.900135608",
      "0.185485200"
    ],
    [
      "1.046591438",
      "0.189372747"
    ],
    [
      "0.539897154",
      "0.191056606"
    ],
    [
      "1.192077455",
      "0.202517520"
    ],
    [
      "1.549051044",
      "0.248583342"
    ],
    [
      "0.424757687",
      "0.268439801"
    ],
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      "0.653567329",
      "0.270582370"
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      "0.272731864"
    ],
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      "1.288559737",
      "0.302199138"
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      "0.970237462",
      "0.305197295"
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      "0.313409504"
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      "0.314110120"
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      "0.338549756"
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      "0.322400681",
      "0.362078997"
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      "0.478712598",
      "0.396244822"
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      "0.617196587",
      "0.404457031"
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      "0.755907204",
      "0.406606524"
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      "0.893883486",
      "0.421021843"
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      "0.443214257"
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      "0.451237772"
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      "1.586276292",
      "0.459256789"
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      "0.470811401"
    ],
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      "0.376355592",
      "0.489884017"
    ],
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      "0.540561531",
      "0.520421974"
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      "0.564538151"
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      "0.569695227"
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      "0.605117483"
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      "0.614061170"
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      "0.669095358"
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      "0.685519776"
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      "0.727480501"
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      "0.734920962"
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      "0.743622263"
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      "0.526337568",
      "0.760283279"
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      "0.664821557",
      "0.768495488"
    ],
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  "radius": "0.069363635"
}

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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