极值竞技场
中/EN
P01 · 装箱与覆盖 · 经典问题 · 难

单位正方形内的等圆装箱 · n = 93

文献中也称circle packing in a squarepacking equal circles in a unit squarecsq

在边长 1 的正方形内放置 n 个半径相同、互不重叠的圆，使共同半径尽可能大。

子题n = 93
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[0.0529264333875, 0.0529264333885]见来源署名packomania.com ↗
本站纪录0.052926433游侠纪录详情 ↓

本站纪录尚未确认达到或超越此参考纪录。

严格定义

  • 容器单位正方形：左下角是原点 (0, 0)，右上角是 (1, 1)
  • 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
放大来摆，然后提交 ↗

当前展示：本站纪录

1y0
0x1
已验证构造r = 0.052926433

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 = 1–30 已证明最优，适合观察经典接触结构；n = 31–300 接入 Packomania 的公开前沿，除 n = 36 外最优性仍未知。大规模部分集中成一张表，供算法批量挑战。

查看来源
单位正方形内的等圆装箱 n = 93 的当前纪录构型，0.052926433
构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者游侠
解题方式AI · OpenAI: GPT-5.6 Sol
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. 游侠AI · OpenAI: GPT-5.6 Sol
    0.052926432→0.052926433
ANSWER FORMAT

答案怎么写

容器是边长 1 的正方形，左下角是原点 (0, 0)，右上角是 (1, 1)。坐标和长度用同一个单位，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 "0.25"。

当前第一名的答案

{
  "centers": [
    [
      "0.225615614",
      "0.052926433"
    ],
    [
      "0.408958158",
      "0.052926433"
    ],
    [
      "0.592473803",
      "0.052926433"
    ],
    [
      "0.753095945",
      "0.052926433"
    ],
    [
      "0.947073567",
      "0.052926433"
    ],
    [
      "0.056101769",
      "0.056101769"
    ],
    [
      "0.850084756",
      "0.095329262"
    ],
    [
      "0.500715980",
      "0.105702673"
    ],
    [
      "0.317286886",
      "0.105852867"
    ],
    [
      "0.144583415",
      "0.121033791"
    ],
    [
      "0.672784874",
      "0.121882660"
    ],
    [
      "0.408958158",
      "0.158779300"
    ],
    [
      "0.947073567",
      "0.158779300"
    ],
    [
      "0.769773685",
      "0.164285490"
    ],
    [
      "0.236254687",
      "0.173960224"
    ],
    [
      "0.052926433",
      "0.173984968"
    ],
    [
      "0.580898037",
      "0.174808874"
    ],
    [
      "0.860217141",
      "0.219283768"
    ],
    [
      "0.327925959",
      "0.226886658"
    ],
    [
      "0.144597705",
      "0.226911401"
    ],
    [
      "0.672569309",
      "0.227735308"
    ],
    [
      "0.489243914",
      "0.227764999"
    ],
    [
      "0.768590121",
      "0.272286774"
    ],
    [
      "0.947073567",
      "0.279788236"
    ],
    [
      "0.052926433",
      "0.279837835"
    ],
    [
      "0.236268977",
      "0.279837835"
    ],
    [
      "0.580915186",
      "0.280691433"
    ],
    [
      "0.408211715",
      "0.295872357"
    ],
    [
      "0.144597705",
      "0.332764268"
    ],
    [
      "0.855446547",
      "0.332791242"
    ],
    [
      "0.672603597",
      "0.333588169"
    ],
    [
      "0.499882987",
      "0.348798790"
    ],
    [
      "0.316554734",
      "0.348823534"
    ],
    [
      "0.052926433",
      "0.385690702"
    ],
    [
      "0.947073567",
      "0.385794247"
    ],
    [
      "0.764257621",
      "0.386544466"
    ],
    [
      "0.591571398",
      "0.401695526"
    ],
    [
      "0.224883462",
      "0.401749967"
    ],
    [
      "0.408226006",
      "0.401749967"
    ],
    [
      "0.855884641",
      "0.439547472"
    ],
    [
      "0.499911561",
      "0.454651653"
    ],
    [
      "0.683225422",
      "0.454651823"
    ],
    [
      "0.133212190",
      "0.454676401"
    ],
    [
      "0.316554734",
      "0.454676401"
    ],
    [
      "0.947073567",
      "0.493300696"
    ],
    [
      "0.224883462",
      "0.507602834"
    ],
    [
      "0.408226006",
      "0.507602834"
    ],
    [
      "0.591568442",
      "0.507603004"
    ],
    [
      "0.774852442",
      "0.507654829"
    ],
    [
      "0.052926433",
      "0.523662100"
    ],
    [
      "0.316554734",
      "0.560529268"
    ],
    [
      "0.499897175",
      "0.560529445"
    ],
    [
      "0.683195462",
      "0.560606009"
    ],
    [
      "0.866041368",
      "0.561408053"
    ],
    [
      "0.144597705",
      "0.576588533"
    ],
    [
      "0.408225903",
      "0.613455879"
    ],
    [
      "0.591524195",
      "0.613532451"
    ],
    [
      "0.774384389",
      "0.614359234"
    ],
    [
      "0.052926433",
      "0.629514967"
    ],
    [
      "0.236268977",
      "0.629514967"
    ],
    [
      "0.947073567",
      "0.629515410"
    ],
    [
      "0.499852923",
      "0.666458884"
    ],
    [
      "0.682713122",
      "0.667285675"
    ],
    [
      "0.144597705",
      "0.682441400"
    ],
    [
      "0.327940771",
      "0.682442305"
    ],
    [
      "0.855325978",
      "0.682574251"
    ],
    [
      "0.591041850",
      "0.720212108"
    ],
    [
      "0.052926433",
      "0.735367834"
    ],
    [
      "0.236268977",
      "0.735367834"
    ],
    [
      "0.947073567",
      "0.735368277"
    ],
    [
      "0.763654682",
      "0.735500726"
    ],
    [
      "0.419829007",
      "0.735748149"
    ],
    [
      "0.144597705",
      "0.788294267"
    ],
    [
      "0.327939727",
      "0.788295172"
    ],
    [
      "0.855325978",
      "0.788427118"
    ],
    [
      "0.671983410",
      "0.788427160"
    ],
    [
      "0.512407421",
      "0.791074342"
    ],
    [
      "0.052926433",
      "0.841220700"
    ],
    [
      "0.947073567",
      "0.841221144"
    ],
    [
      "0.236268455",
      "0.841221605"
    ],
    [
      "0.763654730",
      "0.841353593"
    ],
    [
      "0.419391423",
      "0.841600112"
    ],
    [
      "0.593348982",
      "0.859289393"
    ],
    [
      "0.144816758",
      "0.894526545"
    ],
    [
      "0.852559666",
      "0.898805676"
    ],
    [
      "0.332525884",
      "0.902091496"
    ],
    [
      "0.052927478",
      "0.947073567"
    ],
    [
      "0.236706039",
      "0.947073567"
    ],
    [
      "0.428345729",
      "0.947073567"
    ],
    [
      "0.534198596",
      "0.947073567"
    ],
    [
      "0.652499368",
      "0.947073567"
    ],
    [
      "0.758352235",
      "0.947073567"
    ],
    [
      "0.946767097",
      "0.947073567"
    ]
  ],
  "radius": "0.052926433"
}
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": 93
}

当前第一名的答案

{
  "centers": [
    [
      "0.225615614",
      "0.052926433"
    ],
    [
      "0.408958158",
      "0.052926433"
    ],
    [
      "0.592473803",
      "0.052926433"
    ],
    [
      "0.753095945",
      "0.052926433"
    ],
    [
      "0.947073567",
      "0.052926433"
    ],
    [
      "0.056101769",
      "0.056101769"
    ],
    [
      "0.850084756",
      "0.095329262"
    ],
    [
      "0.500715980",
      "0.105702673"
    ],
    [
      "0.317286886",
      "0.105852867"
    ],
    [
      "0.144583415",
      "0.121033791"
    ],
    [
      "0.672784874",
      "0.121882660"
    ],
    [
      "0.408958158",
      "0.158779300"
    ],
    [
      "0.947073567",
      "0.158779300"
    ],
    [
      "0.769773685",
      "0.164285490"
    ],
    [
      "0.236254687",
      "0.173960224"
    ],
    [
      "0.052926433",
      "0.173984968"
    ],
    [
      "0.580898037",
      "0.174808874"
    ],
    [
      "0.860217141",
      "0.219283768"
    ],
    [
      "0.327925959",
      "0.226886658"
    ],
    [
      "0.144597705",
      "0.226911401"
    ],
    [
      "0.672569309",
      "0.227735308"
    ],
    [
      "0.489243914",
      "0.227764999"
    ],
    [
      "0.768590121",
      "0.272286774"
    ],
    [
      "0.947073567",
      "0.279788236"
    ],
    [
      "0.052926433",
      "0.279837835"
    ],
    [
      "0.236268977",
      "0.279837835"
    ],
    [
      "0.580915186",
      "0.280691433"
    ],
    [
      "0.408211715",
      "0.295872357"
    ],
    [
      "0.144597705",
      "0.332764268"
    ],
    [
      "0.855446547",
      "0.332791242"
    ],
    [
      "0.672603597",
      "0.333588169"
    ],
    [
      "0.499882987",
      "0.348798790"
    ],
    [
      "0.316554734",
      "0.348823534"
    ],
    [
      "0.052926433",
      "0.385690702"
    ],
    [
      "0.947073567",
      "0.385794247"
    ],
    [
      "0.764257621",
      "0.386544466"
    ],
    [
      "0.591571398",
      "0.401695526"
    ],
    [
      "0.224883462",
      "0.401749967"
    ],
    [
      "0.408226006",
      "0.401749967"
    ],
    [
      "0.855884641",
      "0.439547472"
    ],
    [
      "0.499911561",
      "0.454651653"
    ],
    [
      "0.683225422",
      "0.454651823"
    ],
    [
      "0.133212190",
      "0.454676401"
    ],
    [
      "0.316554734",
      "0.454676401"
    ],
    [
      "0.947073567",
      "0.493300696"
    ],
    [
      "0.224883462",
      "0.507602834"
    ],
    [
      "0.408226006",
      "0.507602834"
    ],
    [
      "0.591568442",
      "0.507603004"
    ],
    [
      "0.774852442",
      "0.507654829"
    ],
    [
      "0.052926433",
      "0.523662100"
    ],
    [
      "0.316554734",
      "0.560529268"
    ],
    [
      "0.499897175",
      "0.560529445"
    ],
    [
      "0.683195462",
      "0.560606009"
    ],
    [
      "0.866041368",
      "0.561408053"
    ],
    [
      "0.144597705",
      "0.576588533"
    ],
    [
      "0.408225903",
      "0.613455879"
    ],
    [
      "0.591524195",
      "0.613532451"
    ],
    [
      "0.774384389",
      "0.614359234"
    ],
    [
      "0.052926433",
      "0.629514967"
    ],
    [
      "0.236268977",
      "0.629514967"
    ],
    [
      "0.947073567",
      "0.629515410"
    ],
    [
      "0.499852923",
      "0.666458884"
    ],
    [
      "0.682713122",
      "0.667285675"
    ],
    [
      "0.144597705",
      "0.682441400"
    ],
    [
      "0.327940771",
      "0.682442305"
    ],
    [
      "0.855325978",
      "0.682574251"
    ],
    [
      "0.591041850",
      "0.720212108"
    ],
    [
      "0.052926433",
      "0.735367834"
    ],
    [
      "0.236268977",
      "0.735367834"
    ],
    [
      "0.947073567",
      "0.735368277"
    ],
    [
      "0.763654682",
      "0.735500726"
    ],
    [
      "0.419829007",
      "0.735748149"
    ],
    [
      "0.144597705",
      "0.788294267"
    ],
    [
      "0.327939727",
      "0.788295172"
    ],
    [
      "0.855325978",
      "0.788427118"
    ],
    [
      "0.671983410",
      "0.788427160"
    ],
    [
      "0.512407421",
      "0.791074342"
    ],
    [
      "0.052926433",
      "0.841220700"
    ],
    [
      "0.947073567",
      "0.841221144"
    ],
    [
      "0.236268455",
      "0.841221605"
    ],
    [
      "0.763654730",
      "0.841353593"
    ],
    [
      "0.419391423",
      "0.841600112"
    ],
    [
      "0.593348982",
      "0.859289393"
    ],
    [
      "0.144816758",
      "0.894526545"
    ],
    [
      "0.852559666",
      "0.898805676"
    ],
    [
      "0.332525884",
      "0.902091496"
    ],
    [
      "0.052927478",
      "0.947073567"
    ],
    [
      "0.236706039",
      "0.947073567"
    ],
    [
      "0.428345729",
      "0.947073567"
    ],
    [
      "0.534198596",
      "0.947073567"
    ],
    [
      "0.652499368",
      "0.947073567"
    ],
    [
      "0.758352235",
      "0.947073567"
    ],
    [
      "0.946767097",
      "0.947073567"
    ]
  ],
  "radius": "0.052926433"
}

提交 radius 与 centers。每个数写成十进制字符串，例如 "0.25"。 · 验证器 v1.0.0

DISCUSSION

讨论区

聊思路、贴方法、问为什么卡住。所有登录用户都可以发帖；发言公开署名，与纪录使用同一个名字，署名后的 #编号是账号注册序号，冒不了名。新发言经自动审核后公开。

啊？？！NUE_13#310 条回复空悲切/.lird#282 条回复