极值竞技场
中/EN
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 = 94

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 = 94
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[0.0927813152838669272073287151895, 0.0927813152838669272073287151905]见来源署名packomania.com ↗
本站纪录0.092781315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 = 0.092781315

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 = 94 的当前纪录构型，0.092781315
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
    0.092781314→0.092781315
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": [
    [
      "1.092613385",
      "0.097520913"
    ],
    [
      "0.907050758",
      "0.097555442"
    ],
    [
      "1.327939471",
      "0.154126873"
    ],
    [
      "0.671745441",
      "0.154249099"
    ],
    [
      "1.506819728",
      "0.247550995"
    ],
    [
      "0.492914107",
      "0.247730342"
    ],
    [
      "1.177377512",
      "0.262592208"
    ],
    [
      "0.822347891",
      "0.262658180"
    ],
    [
      "0.998734438",
      "0.327805949"
    ],
    [
      "1.345980456",
      "0.340094060"
    ],
    [
      "0.653777668",
      "0.340231177"
    ],
    [
      "1.649316944",
      "0.366412399"
    ],
    [
      "0.350458950",
      "0.366642150"
    ],
    [
      "1.160588673",
      "0.447393790"
    ],
    [
      "0.839184775",
      "0.447455391"
    ],
    [
      "1.488477672",
      "0.458955463"
    ],
    [
      "0.511322511",
      "0.459142985"
    ],
    [
      "1.321509871",
      "0.539924682"
    ],
    [
      "0.678302847",
      "0.540142224"
    ],
    [
      "0.999904507",
      "0.540205904"
    ],
    [
      "1.649041952",
      "0.551974826"
    ],
    [
      "0.350782658",
      "0.552204499"
    ],
    [
      "1.822646612",
      "0.617510402"
    ],
    [
      "0.177208783",
      "0.617821567"
    ],
    [
      "1.160624239",
      "0.632956417"
    ],
    [
      "0.839220341",
      "0.633018018"
    ],
    [
      "1.488202681",
      "0.644517890"
    ],
    [
      "0.511646219",
      "0.644705334"
    ],
    [
      "1.321343970",
      "0.725706931"
    ],
    [
      "0.999940073",
      "0.725768531"
    ],
    [
      "0.678536174",
      "0.725830131"
    ],
    [
      "1.649077518",
      "0.737537453"
    ],
    [
      "0.350818223",
      "0.737767126"
    ],
    [
      "1.883262481",
      "0.792893429"
    ],
    [
      "0.116659251",
      "0.793227508"
    ],
    [
      "1.160659804",
      "0.818519045"
    ],
    [
      "0.839255905",
      "0.818580645"
    ],
    [
      "1.488238247",
      "0.830080517"
    ],
    [
      "0.511681784",
      "0.830267961"
    ],
    [
      "1.733672904",
      "0.902695288"
    ],
    [
      "0.266297417",
      "0.902963140"
    ],
    [
      "1.321379536",
      "0.911269558"
    ],
    [
      "0.999975637",
      "0.911331158"
    ],
    [
      "0.678571739",
      "0.911392759"
    ],
    [
      "0.096454992",
      "0.977734151"
    ],
    [
      "1.572833633",
      "0.995238352"
    ],
    [
      "0.427160978",
      "0.995463975"
    ],
    [
      "1.160695369",
      "1.004081672"
    ],
    [
      "0.839291471",
      "1.004143272"
    ],
    [
      "1.907176687",
      "1.008714448"
    ],
    [
      "1.734635963",
      "1.088255420"
    ],
    [
      "0.265400594",
      "1.088523603"
    ],
    [
      "1.321415101",
      "1.096832185"
    ],
    [
      "1.000011203",
      "1.096893786"
    ],
    [
      "0.678607304",
      "1.096955386"
    ],
    [
      "1.494588942",
      "1.163497848"
    ],
    [
      "0.505462697",
      "1.163696940"
    ],
    [
      "1.160730935",
      "1.189644299"
    ],
    [
      "0.839327037",
      "1.189705900"
    ],
    [
      "1.885828950",
      "1.195838739"
    ],
    [
      "0.114240665",
      "1.196153365"
    ],
    [
      "1.655417896",
      "1.256058840"
    ],
    [
      "0.344661792",
      "1.256306654"
    ],
    [
      "1.321450667",
      "1.282394812"
    ],
    [
      "1.000046768",
      "1.282456414"
    ],
    [
      "0.678642870",
      "1.282518013"
    ],
    [
      "1.494624508",
      "1.349060475"
    ],
    [
      "0.505498263",
      "1.349259567"
    ],
    [
      "1.827452035",
      "1.371979664"
    ],
    [
      "0.172680141",
      "1.372273545"
    ],
    [
      "1.160766500",
      "1.375206926"
    ],
    [
      "0.839362601",
      "1.375268527"
    ],
    [
      "1.655453462",
      "1.441621467"
    ],
    [
      "0.344697359",
      "1.441869281"
    ],
    [
      "1.321486232",
      "1.467957439"
    ],
    [
      "1.000082332",
      "1.468019041"
    ],
    [
      "0.678678434",
      "1.468080640"
    ],
    [
      "1.494675322",
      "1.534623099"
    ],
    [
      "0.505533827",
      "1.534822194"
    ],
    [
      "1.160802064",
      "1.560769553"
    ],
    [
      "0.839398165",
      "1.560831154"
    ],
    [
      "1.655504277",
      "1.627184091"
    ],
    [
      "0.344732923",
      "1.627431908"
    ],
    [
      "1.350352524",
      "1.651261086"
    ],
    [
      "0.649906011",
      "1.651399045"
    ],
    [
      "0.998315432",
      "1.716192687"
    ],
    [
      "1.189668357",
      "1.744073200"
    ],
    [
      "0.810625743",
      "1.744149559"
    ],
    [
      "1.511197673",
      "1.744198499"
    ],
    [
      "0.486484956",
      "1.747895388"
    ],
    [
      "1.350388089",
      "1.836823713"
    ],
    [
      "0.649941576",
      "1.836961672"
    ],
    [
      "1.092960674",
      "1.902443380"
    ],
    [
      "0.907398047",
      "1.902480260"
    ]
  ],
  "radius": "0.092781315"
}
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": 94
}

当前第一名的答案

{
  "centers": [
    [
      "1.092613385",
      "0.097520913"
    ],
    [
      "0.907050758",
      "0.097555442"
    ],
    [
      "1.327939471",
      "0.154126873"
    ],
    [
      "0.671745441",
      "0.154249099"
    ],
    [
      "1.506819728",
      "0.247550995"
    ],
    [
      "0.492914107",
      "0.247730342"
    ],
    [
      "1.177377512",
      "0.262592208"
    ],
    [
      "0.822347891",
      "0.262658180"
    ],
    [
      "0.998734438",
      "0.327805949"
    ],
    [
      "1.345980456",
      "0.340094060"
    ],
    [
      "0.653777668",
      "0.340231177"
    ],
    [
      "1.649316944",
      "0.366412399"
    ],
    [
      "0.350458950",
      "0.366642150"
    ],
    [
      "1.160588673",
      "0.447393790"
    ],
    [
      "0.839184775",
      "0.447455391"
    ],
    [
      "1.488477672",
      "0.458955463"
    ],
    [
      "0.511322511",
      "0.459142985"
    ],
    [
      "1.321509871",
      "0.539924682"
    ],
    [
      "0.678302847",
      "0.540142224"
    ],
    [
      "0.999904507",
      "0.540205904"
    ],
    [
      "1.649041952",
      "0.551974826"
    ],
    [
      "0.350782658",
      "0.552204499"
    ],
    [
      "1.822646612",
      "0.617510402"
    ],
    [
      "0.177208783",
      "0.617821567"
    ],
    [
      "1.160624239",
      "0.632956417"
    ],
    [
      "0.839220341",
      "0.633018018"
    ],
    [
      "1.488202681",
      "0.644517890"
    ],
    [
      "0.511646219",
      "0.644705334"
    ],
    [
      "1.321343970",
      "0.725706931"
    ],
    [
      "0.999940073",
      "0.725768531"
    ],
    [
      "0.678536174",
      "0.725830131"
    ],
    [
      "1.649077518",
      "0.737537453"
    ],
    [
      "0.350818223",
      "0.737767126"
    ],
    [
      "1.883262481",
      "0.792893429"
    ],
    [
      "0.116659251",
      "0.793227508"
    ],
    [
      "1.160659804",
      "0.818519045"
    ],
    [
      "0.839255905",
      "0.818580645"
    ],
    [
      "1.488238247",
      "0.830080517"
    ],
    [
      "0.511681784",
      "0.830267961"
    ],
    [
      "1.733672904",
      "0.902695288"
    ],
    [
      "0.266297417",
      "0.902963140"
    ],
    [
      "1.321379536",
      "0.911269558"
    ],
    [
      "0.999975637",
      "0.911331158"
    ],
    [
      "0.678571739",
      "0.911392759"
    ],
    [
      "0.096454992",
      "0.977734151"
    ],
    [
      "1.572833633",
      "0.995238352"
    ],
    [
      "0.427160978",
      "0.995463975"
    ],
    [
      "1.160695369",
      "1.004081672"
    ],
    [
      "0.839291471",
      "1.004143272"
    ],
    [
      "1.907176687",
      "1.008714448"
    ],
    [
      "1.734635963",
      "1.088255420"
    ],
    [
      "0.265400594",
      "1.088523603"
    ],
    [
      "1.321415101",
      "1.096832185"
    ],
    [
      "1.000011203",
      "1.096893786"
    ],
    [
      "0.678607304",
      "1.096955386"
    ],
    [
      "1.494588942",
      "1.163497848"
    ],
    [
      "0.505462697",
      "1.163696940"
    ],
    [
      "1.160730935",
      "1.189644299"
    ],
    [
      "0.839327037",
      "1.189705900"
    ],
    [
      "1.885828950",
      "1.195838739"
    ],
    [
      "0.114240665",
      "1.196153365"
    ],
    [
      "1.655417896",
      "1.256058840"
    ],
    [
      "0.344661792",
      "1.256306654"
    ],
    [
      "1.321450667",
      "1.282394812"
    ],
    [
      "1.000046768",
      "1.282456414"
    ],
    [
      "0.678642870",
      "1.282518013"
    ],
    [
      "1.494624508",
      "1.349060475"
    ],
    [
      "0.505498263",
      "1.349259567"
    ],
    [
      "1.827452035",
      "1.371979664"
    ],
    [
      "0.172680141",
      "1.372273545"
    ],
    [
      "1.160766500",
      "1.375206926"
    ],
    [
      "0.839362601",
      "1.375268527"
    ],
    [
      "1.655453462",
      "1.441621467"
    ],
    [
      "0.344697359",
      "1.441869281"
    ],
    [
      "1.321486232",
      "1.467957439"
    ],
    [
      "1.000082332",
      "1.468019041"
    ],
    [
      "0.678678434",
      "1.468080640"
    ],
    [
      "1.494675322",
      "1.534623099"
    ],
    [
      "0.505533827",
      "1.534822194"
    ],
    [
      "1.160802064",
      "1.560769553"
    ],
    [
      "0.839398165",
      "1.560831154"
    ],
    [
      "1.655504277",
      "1.627184091"
    ],
    [
      "0.344732923",
      "1.627431908"
    ],
    [
      "1.350352524",
      "1.651261086"
    ],
    [
      "0.649906011",
      "1.651399045"
    ],
    [
      "0.998315432",
      "1.716192687"
    ],
    [
      "1.189668357",
      "1.744073200"
    ],
    [
      "0.810625743",
      "1.744149559"
    ],
    [
      "1.511197673",
      "1.744198499"
    ],
    [
      "0.486484956",
      "1.747895388"
    ],
    [
      "1.350388089",
      "1.836823713"
    ],
    [
      "0.649941576",
      "1.836961672"
    ],
    [
      "1.092960674",
      "1.902443380"
    ],
    [
      "0.907398047",
      "1.902480260"
    ]
  ],
  "radius": "0.092781315"
}

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

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。

还没有帖子。第一个聊聊这道题的思路？