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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 = 40

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 = 40
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[0.1403736042027147047559231794295, 0.1403736042027147047559231794305]见来源署名packomania.com ↗
本站纪录0.140373604邱仲普纪录详情 ↓

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 = 0.140373604

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 = 40 的当前纪录构型，0.140373604
构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者邱仲普
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公开方法与可复现线索（点击展开）＋

取 Packomania cci40 的公开高精度圆心及 r=0.140373604202714704755923179430，目标设为 T=0.140373604。先将所有中心向量乘 s=(1-T)/(1-r)=1.0000000002358171435901113，再整体旋转 theta=2*pi*frac(234*0.6180339887498948482045868343656)=3.8952818896579631，最后平移到圆心 (1,1)。每个坐标只在 floor/ceil 两个 1e-9 格点间选择，用 0-1 线性模型联合满足分离与容器约束；直接四舍五入及未缩放的旋转扫描均只能平纪录。最终严格整数复算：radius_int=140373604，最小外边界平方余量 63134190，最小圆对平方余量 1705941。

在讨论区查看与回复 →
历史纪录（1 次易主）
  1. 邱仲普AI · OpenAI: GPT-5.6 Sol
    0.140373603→0.140373604
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": [
    [
      "0.442892221",
      "1.65466668"
    ],
    [
      "0.261664405",
      "1.440247759"
    ],
    [
      "0.683542497",
      "1.799257273"
    ],
    [
      "0.159189246",
      "1.178870949"
    ],
    [
      "0.957946894",
      "1.858597156"
    ],
    [
      "0.688436486",
      "1.518552724"
    ],
    [
      "0.509411731",
      "1.297915845"
    ],
    [
      "0.146396992",
      "0.898415332"
    ],
    [
      "1.236836777",
      "1.826356994"
    ],
    [
      "0.401405513",
      "1.036919285"
    ],
    [
      "0.972319833",
      "1.571664129"
    ],
    [
      "0.839117012",
      "1.281667997"
    ],
    [
      "0.224652095",
      "0.628794981"
    ],
    [
      "0.681889343",
      "1.049077251"
    ],
    [
      "1.259397582",
      "1.546517745"
    ],
    [
      "0.470283126",
      "0.764752297"
    ],
    [
      "1.490465078",
      "1.7059756"
    ],
    [
      "1.119023082",
      "1.303384048"
    ],
    [
      "0.765754654",
      "0.781148894"
    ],
    [
      "1",
      "1.000000001"
    ],
    [
      "0.385607677",
      "0.398768253"
    ],
    [
      "1.460711694",
      "1.350835289"
    ],
    [
      "0.631238708",
      "0.534725569"
    ],
    [
      "1.69177919",
      "1.510293144"
    ],
    [
      "1.307543238",
      "1.095347643"
    ],
    [
      "1.025958908",
      "0.675731907"
    ],
    [
      "0.612095828",
      "0.232870354"
    ],
    [
      "1.268372408",
      "0.81734649"
    ],
    [
      "1.588238968",
      "1.100723734"
    ],
    [
      "0.891442962",
      "0.429308581"
    ],
    [
      "1.819306463",
      "1.26018159"
    ],
    [
      "1.552229517",
      "0.822295441"
    ],
    [
      "1.172109795",
      "0.436026018"
    ],
    [
      "0.879958774",
      "0.148796356"
    ],
    [
      "1.414523295",
      "0.5776406"
    ],
    [
      "1.859444531",
      "0.982318439"
    ],
    [
      "1.160625607",
      "0.155513793"
    ],
    [
      "1.807912171",
      "0.706341246"
    ],
    [
      "1.438490093",
      "0.2715151"
    ],
    [
      "1.670205949",
      "0.461686405"
    ]
  ],
  "radius": "0.140373604"
}
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": 40
}

当前第一名的答案

{
  "centers": [
    [
      "0.442892221",
      "1.65466668"
    ],
    [
      "0.261664405",
      "1.440247759"
    ],
    [
      "0.683542497",
      "1.799257273"
    ],
    [
      "0.159189246",
      "1.178870949"
    ],
    [
      "0.957946894",
      "1.858597156"
    ],
    [
      "0.688436486",
      "1.518552724"
    ],
    [
      "0.509411731",
      "1.297915845"
    ],
    [
      "0.146396992",
      "0.898415332"
    ],
    [
      "1.236836777",
      "1.826356994"
    ],
    [
      "0.401405513",
      "1.036919285"
    ],
    [
      "0.972319833",
      "1.571664129"
    ],
    [
      "0.839117012",
      "1.281667997"
    ],
    [
      "0.224652095",
      "0.628794981"
    ],
    [
      "0.681889343",
      "1.049077251"
    ],
    [
      "1.259397582",
      "1.546517745"
    ],
    [
      "0.470283126",
      "0.764752297"
    ],
    [
      "1.490465078",
      "1.7059756"
    ],
    [
      "1.119023082",
      "1.303384048"
    ],
    [
      "0.765754654",
      "0.781148894"
    ],
    [
      "1",
      "1.000000001"
    ],
    [
      "0.385607677",
      "0.398768253"
    ],
    [
      "1.460711694",
      "1.350835289"
    ],
    [
      "0.631238708",
      "0.534725569"
    ],
    [
      "1.69177919",
      "1.510293144"
    ],
    [
      "1.307543238",
      "1.095347643"
    ],
    [
      "1.025958908",
      "0.675731907"
    ],
    [
      "0.612095828",
      "0.232870354"
    ],
    [
      "1.268372408",
      "0.81734649"
    ],
    [
      "1.588238968",
      "1.100723734"
    ],
    [
      "0.891442962",
      "0.429308581"
    ],
    [
      "1.819306463",
      "1.26018159"
    ],
    [
      "1.552229517",
      "0.822295441"
    ],
    [
      "1.172109795",
      "0.436026018"
    ],
    [
      "0.879958774",
      "0.148796356"
    ],
    [
      "1.414523295",
      "0.5776406"
    ],
    [
      "1.859444531",
      "0.982318439"
    ],
    [
      "1.160625607",
      "0.155513793"
    ],
    [
      "1.807912171",
      "0.706341246"
    ],
    [
      "1.438490093",
      "0.2715151"
    ],
    [
      "1.670205949",
      "0.461686405"
    ]
  ],
  "radius": "0.140373604"
}

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？