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 = 98

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 = 98
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[0.0910797982293662903620125690145, 0.0910797982293662903620125690155]见来源署名packomania.com ↗
本站纪录0.091079798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 = 0.091079798

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 = 98 的当前纪录构型,0.091079798
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
    0.091079797→0.091079798
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": [
    [
      "1.121649318",
      "0.099257320"
    ],
    [
      "0.878120455",
      "0.099288443"
    ],
    [
      "1.298818359",
      "0.141604100"
    ],
    [
      "0.700962244",
      "0.141680506"
    ],
    [
      "1.463985226",
      "0.218428735"
    ],
    [
      "0.535815019",
      "0.218547356"
    ],
    [
      "0.999902201",
      "0.234755505"
    ],
    [
      "1.177071241",
      "0.277102284"
    ],
    [
      "0.822743990",
      "0.277147568"
    ],
    [
      "1.610515920",
      "0.326645528"
    ],
    [
      "0.389311990",
      "0.326801599"
    ],
    [
      "0.540240883",
      "0.428794676"
    ],
    [
      "1.277831692",
      "0.428856856"
    ],
    [
      "0.722400076",
      "0.429177886"
    ],
    [
      "1.459990899",
      "0.429233717"
    ],
    [
      "0.909247671",
      "0.437457367"
    ],
    [
      "1.091406818",
      "0.437862121"
    ],
    [
      "1.732524968",
      "0.461907905"
    ],
    [
      "0.267337520",
      "0.462095158"
    ],
    [
      "1.580680113",
      "0.566254772"
    ],
    [
      "0.418577837",
      "0.566509039"
    ],
    [
      "0.808903757",
      "0.589487685"
    ],
    [
      "1.192167269",
      "0.589616693"
    ],
    [
      "0.619960279",
      "0.592583981"
    ],
    [
      "1.380949183",
      "0.593351131"
    ],
    [
      "0.999976716",
      "0.595414193"
    ],
    [
      "1.825111826",
      "0.618782991"
    ],
    [
      "0.174790762",
      "0.618993903"
    ],
    [
      "1.680681292",
      "0.729787222"
    ],
    [
      "0.319249664",
      "0.729961214"
    ],
    [
      "0.501406917",
      "0.730885172"
    ],
    [
      "1.498541458",
      "0.732470417"
    ],
    [
      "0.716983845",
      "0.746754491"
    ],
    [
      "1.100737168",
      "0.747168764"
    ],
    [
      "1.283477127",
      "0.747238482"
    ],
    [
      "0.899632802",
      "0.747444511"
    ],
    [
      "1.884557697",
      "0.790969828"
    ],
    [
      "0.115388903",
      "0.791195929"
    ],
    [
      "1.000393254",
      "0.899199082"
    ],
    [
      "1.740127163",
      "0.901974059"
    ],
    [
      "0.259847805",
      "0.902163240"
    ],
    [
      "0.442005058",
      "0.903087199"
    ],
    [
      "0.624164205",
      "0.903491956"
    ],
    [
      "1.375828182",
      "0.904252498"
    ],
    [
      "1.557987329",
      "0.904657254"
    ],
    [
      "0.807712890",
      "0.904711317"
    ],
    [
      "1.192047026",
      "0.904790553"
    ],
    [
      "1.908474914",
      "0.971552455"
    ],
    [
      "0.091517844",
      "0.971784664"
    ],
    [
      "0.908473342",
      "1.056465888"
    ],
    [
      "1.091703112",
      "1.056820871"
    ],
    [
      "0.532734102",
      "1.061044026"
    ],
    [
      "0.350340563",
      "1.061071153"
    ],
    [
      "0.714893250",
      "1.061448782"
    ],
    [
      "1.284398081",
      "1.061804569"
    ],
    [
      "1.466557228",
      "1.062209325"
    ],
    [
      "1.649928392",
      "1.062432397"
    ],
    [
      "1.895902831",
      "1.153277691"
    ],
    [
      "0.104136377",
      "1.153506680"
    ],
    [
      "0.999783202",
      "1.214087678"
    ],
    [
      "0.817623799",
      "1.214353440"
    ],
    [
      "1.183012970",
      "1.214442660"
    ],
    [
      "0.453649199",
      "1.225140634"
    ],
    [
      "1.364852639",
      "1.225234011"
    ],
    [
      "0.271489855",
      "1.225443679"
    ],
    [
      "0.635808346",
      "1.225545389"
    ],
    [
      "1.547011786",
      "1.225638767"
    ],
    [
      "1.729168638",
      "1.226638630"
    ],
    [
      "1.091093060",
      "1.371709467"
    ],
    [
      "0.908933657",
      "1.371975229"
    ],
    [
      "1.272932729",
      "1.382500818"
    ],
    [
      "0.544378244",
      "1.383097460"
    ],
    [
      "0.727118204",
      "1.383167178"
    ],
    [
      "1.455581684",
      "1.383190838"
    ],
    [
      "0.360387426",
      "1.384438460"
    ],
    [
      "0.177377098",
      "1.384742609"
    ],
    [
      "1.823010566",
      "1.384783932"
    ],
    [
      "1.639336283",
      "1.385107133"
    ],
    [
      "1.172667309",
      "1.534582916"
    ],
    [
      "0.827718441",
      "1.535028005"
    ],
    [
      "1.363661773",
      "1.540457645"
    ],
    [
      "0.635688102",
      "1.540719249"
    ],
    [
      "0.453528697",
      "1.540985010"
    ],
    [
      "1.545818778",
      "1.541429220"
    ],
    [
      "0.271384773",
      "1.543374442"
    ],
    [
      "1.727956429",
      "1.544256714"
    ],
    [
      "1.000268494",
      "1.593411202"
    ],
    [
      "1.263396354",
      "1.692539742"
    ],
    [
      "0.736288339",
      "1.692580076"
    ],
    [
      "0.449301576",
      "1.723095553"
    ],
    [
      "1.550116847",
      "1.723538103"
    ],
    [
      "0.908838392",
      "1.750963273"
    ],
    [
      "1.090997539",
      "1.751368028"
    ],
    [
      "0.604549564",
      "1.818385536"
    ],
    [
      "1.394792293",
      "1.818703230"
    ],
    [
      "0.777485833",
      "1.880759646"
    ],
    [
      "1.220010578",
      "1.881388366"
    ],
    [
      "0.999567437",
      "1.908920099"
    ]
  ],
  "radius": "0.091079798"
}
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": 98
}

当前第一名的答案

{
  "centers": [
    [
      "1.121649318",
      "0.099257320"
    ],
    [
      "0.878120455",
      "0.099288443"
    ],
    [
      "1.298818359",
      "0.141604100"
    ],
    [
      "0.700962244",
      "0.141680506"
    ],
    [
      "1.463985226",
      "0.218428735"
    ],
    [
      "0.535815019",
      "0.218547356"
    ],
    [
      "0.999902201",
      "0.234755505"
    ],
    [
      "1.177071241",
      "0.277102284"
    ],
    [
      "0.822743990",
      "0.277147568"
    ],
    [
      "1.610515920",
      "0.326645528"
    ],
    [
      "0.389311990",
      "0.326801599"
    ],
    [
      "0.540240883",
      "0.428794676"
    ],
    [
      "1.277831692",
      "0.428856856"
    ],
    [
      "0.722400076",
      "0.429177886"
    ],
    [
      "1.459990899",
      "0.429233717"
    ],
    [
      "0.909247671",
      "0.437457367"
    ],
    [
      "1.091406818",
      "0.437862121"
    ],
    [
      "1.732524968",
      "0.461907905"
    ],
    [
      "0.267337520",
      "0.462095158"
    ],
    [
      "1.580680113",
      "0.566254772"
    ],
    [
      "0.418577837",
      "0.566509039"
    ],
    [
      "0.808903757",
      "0.589487685"
    ],
    [
      "1.192167269",
      "0.589616693"
    ],
    [
      "0.619960279",
      "0.592583981"
    ],
    [
      "1.380949183",
      "0.593351131"
    ],
    [
      "0.999976716",
      "0.595414193"
    ],
    [
      "1.825111826",
      "0.618782991"
    ],
    [
      "0.174790762",
      "0.618993903"
    ],
    [
      "1.680681292",
      "0.729787222"
    ],
    [
      "0.319249664",
      "0.729961214"
    ],
    [
      "0.501406917",
      "0.730885172"
    ],
    [
      "1.498541458",
      "0.732470417"
    ],
    [
      "0.716983845",
      "0.746754491"
    ],
    [
      "1.100737168",
      "0.747168764"
    ],
    [
      "1.283477127",
      "0.747238482"
    ],
    [
      "0.899632802",
      "0.747444511"
    ],
    [
      "1.884557697",
      "0.790969828"
    ],
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      "0.115388903",
      "0.791195929"
    ],
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      "1.000393254",
      "0.899199082"
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      "1.740127163",
      "0.901974059"
    ],
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      "0.902163240"
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      "0.442005058",
      "0.903087199"
    ],
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      "0.624164205",
      "0.903491956"
    ],
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      "1.375828182",
      "0.904252498"
    ],
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      "1.557987329",
      "0.904657254"
    ],
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      "0.807712890",
      "0.904711317"
    ],
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      "1.192047026",
      "0.904790553"
    ],
    [
      "1.908474914",
      "0.971552455"
    ],
    [
      "0.091517844",
      "0.971784664"
    ],
    [
      "0.908473342",
      "1.056465888"
    ],
    [
      "1.091703112",
      "1.056820871"
    ],
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      "0.532734102",
      "1.061044026"
    ],
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      "0.350340563",
      "1.061071153"
    ],
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      "0.714893250",
      "1.061448782"
    ],
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      "1.284398081",
      "1.061804569"
    ],
    [
      "1.466557228",
      "1.062209325"
    ],
    [
      "1.649928392",
      "1.062432397"
    ],
    [
      "1.895902831",
      "1.153277691"
    ],
    [
      "0.104136377",
      "1.153506680"
    ],
    [
      "0.999783202",
      "1.214087678"
    ],
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      "0.817623799",
      "1.214353440"
    ],
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      "1.183012970",
      "1.214442660"
    ],
    [
      "0.453649199",
      "1.225140634"
    ],
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      "1.364852639",
      "1.225234011"
    ],
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      "0.271489855",
      "1.225443679"
    ],
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      "0.635808346",
      "1.225545389"
    ],
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      "1.547011786",
      "1.225638767"
    ],
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      "1.729168638",
      "1.226638630"
    ],
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      "1.091093060",
      "1.371709467"
    ],
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      "0.908933657",
      "1.371975229"
    ],
    [
      "1.272932729",
      "1.382500818"
    ],
    [
      "0.544378244",
      "1.383097460"
    ],
    [
      "0.727118204",
      "1.383167178"
    ],
    [
      "1.455581684",
      "1.383190838"
    ],
    [
      "0.360387426",
      "1.384438460"
    ],
    [
      "0.177377098",
      "1.384742609"
    ],
    [
      "1.823010566",
      "1.384783932"
    ],
    [
      "1.639336283",
      "1.385107133"
    ],
    [
      "1.172667309",
      "1.534582916"
    ],
    [
      "0.827718441",
      "1.535028005"
    ],
    [
      "1.363661773",
      "1.540457645"
    ],
    [
      "0.635688102",
      "1.540719249"
    ],
    [
      "0.453528697",
      "1.540985010"
    ],
    [
      "1.545818778",
      "1.541429220"
    ],
    [
      "0.271384773",
      "1.543374442"
    ],
    [
      "1.727956429",
      "1.544256714"
    ],
    [
      "1.000268494",
      "1.593411202"
    ],
    [
      "1.263396354",
      "1.692539742"
    ],
    [
      "0.736288339",
      "1.692580076"
    ],
    [
      "0.449301576",
      "1.723095553"
    ],
    [
      "1.550116847",
      "1.723538103"
    ],
    [
      "0.908838392",
      "1.750963273"
    ],
    [
      "1.090997539",
      "1.751368028"
    ],
    [
      "0.604549564",
      "1.818385536"
    ],
    [
      "1.394792293",
      "1.818703230"
    ],
    [
      "0.777485833",
      "1.880759646"
    ],
    [
      "1.220010578",
      "1.881388366"
    ],
    [
      "0.999567437",
      "1.908920099"
    ]
  ],
  "radius": "0.091079798"
}

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

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