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 = 106

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 = 106
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[0.0875516110593197166678008141555, 0.0875516110593197166678008141565]见来源署名packomania.com ↗
本站纪录暂无选手纪录——

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:起步布局

2y0
0x2
已验证构造r = 0.087551610

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 = 106 的当前构型
构型与历史

起步布局

查看构型、求解笔记与纪录历史。

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。
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"。

起步布局答案

{
  "centers": [
    [
      "0.93158944",
      "0.090119756"
    ],
    [
      "1.10665385",
      "0.093806302"
    ],
    [
      "0.759044416",
      "0.119941781"
    ],
    [
      "1.277790471",
      "0.130865651"
    ],
    [
      "0.587490522",
      "0.186121635"
    ],
    [
      "1.438696777",
      "0.199933003"
    ],
    [
      "0.924230318",
      "0.265102299"
    ],
    [
      "1.119017325",
      "0.268472506"
    ],
    [
      "0.439619856",
      "0.279905453"
    ],
    [
      "0.726903873",
      "0.29207"
    ],
    [
      "1.583446999",
      "0.298464782"
    ],
    [
      "1.290153946",
      "0.305531855"
    ],
    [
      "0.311608963",
      "0.401115398"
    ],
    [
      "1.019894936",
      "0.412818931"
    ],
    [
      "0.844999873",
      "0.421354466"
    ],
    [
      "1.706710356",
      "0.422832316"
    ],
    [
      "0.487658664",
      "0.448290165"
    ],
    [
      "1.191031557",
      "0.44987828"
    ],
    [
      "0.670148156",
      "0.457720014"
    ],
    [
      "1.365661255",
      "0.463518515"
    ],
    [
      "1.54023537",
      "0.477120598"
    ],
    [
      "1.803947371",
      "0.568455464"
    ],
    [
      "0.361867322",
      "0.570100168"
    ],
    [
      "0.917379994",
      "0.58079805"
    ],
    [
      "0.187397683",
      "0.584982486"
    ],
    [
      "1.091964751",
      "0.594262858"
    ],
    [
      "0.571194912",
      "0.602182444"
    ],
    [
      "1.266538866",
      "0.60786494"
    ],
    [
      "0.745779669",
      "0.615647252"
    ],
    [
      "1.441168563",
      "0.621505175"
    ],
    [
      "1.637472386",
      "0.622743746"
    ],
    [
      "1.871577055",
      "0.729971297"
    ],
    [
      "0.29755399",
      "0.732964939"
    ],
    [
      "0.992887302",
      "0.73878471"
    ],
    [
      "0.472128105",
      "0.746567022"
    ],
    [
      "0.12308435",
      "0.747847257"
    ],
    [
      "1.167472059",
      "0.752249518"
    ],
    [
      "0.64670222",
      "0.760169105"
    ],
    [
      "1.342046174",
      "0.7658516"
    ],
    [
      "1.538405579",
      "0.767128323"
    ],
    [
      "0.821286977",
      "0.773633912"
    ],
    [
      "1.715057903",
      "0.808688473"
    ],
    [
      "0.373061298",
      "0.890951599"
    ],
    [
      "1.068394611",
      "0.89677137"
    ],
    [
      "1.907108777",
      "0.901431599"
    ],
    [
      "0.547635413",
      "0.904553682"
    ],
    [
      "1.242979368",
      "0.910236178"
    ],
    [
      "0.722209529",
      "0.918155765"
    ],
    [
      "0.091065586",
      "0.919998171"
    ],
    [
      "1.417553483",
      "0.92383826"
    ],
    [
      "0.896794286",
      "0.931620572"
    ],
    [
      "1.592378451",
      "0.933705835"
    ],
    [
      "1.757402533",
      "0.992256589"
    ],
    [
      "0.242597467",
      "1.007743411"
    ],
    [
      "0.407621549",
      "1.066294165"
    ],
    [
      "1.103205714",
      "1.068379428"
    ],
    [
      "0.582446517",
      "1.07616174"
    ],
    [
      "1.908934414",
      "1.080001829"
    ],
    [
      "1.277790471",
      "1.081844235"
    ],
    [
      "0.757020632",
      "1.089763822"
    ],
    [
      "1.452364587",
      "1.095446318"
    ],
    [
      "0.092891223",
      "1.098568401"
    ],
    [
      "0.931605389",
      "1.10322863"
    ],
    [
      "1.626938702",
      "1.109048401"
    ],
    [
      "0.287202605",
      "1.193633835"
    ],
    [
      "1.178713023",
      "1.226366088"
    ],
    [
      "0.461594421",
      "1.232871677"
    ],
    [
      "0.657953826",
      "1.2341484"
    ],
    [
      "1.35329778",
      "1.239830895"
    ],
    [
      "0.832527941",
      "1.247750482"
    ],
    [
      "1.87691565",
      "1.252152743"
    ],
    [
      "1.527871895",
      "1.253432978"
    ],
    [
      "1.007112698",
      "1.26121529"
    ],
    [
      "1.70244601",
      "1.267035061"
    ],
    [
      "0.128422945",
      "1.270028703"
    ],
    [
      "0.362527614",
      "1.377256254"
    ],
    [
      "0.558831437",
      "1.378494825"
    ],
    [
      "1.254220331",
      "1.384352748"
    ],
    [
      "0.733461134",
      "1.39213506"
    ],
    [
      "1.428805088",
      "1.397817556"
    ],
    [
      "0.908035249",
      "1.405737142"
    ],
    [
      "1.812602317",
      "1.415017514"
    ],
    [
      "1.082620006",
      "1.41920195"
    ],
    [
      "1.638132678",
      "1.429899832"
    ],
    [
      "0.196052629",
      "1.431544536"
    ],
    [
      "0.45976463",
      "1.522879402"
    ],
    [
      "0.634338745",
      "1.536481485"
    ],
    [
      "1.329851844",
      "1.542279986"
    ],
    [
      "0.808968443",
      "1.55012172"
    ],
    [
      "1.512341336",
      "1.551709835"
    ],
    [
      "0.293289644",
      "1.577167684"
    ],
    [
      "1.155000127",
      "1.578645534"
    ],
    [
      "0.980105064",
      "1.587181069"
    ],
    [
      "1.688100251",
      "1.599134506"
    ],
    [
      "0.709846054",
      "1.694468145"
    ],
    [
      "0.416553001",
      "1.701535218"
    ],
    [
      "1.273096127",
      "1.70793"
    ],
    [
      "1.560380144",
      "1.720094547"
    ],
    [
      "0.880982675",
      "1.731527494"
    ],
    [
      "1.074826506",
      "1.734866348"
    ],
    [
      "0.561303223",
      "1.800066997"
    ],
    [
      "1.412509478",
      "1.813878365"
    ],
    [
      "0.722209529",
      "1.869134349"
    ],
    [
      "1.240955584",
      "1.880058219"
    ],
    [
      "0.89334615",
      "1.906193698"
    ],
    [
      "1.06841056",
      "1.909880244"
    ]
  ],
  "radius": "0.08755161"
}
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": 106
}

起步布局答案

{
  "centers": [
    [
      "0.93158944",
      "0.090119756"
    ],
    [
      "1.10665385",
      "0.093806302"
    ],
    [
      "0.759044416",
      "0.119941781"
    ],
    [
      "1.277790471",
      "0.130865651"
    ],
    [
      "0.587490522",
      "0.186121635"
    ],
    [
      "1.438696777",
      "0.199933003"
    ],
    [
      "0.924230318",
      "0.265102299"
    ],
    [
      "1.119017325",
      "0.268472506"
    ],
    [
      "0.439619856",
      "0.279905453"
    ],
    [
      "0.726903873",
      "0.29207"
    ],
    [
      "1.583446999",
      "0.298464782"
    ],
    [
      "1.290153946",
      "0.305531855"
    ],
    [
      "0.311608963",
      "0.401115398"
    ],
    [
      "1.019894936",
      "0.412818931"
    ],
    [
      "0.844999873",
      "0.421354466"
    ],
    [
      "1.706710356",
      "0.422832316"
    ],
    [
      "0.487658664",
      "0.448290165"
    ],
    [
      "1.191031557",
      "0.44987828"
    ],
    [
      "0.670148156",
      "0.457720014"
    ],
    [
      "1.365661255",
      "0.463518515"
    ],
    [
      "1.54023537",
      "0.477120598"
    ],
    [
      "1.803947371",
      "0.568455464"
    ],
    [
      "0.361867322",
      "0.570100168"
    ],
    [
      "0.917379994",
      "0.58079805"
    ],
    [
      "0.187397683",
      "0.584982486"
    ],
    [
      "1.091964751",
      "0.594262858"
    ],
    [
      "0.571194912",
      "0.602182444"
    ],
    [
      "1.266538866",
      "0.60786494"
    ],
    [
      "0.745779669",
      "0.615647252"
    ],
    [
      "1.441168563",
      "0.621505175"
    ],
    [
      "1.637472386",
      "0.622743746"
    ],
    [
      "1.871577055",
      "0.729971297"
    ],
    [
      "0.29755399",
      "0.732964939"
    ],
    [
      "0.992887302",
      "0.73878471"
    ],
    [
      "0.472128105",
      "0.746567022"
    ],
    [
      "0.12308435",
      "0.747847257"
    ],
    [
      "1.167472059",
      "0.752249518"
    ],
    [
      "0.64670222",
      "0.760169105"
    ],
    [
      "1.342046174",
      "0.7658516"
    ],
    [
      "1.538405579",
      "0.767128323"
    ],
    [
      "0.821286977",
      "0.773633912"
    ],
    [
      "1.715057903",
      "0.808688473"
    ],
    [
      "0.373061298",
      "0.890951599"
    ],
    [
      "1.068394611",
      "0.89677137"
    ],
    [
      "1.907108777",
      "0.901431599"
    ],
    [
      "0.547635413",
      "0.904553682"
    ],
    [
      "1.242979368",
      "0.910236178"
    ],
    [
      "0.722209529",
      "0.918155765"
    ],
    [
      "0.091065586",
      "0.919998171"
    ],
    [
      "1.417553483",
      "0.92383826"
    ],
    [
      "0.896794286",
      "0.931620572"
    ],
    [
      "1.592378451",
      "0.933705835"
    ],
    [
      "1.757402533",
      "0.992256589"
    ],
    [
      "0.242597467",
      "1.007743411"
    ],
    [
      "0.407621549",
      "1.066294165"
    ],
    [
      "1.103205714",
      "1.068379428"
    ],
    [
      "0.582446517",
      "1.07616174"
    ],
    [
      "1.908934414",
      "1.080001829"
    ],
    [
      "1.277790471",
      "1.081844235"
    ],
    [
      "0.757020632",
      "1.089763822"
    ],
    [
      "1.452364587",
      "1.095446318"
    ],
    [
      "0.092891223",
      "1.098568401"
    ],
    [
      "0.931605389",
      "1.10322863"
    ],
    [
      "1.626938702",
      "1.109048401"
    ],
    [
      "0.287202605",
      "1.193633835"
    ],
    [
      "1.178713023",
      "1.226366088"
    ],
    [
      "0.461594421",
      "1.232871677"
    ],
    [
      "0.657953826",
      "1.2341484"
    ],
    [
      "1.35329778",
      "1.239830895"
    ],
    [
      "0.832527941",
      "1.247750482"
    ],
    [
      "1.87691565",
      "1.252152743"
    ],
    [
      "1.527871895",
      "1.253432978"
    ],
    [
      "1.007112698",
      "1.26121529"
    ],
    [
      "1.70244601",
      "1.267035061"
    ],
    [
      "0.128422945",
      "1.270028703"
    ],
    [
      "0.362527614",
      "1.377256254"
    ],
    [
      "0.558831437",
      "1.378494825"
    ],
    [
      "1.254220331",
      "1.384352748"
    ],
    [
      "0.733461134",
      "1.39213506"
    ],
    [
      "1.428805088",
      "1.397817556"
    ],
    [
      "0.908035249",
      "1.405737142"
    ],
    [
      "1.812602317",
      "1.415017514"
    ],
    [
      "1.082620006",
      "1.41920195"
    ],
    [
      "1.638132678",
      "1.429899832"
    ],
    [
      "0.196052629",
      "1.431544536"
    ],
    [
      "0.45976463",
      "1.522879402"
    ],
    [
      "0.634338745",
      "1.536481485"
    ],
    [
      "1.329851844",
      "1.542279986"
    ],
    [
      "0.808968443",
      "1.55012172"
    ],
    [
      "1.512341336",
      "1.551709835"
    ],
    [
      "0.293289644",
      "1.577167684"
    ],
    [
      "1.155000127",
      "1.578645534"
    ],
    [
      "0.980105064",
      "1.587181069"
    ],
    [
      "1.688100251",
      "1.599134506"
    ],
    [
      "0.709846054",
      "1.694468145"
    ],
    [
      "0.416553001",
      "1.701535218"
    ],
    [
      "1.273096127",
      "1.70793"
    ],
    [
      "1.560380144",
      "1.720094547"
    ],
    [
      "0.880982675",
      "1.731527494"
    ],
    [
      "1.074826506",
      "1.734866348"
    ],
    [
      "0.561303223",
      "1.800066997"
    ],
    [
      "1.412509478",
      "1.813878365"
    ],
    [
      "0.722209529",
      "1.869134349"
    ],
    [
      "1.240955584",
      "1.880058219"
    ],
    [
      "0.89334615",
      "1.906193698"
    ],
    [
      "1.06841056",
      "1.909880244"
    ]
  ],
  "radius": "0.08755161"
}

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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