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P01 · 装箱与覆盖 · 经典问题 · 难

单位正方形内的等圆装箱 · n = 91

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 = 91
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[0.0534967198835, 0.0534967198845]见来源署名packomania.com ↗
本站纪录暂无选手纪录——

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：起步布局

1y0
0x1
已验证构造r = 0.053496719

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 = 91 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"。

起步布局答案

{
  "centers": [
    [
      "0.05349672",
      "0.05349672"
    ],
    [
      "0.238817929",
      "0.05349672"
    ],
    [
      "0.424893549",
      "0.05349672"
    ],
    [
      "0.619214583",
      "0.05349672"
    ],
    [
      "0.83306954",
      "0.05349672"
    ],
    [
      "0.94650328",
      "0.05349672"
    ],
    [
      "0.726142061",
      "0.057253112"
    ],
    [
      "0.522054066",
      "0.098301074"
    ],
    [
      "0.331855739",
      "0.106331955"
    ],
    [
      "0.146157324",
      "0.106990725"
    ],
    [
      "0.88978641",
      "0.144220437"
    ],
    [
      "0.782858931",
      "0.147976829"
    ],
    [
      "0.609436033",
      "0.160042373"
    ],
    [
      "0.4241313",
      "0.160487444"
    ],
    [
      "0.238814794",
      "0.16049016"
    ],
    [
      "0.05349672",
      "0.16049016"
    ],
    [
      "0.516912183",
      "0.213772563"
    ],
    [
      "0.33147383",
      "0.21398688"
    ],
    [
      "0.146155757",
      "0.21398688"
    ],
    [
      "0.700477568",
      "0.216247666"
    ],
    [
      "0.94650328",
      "0.234944154"
    ],
    [
      "0.803313144",
      "0.253830735"
    ],
    [
      "0.424134435",
      "0.267480884"
    ],
    [
      "0.05349672",
      "0.2674836"
    ],
    [
      "0.238814794",
      "0.2674836"
    ],
    [
      "0.607953718",
      "0.269977856"
    ],
    [
      "0.331473831",
      "0.320980319"
    ],
    [
      "0.146155757",
      "0.320980319"
    ],
    [
      "0.886076308",
      "0.323240117"
    ],
    [
      "0.700757876",
      "0.323240738"
    ],
    [
      "0.515300295",
      "0.323484297"
    ],
    [
      "0.238814794",
      "0.374477039"
    ],
    [
      "0.05349672",
      "0.374477039"
    ],
    [
      "0.793417271",
      "0.376736837"
    ],
    [
      "0.607964943",
      "0.376971295"
    ],
    [
      "0.422641306",
      "0.376981101"
    ],
    [
      "0.94650328",
      "0.411536081"
    ],
    [
      "0.146155757",
      "0.427973759"
    ],
    [
      "0.700758593",
      "0.430234178"
    ],
    [
      "0.515300392",
      "0.430477736"
    ],
    [
      "0.329982269",
      "0.430477821"
    ],
    [
      "0.853844243",
      "0.465032801"
    ],
    [
      "0.05349672",
      "0.481470479"
    ],
    [
      "0.607965041",
      "0.483964735"
    ],
    [
      "0.422641306",
      "0.48397454"
    ],
    [
      "0.237323233",
      "0.483974541"
    ],
    [
      "0.94650328",
      "0.518529521"
    ],
    [
      "0.76118438",
      "0.518530953"
    ],
    [
      "0.515300392",
      "0.537471176"
    ],
    [
      "0.32998227",
      "0.53747126"
    ],
    [
      "0.144664196",
      "0.53747126"
    ],
    [
      "0.853844243",
      "0.572026241"
    ],
    [
      "0.668390667",
      "0.57226162"
    ],
    [
      "0.422641404",
      "0.59096798"
    ],
    [
      "0.237323233",
      "0.59096798"
    ],
    [
      "0.05349672",
      "0.593472042"
    ],
    [
      "0.94650328",
      "0.625522961"
    ],
    [
      "0.76118438",
      "0.625524393"
    ],
    [
      "0.575726018",
      "0.625768061"
    ],
    [
      "0.329982367",
      "0.6444647"
    ],
    [
      "0.146155757",
      "0.646968762"
    ],
    [
      "0.853844243",
      "0.679019681"
    ],
    [
      "0.668390667",
      "0.679255059"
    ],
    [
      "0.48306703",
      "0.679264865"
    ],
    [
      "0.238814841",
      "0.7004654"
    ],
    [
      "0.05349672",
      "0.700465481"
    ],
    [
      "0.94650328",
      "0.732516401"
    ],
    [
      "0.76118438",
      "0.732517833"
    ],
    [
      "0.575726115",
      "0.732761501"
    ],
    [
      "0.390407993",
      "0.732761585"
    ],
    [
      "0.146155757",
      "0.753962201"
    ],
    [
      "0.853844243",
      "0.78601312"
    ],
    [
      "0.66839232",
      "0.786248499"
    ],
    [
      "0.481325044",
      "0.789286723"
    ],
    [
      "0.289917488",
      "0.794466014"
    ],
    [
      "0.05349672",
      "0.807458921"
    ],
    [
      "0.94650328",
      "0.83950984"
    ],
    [
      "0.761186033",
      "0.839511272"
    ],
    [
      "0.386880201",
      "0.83969685"
    ],
    [
      "0.575738897",
      "0.83975494"
    ],
    [
      "0.197258476",
      "0.847962777"
    ],
    [
      "0.853567036",
      "0.893486697"
    ],
    [
      "0.664566695",
      "0.899397361"
    ],
    [
      "0.468218289",
      "0.909207365"
    ],
    [
      "0.05349672",
      "0.914452361"
    ],
    [
      "0.261756197",
      "0.933330453"
    ],
    [
      "0.155576761",
      "0.94650328"
    ],
    [
      "0.367935634",
      "0.94650328"
    ],
    [
      "0.568500944",
      "0.94650328"
    ],
    [
      "0.760632446",
      "0.94650328"
    ],
    [
      "0.946501626",
      "0.94650328"
    ]
  ],
  "radius": "0.053496719"
}
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": 91
}

起步布局答案

{
  "centers": [
    [
      "0.05349672",
      "0.05349672"
    ],
    [
      "0.238817929",
      "0.05349672"
    ],
    [
      "0.424893549",
      "0.05349672"
    ],
    [
      "0.619214583",
      "0.05349672"
    ],
    [
      "0.83306954",
      "0.05349672"
    ],
    [
      "0.94650328",
      "0.05349672"
    ],
    [
      "0.726142061",
      "0.057253112"
    ],
    [
      "0.522054066",
      "0.098301074"
    ],
    [
      "0.331855739",
      "0.106331955"
    ],
    [
      "0.146157324",
      "0.106990725"
    ],
    [
      "0.88978641",
      "0.144220437"
    ],
    [
      "0.782858931",
      "0.147976829"
    ],
    [
      "0.609436033",
      "0.160042373"
    ],
    [
      "0.4241313",
      "0.160487444"
    ],
    [
      "0.238814794",
      "0.16049016"
    ],
    [
      "0.05349672",
      "0.16049016"
    ],
    [
      "0.516912183",
      "0.213772563"
    ],
    [
      "0.33147383",
      "0.21398688"
    ],
    [
      "0.146155757",
      "0.21398688"
    ],
    [
      "0.700477568",
      "0.216247666"
    ],
    [
      "0.94650328",
      "0.234944154"
    ],
    [
      "0.803313144",
      "0.253830735"
    ],
    [
      "0.424134435",
      "0.267480884"
    ],
    [
      "0.05349672",
      "0.2674836"
    ],
    [
      "0.238814794",
      "0.2674836"
    ],
    [
      "0.607953718",
      "0.269977856"
    ],
    [
      "0.331473831",
      "0.320980319"
    ],
    [
      "0.146155757",
      "0.320980319"
    ],
    [
      "0.886076308",
      "0.323240117"
    ],
    [
      "0.700757876",
      "0.323240738"
    ],
    [
      "0.515300295",
      "0.323484297"
    ],
    [
      "0.238814794",
      "0.374477039"
    ],
    [
      "0.05349672",
      "0.374477039"
    ],
    [
      "0.793417271",
      "0.376736837"
    ],
    [
      "0.607964943",
      "0.376971295"
    ],
    [
      "0.422641306",
      "0.376981101"
    ],
    [
      "0.94650328",
      "0.411536081"
    ],
    [
      "0.146155757",
      "0.427973759"
    ],
    [
      "0.700758593",
      "0.430234178"
    ],
    [
      "0.515300392",
      "0.430477736"
    ],
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      "0.329982269",
      "0.430477821"
    ],
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      "0.853844243",
      "0.465032801"
    ],
    [
      "0.05349672",
      "0.481470479"
    ],
    [
      "0.607965041",
      "0.483964735"
    ],
    [
      "0.422641306",
      "0.48397454"
    ],
    [
      "0.237323233",
      "0.483974541"
    ],
    [
      "0.94650328",
      "0.518529521"
    ],
    [
      "0.76118438",
      "0.518530953"
    ],
    [
      "0.515300392",
      "0.537471176"
    ],
    [
      "0.32998227",
      "0.53747126"
    ],
    [
      "0.144664196",
      "0.53747126"
    ],
    [
      "0.853844243",
      "0.572026241"
    ],
    [
      "0.668390667",
      "0.57226162"
    ],
    [
      "0.422641404",
      "0.59096798"
    ],
    [
      "0.237323233",
      "0.59096798"
    ],
    [
      "0.05349672",
      "0.593472042"
    ],
    [
      "0.94650328",
      "0.625522961"
    ],
    [
      "0.76118438",
      "0.625524393"
    ],
    [
      "0.575726018",
      "0.625768061"
    ],
    [
      "0.329982367",
      "0.6444647"
    ],
    [
      "0.146155757",
      "0.646968762"
    ],
    [
      "0.853844243",
      "0.679019681"
    ],
    [
      "0.668390667",
      "0.679255059"
    ],
    [
      "0.48306703",
      "0.679264865"
    ],
    [
      "0.238814841",
      "0.7004654"
    ],
    [
      "0.05349672",
      "0.700465481"
    ],
    [
      "0.94650328",
      "0.732516401"
    ],
    [
      "0.76118438",
      "0.732517833"
    ],
    [
      "0.575726115",
      "0.732761501"
    ],
    [
      "0.390407993",
      "0.732761585"
    ],
    [
      "0.146155757",
      "0.753962201"
    ],
    [
      "0.853844243",
      "0.78601312"
    ],
    [
      "0.66839232",
      "0.786248499"
    ],
    [
      "0.481325044",
      "0.789286723"
    ],
    [
      "0.289917488",
      "0.794466014"
    ],
    [
      "0.05349672",
      "0.807458921"
    ],
    [
      "0.94650328",
      "0.83950984"
    ],
    [
      "0.761186033",
      "0.839511272"
    ],
    [
      "0.386880201",
      "0.83969685"
    ],
    [
      "0.575738897",
      "0.83975494"
    ],
    [
      "0.197258476",
      "0.847962777"
    ],
    [
      "0.853567036",
      "0.893486697"
    ],
    [
      "0.664566695",
      "0.899397361"
    ],
    [
      "0.468218289",
      "0.909207365"
    ],
    [
      "0.05349672",
      "0.914452361"
    ],
    [
      "0.261756197",
      "0.933330453"
    ],
    [
      "0.155576761",
      "0.94650328"
    ],
    [
      "0.367935634",
      "0.94650328"
    ],
    [
      "0.568500944",
      "0.94650328"
    ],
    [
      "0.760632446",
      "0.94650328"
    ],
    [
      "0.946501626",
      "0.94650328"
    ]
  ],
  "radius": "0.053496719"
}

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

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