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 = 159

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 = 159
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[0.0721282496203149720552044883605, 0.0721282496203149720552044883615]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 = 0.072128249

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 = 159 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": [
    [
      "1.641173138",
      "0.77783417"
    ],
    [
      "1.577415422",
      "1.30030142"
    ],
    [
      "0.161888858",
      "1.124028117"
    ],
    [
      "1.71035184",
      "1.356317902"
    ],
    [
      "0.367104583",
      "0.824901963"
    ],
    [
      "0.474422072",
      "0.336232268"
    ],
    [
      "1.482434939",
      "0.538441463"
    ],
    [
      "1.072301857",
      "0.873840431"
    ],
    [
      "1.434908522",
      "1.322700551"
    ],
    [
      "0.86193377",
      "1.414300302"
    ],
    [
      "0.825811505",
      "1.165362408"
    ],
    [
      "1.171656326",
      "1.62590355"
    ],
    [
      "0.422064565",
      "0.691525286"
    ],
    [
      "0.669248736",
      "0.176959962"
    ],
    [
      "0.455608348",
      "1.751387859"
    ],
    [
      "0.682699478",
      "1.188247701"
    ],
    [
      "0.39418625",
      "1.216048411"
    ],
    [
      "0.239913978",
      "0.467820309"
    ],
    [
      "0.304601525",
      "1.102979783"
    ],
    [
      "0.072194011",
      "1.011046827"
    ],
    [
      "1.486763756",
      "1.18808639"
    ],
    [
      "0.113391879",
      "0.726372471"
    ],
    [
      "1.085360658",
      "0.492397079"
    ],
    [
      "1.906589007",
      "1.197591392"
    ],
    [
      "0.515240619",
      "0.581397358"
    ],
    [
      "1.158444908",
      "1.91424351"
    ],
    [
      "0.500416094",
      "0.947817861"
    ],
    [
      "0.929794957",
      "0.896239562"
    ],
    [
      "1.409170891",
      "0.694428008"
    ],
    [
      "1.357315657",
      "0.829042169"
    ],
    [
      "1.384515444",
      "1.47756593"
    ],
    [
      "0.224391917",
      "0.845950297"
    ],
    [
      "1.499822557",
      "0.806643038"
    ],
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      "0.592652939",
      "1.075546509"
    ],
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      "1.119182826",
      "0.079814454"
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      "0.585744203",
      "1.6891416"
    ],
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      "1.678436333",
      "1.632985092"
    ],
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      "1.636212638",
      "0.32458941"
    ],
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      "1.437872897",
      "1.677705868"
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    ],
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      "0.580974357"
    ],
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      "0.422566838",
      "0.470846429"
    ],
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      "1.053472229"
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      "0.466769434",
      "1.607563773"
    ],
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      "1.262618404",
      "1.738237754"
    ],
    [
      "1.298237631",
      "1.878635476"
    ],
    [
      "1.256349093",
      "0.328581048"
    ],
    [
      "1.3982202",
      "0.161926819"
    ],
    [
      "0.251473583",
      "1.237096745"
    ],
    [
      "1.398855992",
      "0.306181917"
    ],
    [
      "0.256089035",
      "0.705219241"
    ],
    [
      "0.302257109",
      "1.372118858"
    ],
    [
      "0.942853758",
      "0.51479621"
    ],
    [
      "0.589061855",
      "1.344185011"
    ],
    [
      "0.735159839",
      "1.053147378"
    ],
    [
      "0.981650191",
      "0.761625401"
    ],
    [
      "0.709695192",
      "0.42498031"
    ],
    [
      "1.511521989",
      "0.397146652"
    ],
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      "1.007114838",
      "1.389792468"
    ],
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      "0.538278329",
      "1.209162897"
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      "1.863459658",
      "1.336296071"
    ],
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      "0.159544442",
      "1.393167192"
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      "1.292401622",
      "1.345099682"
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      "1.113842193",
      "0.350980178"
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    ],
    [
      "1.39611209",
      "1.07587136"
    ],
    [
      "1.124157091",
      "0.73922627"
    ],
    [
      "0.108760917",
      "1.258145079"
    ],
    [
      "0.725914204",
      "1.88646656"
    ],
    [
      "1.020446624",
      "1.008454592"
    ],
    [
      "0.44856086",
      "1.082432022"
    ],
    [
      "1.572124489",
      "1.730492679"
    ],
    [
      "1.705607942",
      "0.648767933"
    ],
    [
      "1.872127751",
      "0.683236411"
    ],
    [
      "1.165697426",
      "0.216366017"
    ],
    [
      "1.813881086",
      "1.087069091"
    ],
    [
      "1.263506287",
      "0.110331285"
    ],
    [
      "0.081694761",
      "0.867103528"
    ],
    [
      "0.916463171",
      "1.277577438"
    ],
    [
      "1.344256856",
      "1.210485521"
    ],
    [
      "0.644508173",
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    ],
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      "0.706743111",
      "1.427617975"
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    [
      "0.825339647",
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      "1.497637722"
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      "0.592065192"
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      "1.733212228",
      "0.431364955"
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      "1.058970071",
      "1.255178307"
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      "1.370374458",
      "0.447598817"
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      "1.120669622"
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    ],
    [
      "1.430821649",
      "1.821789932"
    ]
  ],
  "radius": "0.072128249"
}
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": 159
}

起步布局答案

{
  "centers": [
    [
      "1.641173138",
      "0.77783417"
    ],
    [
      "1.577415422",
      "1.30030142"
    ],
    [
      "0.161888858",
      "1.124028117"
    ],
    [
      "1.71035184",
      "1.356317902"
    ],
    [
      "0.367104583",
      "0.824901963"
    ],
    [
      "0.474422072",
      "0.336232268"
    ],
    [
      "1.482434939",
      "0.538441463"
    ],
    [
      "1.072301857",
      "0.873840431"
    ],
    [
      "1.434908522",
      "1.322700551"
    ],
    [
      "0.86193377",
      "1.414300302"
    ],
    [
      "0.825811505",
      "1.165362408"
    ],
    [
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      "1.62590355"
    ],
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      "0.691525286"
    ],
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      "0.176959962"
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      "1.102979783"
    ],
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      "1.011046827"
    ],
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      "1.344185011"
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      "1.053147378"
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  "radius": "0.072128249"
}

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

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还没有帖子。第一个聊聊这道题的思路?