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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 = 97

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 = 97
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[0.0914194599063643999907623808885, 0.0914194599063643999907623808895]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 = 0.091419459

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 = 97 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": [
    [
      "1.009561339",
      "0.09146977"
    ],
    [
      "0.786574654",
      "0.117126628"
    ],
    [
      "1.226674346",
      "0.120149252"
    ],
    [
      "1.398243578",
      "0.183347901"
    ],
    [
      "0.590032127",
      "0.189170215"
    ],
    [
      "0.916118719",
      "0.248627485"
    ],
    [
      "1.098942603",
      "0.25097231"
    ],
    [
      "1.553685601",
      "0.279617564"
    ],
    [
      "0.435993208",
      "0.287669363"
    ],
    [
      "0.739223553",
      "0.294867844"
    ],
    [
      "1.395692057",
      "0.371639902"
    ],
    [
      "1.21410353",
      "0.392986505"
    ],
    [
      "1.686705659",
      "0.405059721"
    ],
    [
      "0.304794219",
      "0.41501494"
    ],
    [
      "1.033535166",
      "0.421711695"
    ],
    [
      "0.851529753",
      "0.439150165"
    ],
    [
      "0.663480023",
      "0.461279948"
    ],
    [
      "0.480678219",
      "0.464963808"
    ],
    [
      "1.528712115",
      "0.497082059"
    ],
    [
      "1.323384493",
      "0.539573481"
    ],
    [
      "1.791916994",
      "0.554594485"
    ],
    [
      "0.201748174",
      "0.566049979"
    ],
    [
      "1.142089161",
      "0.568837835"
    ],
    [
      "0.957791636",
      "0.588123799"
    ],
    [
      "0.775786223",
      "0.605562269"
    ],
    [
      "0.377632173",
      "0.615998848"
    ],
    [
      "0.572812657",
      "0.622892007"
    ],
    [
      "1.640807936",
      "0.661243938"
    ],
    [
      "1.456404552",
      "0.665015638"
    ],
    [
      "1.257932611",
      "0.710295834"
    ],
    [
      "1.865058983",
      "0.72216632"
    ],
    [
      "0.131028009",
      "0.73465819"
    ],
    [
      "1.06634563",
      "0.735249938"
    ],
    [
      "0.881891087",
      "0.754464345"
    ],
    [
      "0.687216447",
      "0.765516842"
    ],
    [
      "0.469766611",
      "0.773927047"
    ],
    [
      "0.294452005",
      "0.825841303"
    ],
    [
      "1.716641956",
      "0.828948606"
    ],
    [
      "1.533811052",
      "0.830660753"
    ],
    [
      "1.35435046",
      "0.865645984"
    ],
    [
      "1.172450495",
      "0.884152015"
    ],
    [
      "1.903169681",
      "0.900989264"
    ],
    [
      "0.990445082",
      "0.901590484"
    ],
    [
      "0.095497595",
      "0.914011641"
    ],
    [
      "0.793321311",
      "0.914418918"
    ],
    [
      "0.611315898",
      "0.931857388"
    ],
    [
      "0.429623926",
      "0.952304835"
    ],
    [
      "1.642352529",
      "0.996014915"
    ],
    [
      "0.254386975",
      "1.004478406"
    ],
    [
      "1.460455324",
      "1.01454806"
    ],
    [
      "1.278555359",
      "1.033054091"
    ],
    [
      "1.096549946",
      "1.050492561"
    ],
    [
      "0.901875306",
      "1.061545057"
    ],
    [
      "0.720183334",
      "1.081992505"
    ],
    [
      "1.904705762",
      "1.083821732"
    ],
    [
      "0.096595768",
      "1.096847263"
    ],
    [
      "0.538177921",
      "1.099430974"
    ],
    [
      "1.735581739",
      "1.153299323"
    ],
    [
      "1.553684535",
      "1.171832468"
    ],
    [
      "0.292069263",
      "1.183392117"
    ],
    [
      "1.371784569",
      "1.190338499"
    ],
    [
      "1.00798017",
      "1.210447134"
    ],
    [
      "1.19073837",
      "1.215879531"
    ],
    [
      "0.826337956",
      "1.231334057"
    ],
    [
      "0.644282785",
      "1.248333051"
    ],
    [
      "0.456630394",
      "1.263077152"
    ],
    [
      "0.134278057",
      "1.275760973"
    ],
    [
      "1.862241398",
      "1.286344373"
    ],
    [
      "1.661882819",
      "1.320626965"
    ],
    [
      "1.479985614",
      "1.33916011"
    ],
    [
      "1.298760197",
      "1.363396836"
    ],
    [
      "0.935271295",
      "1.37820735"
    ],
    [
      "1.11777104",
      "1.389338993"
    ],
    [
      "0.753728732",
      "1.399941392"
    ],
    [
      "0.571605861",
      "1.416107111"
    ],
    [
      "0.389565056",
      "1.433172158"
    ],
    [
      "0.207018483",
      "1.443507511"
    ],
    [
      "1.787106041",
      "1.453853146"
    ],
    [
      "1.518033074",
      "1.517996519"
    ],
    [
      "1.230732186",
      "1.533109123"
    ],
    [
      "0.863322246",
      "1.546294842"
    ],
    [
      "1.045821991",
      "1.557426486"
    ],
    [
      "0.68441205",
      "1.583993814"
    ],
    [
      "0.501575917",
      "1.585003166"
    ],
    [
      "0.319535111",
      "1.602068213"
    ],
    [
      "1.680300779",
      "1.60225364"
    ],
    [
      "1.38031113",
      "1.638257638"
    ],
    [
      "1.170048384",
      "1.705583894"
    ],
    [
      "0.806515416",
      "1.720085102"
    ],
    [
      "1.545946165",
      "1.726265366"
    ],
    [
      "0.989015162",
      "1.731216746"
    ],
    [
      "0.513637394",
      "1.767443818"
    ],
    [
      "1.389483001",
      "1.820866365"
    ],
    [
      "0.67713864",
      "1.849281543"
    ],
    [
      "1.217247395",
      "1.882225689"
    ],
    [
      "0.853714428",
      "1.896726898"
    ],
    [
      "1.036214173",
      "1.907858542"
    ]
  ],
  "radius": "0.091419459"
}
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": 97
}

起步布局答案

{
  "centers": [
    [
      "1.009561339",
      "0.09146977"
    ],
    [
      "0.786574654",
      "0.117126628"
    ],
    [
      "1.226674346",
      "0.120149252"
    ],
    [
      "1.398243578",
      "0.183347901"
    ],
    [
      "0.590032127",
      "0.189170215"
    ],
    [
      "0.916118719",
      "0.248627485"
    ],
    [
      "1.098942603",
      "0.25097231"
    ],
    [
      "1.553685601",
      "0.279617564"
    ],
    [
      "0.435993208",
      "0.287669363"
    ],
    [
      "0.739223553",
      "0.294867844"
    ],
    [
      "1.395692057",
      "0.371639902"
    ],
    [
      "1.21410353",
      "0.392986505"
    ],
    [
      "1.686705659",
      "0.405059721"
    ],
    [
      "0.304794219",
      "0.41501494"
    ],
    [
      "1.033535166",
      "0.421711695"
    ],
    [
      "0.851529753",
      "0.439150165"
    ],
    [
      "0.663480023",
      "0.461279948"
    ],
    [
      "0.480678219",
      "0.464963808"
    ],
    [
      "1.528712115",
      "0.497082059"
    ],
    [
      "1.323384493",
      "0.539573481"
    ],
    [
      "1.791916994",
      "0.554594485"
    ],
    [
      "0.201748174",
      "0.566049979"
    ],
    [
      "1.142089161",
      "0.568837835"
    ],
    [
      "0.957791636",
      "0.588123799"
    ],
    [
      "0.775786223",
      "0.605562269"
    ],
    [
      "0.377632173",
      "0.615998848"
    ],
    [
      "0.572812657",
      "0.622892007"
    ],
    [
      "1.640807936",
      "0.661243938"
    ],
    [
      "1.456404552",
      "0.665015638"
    ],
    [
      "1.257932611",
      "0.710295834"
    ],
    [
      "1.865058983",
      "0.72216632"
    ],
    [
      "0.131028009",
      "0.73465819"
    ],
    [
      "1.06634563",
      "0.735249938"
    ],
    [
      "0.881891087",
      "0.754464345"
    ],
    [
      "0.687216447",
      "0.765516842"
    ],
    [
      "0.469766611",
      "0.773927047"
    ],
    [
      "0.294452005",
      "0.825841303"
    ],
    [
      "1.716641956",
      "0.828948606"
    ],
    [
      "1.533811052",
      "0.830660753"
    ],
    [
      "1.35435046",
      "0.865645984"
    ],
    [
      "1.172450495",
      "0.884152015"
    ],
    [
      "1.903169681",
      "0.900989264"
    ],
    [
      "0.990445082",
      "0.901590484"
    ],
    [
      "0.095497595",
      "0.914011641"
    ],
    [
      "0.793321311",
      "0.914418918"
    ],
    [
      "0.611315898",
      "0.931857388"
    ],
    [
      "0.429623926",
      "0.952304835"
    ],
    [
      "1.642352529",
      "0.996014915"
    ],
    [
      "0.254386975",
      "1.004478406"
    ],
    [
      "1.460455324",
      "1.01454806"
    ],
    [
      "1.278555359",
      "1.033054091"
    ],
    [
      "1.096549946",
      "1.050492561"
    ],
    [
      "0.901875306",
      "1.061545057"
    ],
    [
      "0.720183334",
      "1.081992505"
    ],
    [
      "1.904705762",
      "1.083821732"
    ],
    [
      "0.096595768",
      "1.096847263"
    ],
    [
      "0.538177921",
      "1.099430974"
    ],
    [
      "1.735581739",
      "1.153299323"
    ],
    [
      "1.553684535",
      "1.171832468"
    ],
    [
      "0.292069263",
      "1.183392117"
    ],
    [
      "1.371784569",
      "1.190338499"
    ],
    [
      "1.00798017",
      "1.210447134"
    ],
    [
      "1.19073837",
      "1.215879531"
    ],
    [
      "0.826337956",
      "1.231334057"
    ],
    [
      "0.644282785",
      "1.248333051"
    ],
    [
      "0.456630394",
      "1.263077152"
    ],
    [
      "0.134278057",
      "1.275760973"
    ],
    [
      "1.862241398",
      "1.286344373"
    ],
    [
      "1.661882819",
      "1.320626965"
    ],
    [
      "1.479985614",
      "1.33916011"
    ],
    [
      "1.298760197",
      "1.363396836"
    ],
    [
      "0.935271295",
      "1.37820735"
    ],
    [
      "1.11777104",
      "1.389338993"
    ],
    [
      "0.753728732",
      "1.399941392"
    ],
    [
      "0.571605861",
      "1.416107111"
    ],
    [
      "0.389565056",
      "1.433172158"
    ],
    [
      "0.207018483",
      "1.443507511"
    ],
    [
      "1.787106041",
      "1.453853146"
    ],
    [
      "1.518033074",
      "1.517996519"
    ],
    [
      "1.230732186",
      "1.533109123"
    ],
    [
      "0.863322246",
      "1.546294842"
    ],
    [
      "1.045821991",
      "1.557426486"
    ],
    [
      "0.68441205",
      "1.583993814"
    ],
    [
      "0.501575917",
      "1.585003166"
    ],
    [
      "0.319535111",
      "1.602068213"
    ],
    [
      "1.680300779",
      "1.60225364"
    ],
    [
      "1.38031113",
      "1.638257638"
    ],
    [
      "1.170048384",
      "1.705583894"
    ],
    [
      "0.806515416",
      "1.720085102"
    ],
    [
      "1.545946165",
      "1.726265366"
    ],
    [
      "0.989015162",
      "1.731216746"
    ],
    [
      "0.513637394",
      "1.767443818"
    ],
    [
      "1.389483001",
      "1.820866365"
    ],
    [
      "0.67713864",
      "1.849281543"
    ],
    [
      "1.217247395",
      "1.882225689"
    ],
    [
      "0.853714428",
      "1.896726898"
    ],
    [
      "1.036214173",
      "1.907858542"
    ]
  ],
  "radius": "0.091419459"
}

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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