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 = 191

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 = 191
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[0.0660399148062368432437722300995, 0.0660399148062368432437722301005]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 = 0.066039914

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 = 191 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": [
    [
      "1.349970495",
      "1.319655289"
    ],
    [
      "1.011054644",
      "0.194399347"
    ],
    [
      "1.886464126",
      "1.294045563"
    ],
    [
      "1.238036537",
      "0.591600183"
    ],
    [
      "0.892338443",
      "1.184499721"
    ],
    [
      "0.074193258",
      "0.876860578"
    ],
    [
      "1.00813309",
      "0.854792018"
    ],
    [
      "1.006453884",
      "1.251027948"
    ],
    [
      "0.650324521",
      "1.738518552"
    ],
    [
      "1.580904838",
      "0.659106061"
    ],
    [
      "1.123935442",
      "0.525071832"
    ],
    [
      "1.355336867",
      "0.261454968"
    ],
    [
      "1.923208124",
      "0.858691114"
    ],
    [
      "1.396346762",
      "1.72016605"
    ],
    [
      "0.436705582",
      "0.387597963"
    ],
    [
      "0.547321118",
      "0.183086453"
    ],
    [
      "1.110891912",
      "0.072646548"
    ],
    [
      "1.235427809",
      "1.516149599"
    ],
    [
      "1.340282099",
      "0.130235932"
    ],
    [
      "0.642672482",
      "1.60109363"
    ],
    [
      "1.016661414",
      "1.778926652"
    ],
    [
      "0.432181054",
      "1.44426593"
    ],
    [
      "1.69331185",
      "1.12187028"
    ],
    [
      "0.549654907",
      "0.718322297"
    ],
    [
      "0.190832976",
      "1.081266946"
    ],
    [
      "0.078206924",
      "1.150262988"
    ],
    [
      "1.122806053",
      "0.789229077"
    ],
    [
      "0.184614005",
      "0.949333607"
    ],
    [
      "1.930728966",
      "0.99114297"
    ],
    [
      "1.125396198",
      "0.391479649"
    ],
    [
      "0.531819738",
      "1.808139013"
    ],
    [
      "1.005324921",
      "1.515185195"
    ],
    [
      "1.694441239",
      "0.857713035"
    ],
    [
      "1.352137739",
      "0.65812835"
    ],
    [
      "0.895161111",
      "0.524106524"
    ],
    [
      "0.188167799",
      "1.223431884"
    ],
    [
      "1.007568609",
      "0.986870641"
    ],
    [
      "0.776469015",
      "1.777667599"
    ],
    [
      "0.780495167",
      "0.58965706"
    ],
    [
      "1.006195172",
      "1.647262157"
    ],
    [
      "1.627687992",
      "0.403271504"
    ],
    [
      "0.660790763",
      "1.870183047"
    ],
    [
      "1.23814189",
      "1.64820154"
    ],
    [
      "1.121676982",
      "1.053386507"
    ],
    [
      "1.472823632",
      "0.195473094"
    ],
    [
      "0.550219388",
      "0.586243673"
    ],
    [
      "0.666880842",
      "0.391183913"
    ],
    [
      "0.90199573",
      "1.844477108"
    ],
    [
      "0.312770428",
      "0.981286896"
    ],
    [
      "0.533938478",
      "1.676073613"
    ],
    [
      "0.547961462",
      "1.114558167"
    ],
    [
      "1.464727385",
      "1.385048727"
    ],
    [
      "1.814254912",
      "1.05619762"
    ],
    [
      "1.808542334",
      "0.924241386"
    ],
    [
      "1.581541827",
      "0.527027768"
    ],
    [
      "1.008697572",
      "0.722713394"
    ],
    [
      "1.142730158",
      "1.739534353"
    ],
    [
      "0.435553704",
      "0.651794129"
    ],
    [
      "1.465109553",
      "0.988813762"
    ],
    [
      "0.192167116",
      "0.817469921"
    ],
    [
      "1.8229509",
      "0.558374302"
    ],
    [
      "1.350443869",
      "1.054364218"
    ],
    [
      "0.314879004",
      "1.634736686"
    ],
    [
      "0.892902925",
      "1.052421098"
    ],
    [
      "0.437548252",
      "1.585774079"
    ],
    [
      "0.891529488",
      "1.712812614"
    ],
    [
      "0.847460293",
      "0.078580943"
    ],
    [
      "0.668037338",
      "0.127026786"
    ],
    [
      "1.241495252",
      "1.90219814"
    ],
    [
      "0.778801722",
      "0.98589293"
    ],
    [
      "1.353339422",
      "0.393971434"
    ],
    [
      "0.546282256",
      "1.510794097"
    ],
    [
      "0.434424741",
      "0.915951376"
    ],
    [
      "0.665187185",
      "0.787419782"
    ],
    [
      "0.664058222",
      "1.051577029"
    ],
    [
      "0.549090425",
      "0.85040092"
    ],
    [
      "0.131131238",
      "1.342561696"
    ],
    [
      "0.777122516",
      "1.382128861"
    ],
    [
      "1.877019648",
      "0.678880119"
    ],
    [
      "1.237471842",
      "0.723678805"
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    [
      "0.779366204",
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    ],
    [
      "0.776558035",
      "1.514207484"
    ],
    [
      "1.695642922",
      "0.593556119"
    ],
    [
      "0.54739698",
      "1.24663679"
    ],
    [
      "0.321452618",
      "0.585265763"
    ],
    [
      "1.795193957",
      "1.489844886"
    ],
    [
      "0.665751666",
      "0.655341158"
    ],
    [
      "0.894596369",
      "0.656185227"
    ],
    [
      "0.999144834",
      "1.933959694"
    ],
    [
      "0.546846737",
      "1.378715473"
    ],
    [
      "1.005889403",
      "1.383106571"
    ],
    [
      "1.211261996",
      "0.158500654"
    ],
    [
      "0.778237241",
      "1.117971554"
    ],
    [
      "0.786935258",
      "1.909332094"
    ],
    [
      "1.35100835",
      "0.922285595"
    ],
    [
      "1.579210755",
      "1.055341929"
    ],
    [
      "0.8917882",
      "1.316578404"
    ],
    [
      "1.626728091",
      "1.692454577"
    ],
    [
      "1.580340144",
      "0.791184684"
    ],
    [
      "1.123370747",
      "0.657150454"
    ],
    [
      "0.782216357",
      "0.193421312"
    ],
    [
      "1.695005933",
      "0.725634412"
    ],
    [
      "0.288682548",
      "0.717317591"
    ],
    [
      "0.067192742",
      "1.01302528"
    ],
    [
      "1.23466346",
      "1.384071981"
    ],
    [
      "1.466238941",
      "0.724656517"
    ],
    [
      "0.777672759",
      "1.250050177"
    ],
    [
      "0.781624563",
      "0.325499815"
    ],
    [
      "1.693876544",
      "0.989791657"
    ],
    [
      "1.663362116",
      "1.435046611"
    ],
    [
      "1.464545071",
      "1.120892385"
    ],
    [
      "1.467440625",
      "0.460499601"
    ],
    [
      "1.584403112",
      "1.329166692"
    ],
    [
      "1.120133449",
      "1.449867163"
    ],
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      "1.515301289",
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      "0.978965766",
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      "1.359829594"
    ],
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      "0.792162763"
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      "1.383449886",
      "1.851614717"
    ],
    [
      "0.781059862",
      "0.457578438"
    ],
    [
      "0.286398747",
      "1.373437087"
    ],
    [
      "1.463980589",
      "1.252971008"
    ],
    [
      "0.191374157",
      "0.608171557"
    ],
    [
      "1.236907148",
      "0.855757428"
    ],
    [
      "1.238601231",
      "0.45952156"
    ],
    [
      "0.667445543",
      "0.25910529"
    ],
    [
      "1.466080737",
      "1.517121623"
    ],
    [
      "0.411269229",
      "1.72503622"
    ],
    [
      "1.235778185",
      "1.119914674"
    ],
    [
      "0.318989399",
      "1.113220234"
    ],
    [
      "1.120555087",
      "1.317556115"
    ],
    [
      "0.43386026",
      "1.04803"
    ],
    [
      "0.891223719",
      "1.448657028"
    ],
    [
      "1.122424514",
      "1.584526033"
    ],
    [
      "0.895739499",
      "0.392004421"
    ],
    [
      "1.007004127",
      "1.118949265"
    ],
    [
      "1.925156582",
      "1.127932556"
    ],
    [
      "1.503999287",
      "1.643641443"
    ],
    [
      "0.305141399",
      "0.375938333"
    ],
    [
      "1.113810518",
      "1.868409237"
    ],
    [
      "0.111189127",
      "0.713126034"
    ],
    [
      "1.465674247",
      "0.856735139"
    ],
    [
      "0.779930685",
      "0.721735684"
    ],
    [
      "1.358428213",
      "1.593646229"
    ],
    [
      "1.349879387",
      "1.186442841"
    ],
    [
      "0.550764915",
      "0.320998038"
    ],
    [
      "0.893467406",
      "0.920342474"
    ],
    [
      "1.235213703",
      "1.251993298"
    ],
    [
      "0.664622703",
      "0.919498405"
    ],
    [
      "1.351573045",
      "0.790206972"
    ],
    [
      "1.595413416",
      "1.548307618"
    ],
    [
      "0.548525944",
      "0.982479543"
    ],
    [
      "0.436096212",
      "0.519676681"
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    [
      "1.010418717",
      "0.326477646"
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    [
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      "0.592577894"
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      "1.240097323",
      "0.325991228"
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      "1.009826961",
      "0.45855615"
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    [
      "1.807581471",
      "1.188108752"
    ],
    [
      "1.351025578",
      "1.452257212"
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    [
      "0.432699669",
      "1.312187118"
    ],
    [
      "1.009262266",
      "0.590634772"
    ],
    [
      "1.469175657",
      "0.328431168"
    ],
    [
      "0.313711738",
      "1.502662015"
    ],
    [
      "1.590618148",
      "0.276500419"
    ],
    [
      "1.692973282",
      "1.253949675"
    ],
    [
      "0.662943497",
      "1.315734335"
    ],
    [
      "0.436058021",
      "0.255519721"
    ],
    [
      "1.121112501",
      "1.185465131"
    ],
    [
      "0.433295778",
      "1.180108623"
    ],
    [
      "0.320323539",
      "0.849423209"
    ],
    [
      "0.550784083",
      "0.454165051"
    ],
    [
      "1.75242365",
      "0.446700722"
    ],
    [
      "1.718142221",
      "1.597120751"
    ],
    [
      "1.352702434",
      "0.526049727"
    ],
    [
      "1.122241464",
      "0.921307884"
    ],
    [
      "1.236342666",
      "0.987836051"
    ],
    [
      "0.318424917",
      "1.245298858"
    ],
    [
      "0.666316148",
      "0.523262535"
    ],
    [
      "0.662379016",
      "1.447812959"
    ],
    [
      "1.57875782",
      "1.187615902"
    ],
    [
      "0.890659237",
      "1.580735651"
    ],
    [
      "0.224056409",
      "0.480199092"
    ],
    [
      "0.188143173",
      "1.461703295"
    ],
    [
      "1.27606044",
      "1.774721361"
    ],
    [
      "0.434989223",
      "0.783872753"
    ],
    [
      "0.66349374",
      "1.183655652"
    ],
    [
      "1.126032125",
      "0.25940135"
    ],
    [
      "0.894031888",
      "0.788263851"
    ],
    [
      "1.579775449",
      "0.923263306"
    ],
    [
      "0.766907727",
      "1.645934296"
    ]
  ],
  "radius": "0.066039914"
}
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": 191
}

起步布局答案

{
  "centers": [
    [
      "1.349970495",
      "1.319655289"
    ],
    [
      "1.011054644",
      "0.194399347"
    ],
    [
      "1.886464126",
      "1.294045563"
    ],
    [
      "1.238036537",
      "0.591600183"
    ],
    [
      "0.892338443",
      "1.184499721"
    ],
    [
      "0.074193258",
      "0.876860578"
    ],
    [
      "1.00813309",
      "0.854792018"
    ],
    [
      "1.006453884",
      "1.251027948"
    ],
    [
      "0.650324521",
      "1.738518552"
    ],
    [
      "1.580904838",
      "0.659106061"
    ],
    [
      "1.123935442",
      "0.525071832"
    ],
    [
      "1.355336867",
      "0.261454968"
    ],
    [
      "1.923208124",
      "0.858691114"
    ],
    [
      "1.396346762",
      "1.72016605"
    ],
    [
      "0.436705582",
      "0.387597963"
    ],
    [
      "0.547321118",
      "0.183086453"
    ],
    [
      "1.110891912",
      "0.072646548"
    ],
    [
      "1.235427809",
      "1.516149599"
    ],
    [
      "1.340282099",
      "0.130235932"
    ],
    [
      "0.642672482",
      "1.60109363"
    ],
    [
      "1.016661414",
      "1.778926652"
    ],
    [
      "0.432181054",
      "1.44426593"
    ],
    [
      "1.69331185",
      "1.12187028"
    ],
    [
      "0.549654907",
      "0.718322297"
    ],
    [
      "0.190832976",
      "1.081266946"
    ],
    [
      "0.078206924",
      "1.150262988"
    ],
    [
      "1.122806053",
      "0.789229077"
    ],
    [
      "0.184614005",
      "0.949333607"
    ],
    [
      "1.930728966",
      "0.99114297"
    ],
    [
      "1.125396198",
      "0.391479649"
    ],
    [
      "0.531819738",
      "1.808139013"
    ],
    [
      "1.005324921",
      "1.515185195"
    ],
    [
      "1.694441239",
      "0.857713035"
    ],
    [
      "1.352137739",
      "0.65812835"
    ],
    [
      "0.895161111",
      "0.524106524"
    ],
    [
      "0.188167799",
      "1.223431884"
    ],
    [
      "1.007568609",
      "0.986870641"
    ],
    [
      "0.776469015",
      "1.777667599"
    ],
    [
      "0.780495167",
      "0.58965706"
    ],
    [
      "1.006195172",
      "1.647262157"
    ],
    [
      "1.627687992",
      "0.403271504"
    ],
    [
      "0.660790763",
      "1.870183047"
    ],
    [
      "1.23814189",
      "1.64820154"
    ],
    [
      "1.121676982",
      "1.053386507"
    ],
    [
      "1.472823632",
      "0.195473094"
    ],
    [
      "0.550219388",
      "0.586243673"
    ],
    [
      "0.666880842",
      "0.391183913"
    ],
    [
      "0.90199573",
      "1.844477108"
    ],
    [
      "0.312770428",
      "0.981286896"
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  "radius": "0.066039914"
}

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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