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 = 168

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 = 168
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[0.0701375056974642817538784673995, 0.0701375056974642817538784674005]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 = 0.070137505

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 = 168 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": [
    [
      "1.202299709",
      "0.936766926"
    ],
    [
      "0.759249127",
      "0.692823887"
    ],
    [
      "0.545326993",
      "0.353335928"
    ],
    [
      "1.019107106",
      "1.929666164"
    ],
    [
      "1.927305073",
      "0.931082952"
    ],
    [
      "1.519345978",
      "1.617669558"
    ],
    [
      "0.53108323",
      "0.630527015"
    ],
    [
      "1.158735564",
      "1.916213555"
    ],
    [
      "1.430386513",
      "0.483163096"
    ],
    [
      "0.680084183",
      "0.419936089"
    ],
    [
      "1.511395747",
      "1.222342395"
    ],
    [
      "1.738068022",
      "0.770077122"
    ],
    [
      "1.60175488",
      "0.291102742"
    ],
    [
      "0.554649028",
      "1.652747468"
    ],
    [
      "1.209157705",
      "0.255331672"
    ],
    [
      "0.975579088",
      "1.388876981"
    ],
    [
      "0.956512657",
      "1.52785018"
    ],
    [
      "1.068345165",
      "0.978399355"
    ],
    [
      "0.429496493",
      "1.049742119"
    ],
    [
      "0.367651129",
      "1.323389646"
    ],
    [
      "0.676281404",
      "0.279712633"
    ],
    [
      "0.095959534",
      "0.782387743"
    ],
    [
      "0.903468092",
      "1.156856036"
    ],
    [
      "0.573715458",
      "1.513774269"
    ],
    [
      "0.822558114",
      "1.56948261"
    ],
    [
      "0.487735461",
      "1.776034343"
    ],
    [
      "0.560577119",
      "0.493387708"
    ],
    [
      "1.398099313",
      "0.621485993"
    ],
    [
      "1.805008699",
      "0.534591305"
    ],
    [
      "1.367176782",
      "0.758310245"
    ],
    [
      "0.336728598",
      "1.460213898"
    ],
    [
      "0.490736624",
      "0.221992884"
    ],
    [
      "1.865858258",
      "0.660981217"
    ],
    [
      "1.637794784",
      "0.671983512"
    ],
    [
      "0.138980872",
      "0.648872788"
    ],
    [
      "1.589825142",
      "1.496385732"
    ],
    [
      "1.081993716",
      "0.195810757"
    ],
    [
      "0.89320367",
      "0.651191458"
    ],
    [
      "1.480473217",
      "1.359166647"
    ],
    [
      "1.710757398",
      "0.400491931"
    ],
    [
      "1.140454648",
      "1.21041543"
    ],
    [
      "1.315596143",
      "1.537623328"
    ],
    [
      "1.233222239",
      "0.799942674"
    ],
    [
      "0.87904382",
      "1.921961963"
    ],
    [
      "0.813840359",
      "0.375107"
    ],
    [
      "1.21256413",
      "1.442431505"
    ],
    [
      "0.501605976",
      "1.281758194"
    ],
    [
      "1.61442776",
      "1.317534218"
    ],
    [
      "1.058080744",
      "0.472734776"
    ],
    [
      "1.532043899",
      "0.579821539"
    ],
    [
      "0.613563809",
      "0.154239201"
    ],
    [
      "0.217042064",
      "0.854867675"
    ],
    [
      "0.436190883",
      "1.559681091"
    ],
    [
      "1.408363734",
      "1.127150572"
    ],
    [
      "0.166929879",
      "1.331250344"
    ],
    [
      "0.635560519",
      "1.240125765"
    ],
    [
      "1.096640069",
      "1.521416889"
    ],
    [
      "0.885321578",
      "0.111969754"
    ],
    [
      "1.750744224",
      "1.284444936"
    ],
    [
      "0.325303827",
      "0.955822115"
    ],
    [
      "0.713429529",
      "1.749175716"
    ],
    [
      "1.089003274",
      "0.335910524"
    ],
    [
      "0.356226358",
      "0.818997863"
    ],
    [
      "1.470208795",
      "0.853502068"
    ],
    [
      "1.385391435",
      "1.659301987"
    ],
    [
      "0.604637988",
      "1.376950017"
    ],
    [
      "0.280103727",
      "1.588552134"
    ],
    [
      "1.541034671",
      "1.756257723"
    ],
    [
      "0.9241262",
      "0.514367206"
    ],
    [
      "0.665037773",
      "0.588894585"
    ],
    [
      "0.996235683",
      "0.74638328"
    ],
    [
      "1.037422635",
      "1.115223607"
    ],
    [
      "1.161112757",
      "0.567926599"
    ],
    [
      "1.604163339",
      "0.811869639"
    ],
    [
      "1.57281255",
      "0.42835951"
    ],
    [
      "1.336254252",
      "0.895134497"
    ],
    [
      "0.470683445",
      "1.418582446"
    ],
    [
      "1.501151505",
      "0.716655971"
    ],
    [
      "0.745185287",
      "0.105732858"
    ],
    [
      "0.741733186",
      "1.893276279"
    ],
    [
      "0.431881718",
      "0.437579352"
    ],
    [
      "0.072971221",
      "1.072538958"
    ],
    [
      "0.807902882",
      "0.23120629"
    ],
    [
      "0.379498964",
      "0.307452007"
    ],
    [
      "1.354932161",
      "0.25221315"
    ],
    [
      "1.130190226",
      "0.704750851"
    ],
    [
      "0.263232154",
      "0.71397807"
    ],
    [
      "1.542318278",
      "1.085518143"
    ],
    [
      "1.844813441",
      "1.388502906"
    ],
    [
      "1.274409191",
      "1.168783001"
    ],
    [
      "0.965313153",
      "0.883207532"
    ],
    [
      "0.294037313",
      "1.092568171"
    ],
    [
      "1.2950673",
      "0.52629417"
    ],
    [
      "1.459328843",
      "0.345906328"
    ],
    [
      "1.346518674",
      "1.400799076"
    ],
    [
      "0.738592532",
      "1.335317588"
    ],
    [
      "1.74443931",
      "1.557184325"
    ],
    [
      "1.874900212",
      "0.800964509"
    ],
    [
      "1.006500105",
      "1.252047859"
    ],
    [
      "1.019276121",
      "0.070337325"
    ],
    [
      "1.573240808",
      "0.948693891"
    ],
    [
      "0.728326596",
      "0.829648139"
    ],
    [
      "0.783851195",
      "0.514326163"
    ],
    [
      "0.282382311",
      "0.408671741"
    ],
    [
      "0.872547075",
      "1.293685158"
    ],
    [
      "0.20159679",
      "0.523348584"
    ],
    [
      "0.625294583",
      "0.734456316"
    ],
    [
      "0.397128834",
      "0.672159919"
    ],
    [
      "0.688603571",
      "1.611115039"
    ],
    [
      "1.377441204",
      "1.263974824"
    ],
    [
      "0.948039173",
      "0.237443186"
    ],
    [
      "0.800436079",
      "1.061664214"
    ],
    [
      "1.795542806",
      "1.15151576"
    ],
    [
      "0.610300039",
      "1.844261926"
    ],
    [
      "1.439286265",
      "0.99032632"
    ],
    [
      "1.137046871",
      "1.777625391"
    ],
    [
      "1.171377178",
      "1.073591178"
    ],
    [
      "1.488271174",
      "0.20864956"
    ],
    [
      "0.263458463",
      "1.229469642"
    ],
    [
      "0.924016781",
      "1.66634962"
    ],
    [
      "1.407080128",
      "1.797890152"
    ],
    [
      "0.532528506",
      "1.144933942"
    ],
    [
      "0.563451036",
      "1.00810969"
    ],
    [
      "0.831358609",
      "0.924839962"
    ],
    [
      "0.491336579",
      "0.776092491"
    ],
    [
      "1.32598983",
      "0.389469918"
    ],
    [
      "0.666483049",
      "1.103301513"
    ],
    [
      "1.676272821",
      "1.043885714"
    ],
    [
      "0.934390622",
      "1.020031784"
    ],
    [
      "0.340360361",
      "0.543885138"
    ],
    [
      "0.849072366",
      "1.784926235"
    ],
    [
      "1.812951314",
      "1.012325161"
    ],
    [
      "1.294751608",
      "1.881910283"
    ],
    [
      "0.196688716",
      "1.468332402"
    ],
    [
      "0.841624545",
      "1.43050941"
    ],
    [
      "0.460419023",
      "0.912917867"
    ],
    [
      "1.168743624",
      "0.085576768"
    ],
    [
      "0.398229979",
      "1.186488175"
    ],
    [
      "1.924789654",
      "1.09699667"
    ],
    [
      "1.708496978",
      "1.421592188"
    ],
    [
      "0.707670001",
      "1.47214184"
    ],
    [
      "1.192035287",
      "0.431102347"
    ],
    [
      "1.099267696",
      "0.841575103"
    ],
    [
      "0.955048731",
      "0.377542954"
    ],
    [
      "1.109532117",
      "1.347239682"
    ],
    [
      "0.769515062",
      "1.198493336"
    ],
    [
      "1.273062916",
      "1.743322118"
    ],
    [
      "1.243486661",
      "1.305607253"
    ],
    [
      "1.064034033",
      "1.657849749"
    ],
    [
      "1.201344077",
      "1.622501406"
    ],
    [
      "1.304376479",
      "0.12136514"
    ],
    [
      "1.449550687",
      "1.495990899"
    ],
    [
      "0.073511832",
      "0.920854994"
    ],
    [
      "0.989592353",
      "1.792124542"
    ],
    [
      "0.094431354",
      "1.211162699"
    ],
    [
      "1.652153594",
      "1.662827239"
    ],
    [
      "1.895681534",
      "1.249764071"
    ],
    [
      "0.191245012",
      "0.997117548"
    ],
    [
      "1.027158213",
      "0.609559028"
    ],
    [
      "1.664733951",
      "0.534319573"
    ],
    [
      "1.264144769",
      "0.663118422"
    ],
    [
      "1.305331722",
      "1.031958749"
    ],
    [
      "0.376828707",
      "1.690146214"
    ],
    [
      "0.86228114",
      "0.78801571"
    ],
    [
      "0.697404066",
      "0.966472391"
    ],
    [
      "1.714652818",
      "0.908649807"
    ],
    [
      "0.594372053",
      "0.871280568"
    ],
    [
      "1.648474289",
      "1.181386611"
    ]
  ],
  "radius": "0.070137505"
}
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": 168
}

起步布局答案

{
  "centers": [
    [
      "1.202299709",
      "0.936766926"
    ],
    [
      "0.759249127",
      "0.692823887"
    ],
    [
      "0.545326993",
      "0.353335928"
    ],
    [
      "1.019107106",
      "1.929666164"
    ],
    [
      "1.927305073",
      "0.931082952"
    ],
    [
      "1.519345978",
      "1.617669558"
    ],
    [
      "0.53108323",
      "0.630527015"
    ],
    [
      "1.158735564",
      "1.916213555"
    ],
    [
      "1.430386513",
      "0.483163096"
    ],
    [
      "0.680084183",
      "0.419936089"
    ],
    [
      "1.511395747",
      "1.222342395"
    ],
    [
      "1.738068022",
      "0.770077122"
    ],
    [
      "1.60175488",
      "0.291102742"
    ],
    [
      "0.554649028",
      "1.652747468"
    ],
    [
      "1.209157705",
      "0.255331672"
    ],
    [
      "0.975579088",
      "1.388876981"
    ],
    [
      "0.956512657",
      "1.52785018"
    ],
    [
      "1.068345165",
      "0.978399355"
    ],
    [
      "0.429496493",
      "1.049742119"
    ],
    [
      "0.367651129",
      "1.323389646"
    ],
    [
      "0.676281404",
      "0.279712633"
    ],
    [
      "0.095959534",
      "0.782387743"
    ],
    [
      "0.903468092",
      "1.156856036"
    ],
    [
      "0.573715458",
      "1.513774269"
    ],
    [
      "0.822558114",
      "1.56948261"
    ],
    [
      "0.487735461",
      "1.776034343"
    ],
    [
      "0.560577119",
      "0.493387708"
    ],
    [
      "1.398099313",
      "0.621485993"
    ],
    [
      "1.805008699",
      "0.534591305"
    ],
    [
      "1.367176782",
      "0.758310245"
    ],
    [
      "0.336728598",
      "1.460213898"
    ],
    [
      "0.490736624",
      "0.221992884"
    ],
    [
      "1.865858258",
      "0.660981217"
    ],
    [
      "1.637794784",
      "0.671983512"
    ],
    [
      "0.138980872",
      "0.648872788"
    ],
    [
      "1.589825142",
      "1.496385732"
    ],
    [
      "1.081993716",
      "0.195810757"
    ],
    [
      "0.89320367",
      "0.651191458"
    ],
    [
      "1.480473217",
      "1.359166647"
    ],
    [
      "1.710757398",
      "0.400491931"
    ],
    [
      "1.140454648",
      "1.21041543"
    ],
    [
      "1.315596143",
      "1.537623328"
    ],
    [
      "1.233222239",
      "0.799942674"
    ],
    [
      "0.87904382",
      "1.921961963"
    ],
    [
      "0.813840359",
      "0.375107"
    ],
    [
      "1.21256413",
      "1.442431505"
    ],
    [
      "0.501605976",
      "1.281758194"
    ],
    [
      "1.61442776",
      "1.317534218"
    ],
    [
      "1.058080744",
      "0.472734776"
    ],
    [
      "1.532043899",
      "0.579821539"
    ],
    [
      "0.613563809",
      "0.154239201"
    ],
    [
      "0.217042064",
      "0.854867675"
    ],
    [
      "0.436190883",
      "1.559681091"
    ],
    [
      "1.408363734",
      "1.127150572"
    ],
    [
      "0.166929879",
      "1.331250344"
    ],
    [
      "0.635560519",
      "1.240125765"
    ],
    [
      "1.096640069",
      "1.521416889"
    ],
    [
      "0.885321578",
      "0.111969754"
    ],
    [
      "1.750744224",
      "1.284444936"
    ],
    [
      "0.325303827",
      "0.955822115"
    ],
    [
      "0.713429529",
      "1.749175716"
    ],
    [
      "1.089003274",
      "0.335910524"
    ],
    [
      "0.356226358",
      "0.818997863"
    ],
    [
      "1.470208795",
      "0.853502068"
    ],
    [
      "1.385391435",
      "1.659301987"
    ],
    [
      "0.604637988",
      "1.376950017"
    ],
    [
      "0.280103727",
      "1.588552134"
    ],
    [
      "1.541034671",
      "1.756257723"
    ],
    [
      "0.9241262",
      "0.514367206"
    ],
    [
      "0.665037773",
      "0.588894585"
    ],
    [
      "0.996235683",
      "0.74638328"
    ],
    [
      "1.037422635",
      "1.115223607"
    ],
    [
      "1.161112757",
      "0.567926599"
    ],
    [
      "1.604163339",
      "0.811869639"
    ],
    [
      "1.57281255",
      "0.42835951"
    ],
    [
      "1.336254252",
      "0.895134497"
    ],
    [
      "0.470683445",
      "1.418582446"
    ],
    [
      "1.501151505",
      "0.716655971"
    ],
    [
      "0.745185287",
      "0.105732858"
    ],
    [
      "0.741733186",
      "1.893276279"
    ],
    [
      "0.431881718",
      "0.437579352"
    ],
    [
      "0.072971221",
      "1.072538958"
    ],
    [
      "0.807902882",
      "0.23120629"
    ],
    [
      "0.379498964",
      "0.307452007"
    ],
    [
      "1.354932161",
      "0.25221315"
    ],
    [
      "1.130190226",
      "0.704750851"
    ],
    [
      "0.263232154",
      "0.71397807"
    ],
    [
      "1.542318278",
      "1.085518143"
    ],
    [
      "1.844813441",
      "1.388502906"
    ],
    [
      "1.274409191",
      "1.168783001"
    ],
    [
      "0.965313153",
      "0.883207532"
    ],
    [
      "0.294037313",
      "1.092568171"
    ],
    [
      "1.2950673",
      "0.52629417"
    ],
    [
      "1.459328843",
      "0.345906328"
    ],
    [
      "1.346518674",
      "1.400799076"
    ],
    [
      "0.738592532",
      "1.335317588"
    ],
    [
      "1.74443931",
      "1.557184325"
    ],
    [
      "1.874900212",
      "0.800964509"
    ],
    [
      "1.006500105",
      "1.252047859"
    ],
    [
      "1.019276121",
      "0.070337325"
    ],
    [
      "1.573240808",
      "0.948693891"
    ],
    [
      "0.728326596",
      "0.829648139"
    ],
    [
      "0.783851195",
      "0.514326163"
    ],
    [
      "0.282382311",
      "0.408671741"
    ],
    [
      "0.872547075",
      "1.293685158"
    ],
    [
      "0.20159679",
      "0.523348584"
    ],
    [
      "0.625294583",
      "0.734456316"
    ],
    [
      "0.397128834",
      "0.672159919"
    ],
    [
      "0.688603571",
      "1.611115039"
    ],
    [
      "1.377441204",
      "1.263974824"
    ],
    [
      "0.948039173",
      "0.237443186"
    ],
    [
      "0.800436079",
      "1.061664214"
    ],
    [
      "1.795542806",
      "1.15151576"
    ],
    [
      "0.610300039",
      "1.844261926"
    ],
    [
      "1.439286265",
      "0.99032632"
    ],
    [
      "1.137046871",
      "1.777625391"
    ],
    [
      "1.171377178",
      "1.073591178"
    ],
    [
      "1.488271174",
      "0.20864956"
    ],
    [
      "0.263458463",
      "1.229469642"
    ],
    [
      "0.924016781",
      "1.66634962"
    ],
    [
      "1.407080128",
      "1.797890152"
    ],
    [
      "0.532528506",
      "1.144933942"
    ],
    [
      "0.563451036",
      "1.00810969"
    ],
    [
      "0.831358609",
      "0.924839962"
    ],
    [
      "0.491336579",
      "0.776092491"
    ],
    [
      "1.32598983",
      "0.389469918"
    ],
    [
      "0.666483049",
      "1.103301513"
    ],
    [
      "1.676272821",
      "1.043885714"
    ],
    [
      "0.934390622",
      "1.020031784"
    ],
    [
      "0.340360361",
      "0.543885138"
    ],
    [
      "0.849072366",
      "1.784926235"
    ],
    [
      "1.812951314",
      "1.012325161"
    ],
    [
      "1.294751608",
      "1.881910283"
    ],
    [
      "0.196688716",
      "1.468332402"
    ],
    [
      "0.841624545",
      "1.43050941"
    ],
    [
      "0.460419023",
      "0.912917867"
    ],
    [
      "1.168743624",
      "0.085576768"
    ],
    [
      "0.398229979",
      "1.186488175"
    ],
    [
      "1.924789654",
      "1.09699667"
    ],
    [
      "1.708496978",
      "1.421592188"
    ],
    [
      "0.707670001",
      "1.47214184"
    ],
    [
      "1.192035287",
      "0.431102347"
    ],
    [
      "1.099267696",
      "0.841575103"
    ],
    [
      "0.955048731",
      "0.377542954"
    ],
    [
      "1.109532117",
      "1.347239682"
    ],
    [
      "0.769515062",
      "1.198493336"
    ],
    [
      "1.273062916",
      "1.743322118"
    ],
    [
      "1.243486661",
      "1.305607253"
    ],
    [
      "1.064034033",
      "1.657849749"
    ],
    [
      "1.201344077",
      "1.622501406"
    ],
    [
      "1.304376479",
      "0.12136514"
    ],
    [
      "1.449550687",
      "1.495990899"
    ],
    [
      "0.073511832",
      "0.920854994"
    ],
    [
      "0.989592353",
      "1.792124542"
    ],
    [
      "0.094431354",
      "1.211162699"
    ],
    [
      "1.652153594",
      "1.662827239"
    ],
    [
      "1.895681534",
      "1.249764071"
    ],
    [
      "0.191245012",
      "0.997117548"
    ],
    [
      "1.027158213",
      "0.609559028"
    ],
    [
      "1.664733951",
      "0.534319573"
    ],
    [
      "1.264144769",
      "0.663118422"
    ],
    [
      "1.305331722",
      "1.031958749"
    ],
    [
      "0.376828707",
      "1.690146214"
    ],
    [
      "0.86228114",
      "0.78801571"
    ],
    [
      "0.697404066",
      "0.966472391"
    ],
    [
      "1.714652818",
      "0.908649807"
    ],
    [
      "0.594372053",
      "0.871280568"
    ],
    [
      "1.648474289",
      "1.181386611"
    ]
  ],
  "radius": "0.070137505"
}

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

讨论区

聊思路、贴方法、问为什么卡住。所有登录用户都可以发帖;发言公开署名,与纪录使用同一个名字,署名后的 #编号是账号注册序号,冒不了名。新发言经自动审核后公开。

还没有帖子。第一个聊聊这道题的思路?