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 = 179

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 = 179
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[0.0680278992525827299460776543205, 0.0680278992525827299460776543215]见来源署名packomania.com ↗
本站纪录0.068027899游侠纪录详情 ↓

本站纪录尚未确认达到或超越此参考纪录。

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:本站纪录

2y0
0x2
已验证构造r = 0.068027899

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 = 179 的当前纪录构型,0.068027899
构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者游侠
解题方式AI · OpenAI: GPT-5.6 Sol
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。
纪录保持者的求解笔记暂无求解笔记(点击展开)+

纪录保持者还没有分享求解过程。

历史纪录(1 次易主)
  1. 游侠AI · OpenAI: GPT-5.6 Sol
    0.068027898→0.068027899
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"。

当前第一名的答案

{
  "centers": [
    [
      "1.067268453",
      "0.320215841"
    ],
    [
      "1.797698647",
      "0.984642355"
    ],
    [
      "0.491681833",
      "1.218436628"
    ],
    [
      "0.485171723",
      "0.946402917"
    ],
    [
      "1.660364521",
      "0.342359752"
    ],
    [
      "0.853201624",
      "1.550021887"
    ],
    [
      "1.197876426",
      "0.358330642"
    ],
    [
      "0.824405955",
      "0.325859147"
    ],
    [
      "0.476136343",
      "0.537032890"
    ],
    [
      "0.949837824",
      "0.663093368"
    ],
    [
      "1.082279621",
      "1.272350258"
    ],
    [
      "0.537162437",
      "0.191078875"
    ],
    [
      "0.304956265",
      "0.621906433"
    ],
    [
      "0.611103412",
      "1.283626096"
    ],
    [
      "0.185019690",
      "0.997730709"
    ],
    [
      "0.615983815",
      "1.558561025"
    ],
    [
      "0.362495090",
      "0.745196594"
    ],
    [
      "1.557576799",
      "0.253220173"
    ],
    [
      "1.264805206",
      "1.765204250"
    ],
    [
      "1.085534674",
      "1.408367113"
    ],
    [
      "0.226953266",
      "0.733381598"
    ],
    [
      "1.544232868",
      "0.710042849"
    ],
    [
      "0.707739610",
      "0.396697577"
    ],
    [
      "1.301729363",
      "0.586377024"
    ],
    [
      "0.614358466",
      "1.419642951"
    ],
    [
      "0.381719694",
      "1.697353181"
    ],
    [
      "1.624474413",
      "0.473596486"
    ],
    [
      "1.688710420",
      "1.413449372"
    ],
    [
      "0.969368149",
      "1.479194500"
    ],
    [
      "0.459069058",
      "0.398883454"
    ],
    [
      "1.430779195",
      "1.059868096"
    ],
    [
      "1.298144310",
      "0.450295482"
    ],
    [
      "0.073581677",
      "0.912651813"
    ],
    [
      "1.159192990",
      "0.219911007"
    ],
    [
      "1.361615802",
      "1.669606715"
    ],
    [
      "1.304847507",
      "0.722644917"
    ],
    [
      "0.375515308",
      "1.289264015"
    ],
    [
      "1.191936037",
      "0.929489160"
    ],
    [
      "0.243786718",
      "1.408233985"
    ],
    [
      "0.489841069",
      "1.779942217"
    ],
    [
      "0.737028972",
      "1.893881313"
    ],
    [
      "1.075769512",
      "1.000316547"
    ],
    [
      "0.710994664",
      "0.532714432"
    ],
    [
      "0.372260253",
      "1.153247160"
    ],
    [
      "1.669855416",
      "1.189818253"
    ],
    [
      "0.230208320",
      "0.869398453"
    ],
    [
      "0.972623203",
      "1.615211357"
    ],
    [
      "0.243480997",
      "1.544289440"
    ],
    [
      "1.198446145",
      "1.201522870"
    ],
    [
      "0.604593302",
      "1.011592385"
    ],
    [
      "1.660631073",
      "0.783481624"
    ],
    [
      "1.453749519",
      "1.496540594"
    ],
    [
      "1.786786189",
      "1.120259826"
    ],
    [
      "0.243030055",
      "1.258298904"
    ],
    [
      "1.364582164",
      "1.857701488"
    ],
    [
      "0.607848358",
      "1.147609240"
    ],
    [
      "0.498191942",
      "1.490470339"
    ],
    [
      "0.494936887",
      "1.354453483"
    ],
    [
      "1.553023829",
      "1.403432916"
    ],
    [
      "0.953092879",
      "0.799110224"
    ],
    [
      "1.434034249",
      "1.195884951"
    ],
    [
      "0.946582769",
      "0.527076512"
    ],
    [
      "0.147811056",
      "0.622712312"
    ],
    [
      "1.429762155",
      "0.484761742"
    ],
    [
      "1.321122779",
      "1.402729194"
    ],
    [
      "1.666600362",
      "1.053801397"
    ],
    [
      "0.874089513",
      "0.199102212"
    ],
    [
      "0.211824231",
      "0.502656101"
    ],
    [
      "1.072514458",
      "0.864299692"
    ],
    [
      "0.733780045",
      "1.484832419"
    ],
    [
      "1.830354274",
      "0.576817090"
    ],
    [
      "1.922998320",
      "0.870980246"
    ],
    [
      "0.609658553",
      "1.845943807"
    ],
    [
      "0.856456678",
      "1.686038744"
    ],
    [
      "1.188680982",
      "0.793472304"
    ],
    [
      "0.347513630",
      "0.492677756"
    ],
    [
      "0.956347933",
      "0.935127079"
    ],
    [
      "0.830416245",
      "0.597903900"
    ],
    [
      "1.759891539",
      "0.460429018"
    ],
    [
      "0.714249720",
      "0.668731287"
    ],
    [
      "1.461392760",
      "1.762103952"
    ],
    [
      "1.071997662",
      "1.929186920"
    ],
    [
      "1.764922436",
      "0.696106045"
    ],
    [
      "1.069259403",
      "0.728282836"
    ],
    [
      "1.921035260",
      "1.142358862"
    ],
    [
      "0.737035099",
      "1.620849274"
    ],
    [
      "0.947261221",
      "0.384320791"
    ],
    [
      "0.606007593",
      "1.709114829"
    ],
    [
      "1.010132998",
      "0.196623902"
    ],
    [
      "0.846691516",
      "1.277988177"
    ],
    [
      "0.097008036",
      "0.774963700"
    ],
    [
      "0.424317543",
      "0.267086837"
    ],
    [
      "1.550200774",
      "1.125057564"
    ],
    [
      "0.334302336",
      "1.017072182"
    ],
    [
      "1.311357616",
      "0.994678628"
    ],
    [
      "1.546945719",
      "0.989040708"
    ],
    [
      "1.755949728",
      "1.295169806"
    ],
    [
      "0.070784922",
      "1.071633334"
    ],
    [
      "0.959602987",
      "1.071143934"
    ],
    [
      "0.724014883",
      "1.076781853"
    ],
    [
      "0.852833576",
      "1.822046293"
    ],
    [
      "1.890493514",
      "1.274942353"
    ],
    [
      "1.182170874",
      "0.521438594"
    ],
    [
      "1.521686691",
      "0.384456907"
    ],
    [
      "1.532549877",
      "0.573901321"
    ],
    [
      "1.390068846",
      "0.349990646"
    ],
    [
      "1.088789728",
      "1.544383968"
    ],
    [
      "0.843436462",
      "1.141971321"
    ],
    [
      "1.678911574",
      "0.918303742"
    ],
    [
      "1.314612670",
      "1.130695483"
    ],
    [
      "1.195191091",
      "1.065506015"
    ],
    [
      "1.425958954",
      "0.218753912"
    ],
    [
      "0.481916669",
      "0.810386062"
    ],
    [
      "0.585030630",
      "0.455465650"
    ],
    [
      "0.720759827",
      "0.940764998"
    ],
    [
      "1.195083098",
      "0.088674273"
    ],
    [
      "0.789829287",
      "0.092035096"
    ],
    [
      "0.121030878",
      "1.198071161"
    ],
    [
      "0.849946570",
      "1.414005032"
    ],
    [
      "0.730524991",
      "1.348815564"
    ],
    [
      "0.836926353",
      "0.869937610"
    ],
    [
      "1.421014031",
      "0.651817528"
    ],
    [
      "0.962858042",
      "1.207160790"
    ],
    [
      "1.325691070",
      "0.126789073"
    ],
    [
      "0.833671299",
      "0.733920755"
    ],
    [
      "0.935943104",
      "1.929768095"
    ],
    [
      "1.590975855",
      "1.720638283"
    ],
    [
      "0.378464640",
      "1.561336326"
    ],
    [
      "1.840973392",
      "1.401666218"
    ],
    [
      "0.727269937",
      "1.212798709"
    ],
    [
      "1.553905863",
      "1.261792968"
    ],
    [
      "1.437289303",
      "1.331901806"
    ],
    [
      "1.221324258",
      "1.629128079"
    ],
    [
      "0.378770361",
      "1.425280871"
    ],
    [
      "0.477997634",
      "0.673269206"
    ],
    [
      "0.598083194",
      "0.739558675"
    ],
    [
      "1.062749294",
      "0.456249125"
    ],
    [
      "1.060317476",
      "0.069981829"
    ],
    [
      "0.129871711",
      "1.333839419"
    ],
    [
      "0.601338248",
      "0.875575530"
    ],
    [
      "0.659871418",
      "0.132310802"
    ],
    [
      "1.665378680",
      "0.603357887"
    ],
    [
      "0.582275395",
      "0.319437752"
    ],
    [
      "0.827161189",
      "0.461887045"
    ],
    [
      "1.773530276",
      "1.519829691"
    ],
    [
      "1.317867724",
      "1.266712338"
    ],
    [
      "0.966113095",
      "1.343177645"
    ],
    [
      "1.308102561",
      "0.858661772"
    ],
    [
      "0.717504774",
      "0.804748142"
    ],
    [
      "1.066004349",
      "0.592265981"
    ],
    [
      "1.589824572",
      "1.534417211"
    ],
    [
      "1.689601528",
      "1.626914450"
    ],
    [
      "1.790562212",
      "0.834088791"
    ],
    [
      "1.289800962",
      "0.258025807"
    ],
    [
      "0.704984376",
      "0.260669679"
    ],
    [
      "0.594828139",
      "0.603541820"
    ],
    [
      "0.993306129",
      "1.785153985"
    ],
    [
      "0.497886219",
      "1.626525794"
    ],
    [
      "1.427524140",
      "0.923851240"
    ],
    [
      "0.924266356",
      "0.071110119"
    ],
    [
      "1.079024566",
      "1.136333402"
    ],
    [
      "1.204956253",
      "1.473556581"
    ],
    [
      "1.201701200",
      "1.337539726"
    ],
    [
      "1.324377832",
      "1.538746049"
    ],
    [
      "0.840181407",
      "1.005954465"
    ],
    [
      "1.424269086",
      "0.787834385"
    ],
    [
      "0.323741709",
      "0.358714791"
    ],
    [
      "0.733411997",
      "1.756856824"
    ],
    [
      "1.543690665",
      "0.853023853"
    ],
    [
      "1.136040943",
      "1.809146767"
    ],
    [
      "0.365750144",
      "0.881213449"
    ],
    [
      "1.931947718",
      "1.006741391"
    ],
    [
      "1.092044783",
      "1.680400823"
    ],
    [
      "1.185425928",
      "0.657455449"
    ],
    [
      "0.239775000",
      "1.122282049"
    ],
    [
      "1.894377797",
      "0.737968795"
    ],
    [
      "0.488426778",
      "1.082419773"
    ],
    [
      "1.491198898",
      "1.628141044"
    ],
    [
      "1.235817901",
      "1.901644006"
    ]
  ],
  "radius": "0.068027899"
}
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": 179
}

当前第一名的答案

{
  "centers": [
    [
      "1.067268453",
      "0.320215841"
    ],
    [
      "1.797698647",
      "0.984642355"
    ],
    [
      "0.491681833",
      "1.218436628"
    ],
    [
      "0.485171723",
      "0.946402917"
    ],
    [
      "1.660364521",
      "0.342359752"
    ],
    [
      "0.853201624",
      "1.550021887"
    ],
    [
      "1.197876426",
      "0.358330642"
    ],
    [
      "0.824405955",
      "0.325859147"
    ],
    [
      "0.476136343",
      "0.537032890"
    ],
    [
      "0.949837824",
      "0.663093368"
    ],
    [
      "1.082279621",
      "1.272350258"
    ],
    [
      "0.537162437",
      "0.191078875"
    ],
    [
      "0.304956265",
      "0.621906433"
    ],
    [
      "0.611103412",
      "1.283626096"
    ],
    [
      "0.185019690",
      "0.997730709"
    ],
    [
      "0.615983815",
      "1.558561025"
    ],
    [
      "0.362495090",
      "0.745196594"
    ],
    [
      "1.557576799",
      "0.253220173"
    ],
    [
      "1.264805206",
      "1.765204250"
    ],
    [
      "1.085534674",
      "1.408367113"
    ],
    [
      "0.226953266",
      "0.733381598"
    ],
    [
      "1.544232868",
      "0.710042849"
    ],
    [
      "0.707739610",
      "0.396697577"
    ],
    [
      "1.301729363",
      "0.586377024"
    ],
    [
      "0.614358466",
      "1.419642951"
    ],
    [
      "0.381719694",
      "1.697353181"
    ],
    [
      "1.624474413",
      "0.473596486"
    ],
    [
      "1.688710420",
      "1.413449372"
    ],
    [
      "0.969368149",
      "1.479194500"
    ],
    [
      "0.459069058",
      "0.398883454"
    ],
    [
      "1.430779195",
      "1.059868096"
    ],
    [
      "1.298144310",
      "0.450295482"
    ],
    [
      "0.073581677",
      "0.912651813"
    ],
    [
      "1.159192990",
      "0.219911007"
    ],
    [
      "1.361615802",
      "1.669606715"
    ],
    [
      "1.304847507",
      "0.722644917"
    ],
    [
      "0.375515308",
      "1.289264015"
    ],
    [
      "1.191936037",
      "0.929489160"
    ],
    [
      "0.243786718",
      "1.408233985"
    ],
    [
      "0.489841069",
      "1.779942217"
    ],
    [
      "0.737028972",
      "1.893881313"
    ],
    [
      "1.075769512",
      "1.000316547"
    ],
    [
      "0.710994664",
      "0.532714432"
    ],
    [
      "0.372260253",
      "1.153247160"
    ],
    [
      "1.669855416",
      "1.189818253"
    ],
    [
      "0.230208320",
      "0.869398453"
    ],
    [
      "0.972623203",
      "1.615211357"
    ],
    [
      "0.243480997",
      "1.544289440"
    ],
    [
      "1.198446145",
      "1.201522870"
    ],
    [
      "0.604593302",
      "1.011592385"
    ],
    [
      "1.660631073",
      "0.783481624"
    ],
    [
      "1.453749519",
      "1.496540594"
    ],
    [
      "1.786786189",
      "1.120259826"
    ],
    [
      "0.243030055",
      "1.258298904"
    ],
    [
      "1.364582164",
      "1.857701488"
    ],
    [
      "0.607848358",
      "1.147609240"
    ],
    [
      "0.498191942",
      "1.490470339"
    ],
    [
      "0.494936887",
      "1.354453483"
    ],
    [
      "1.553023829",
      "1.403432916"
    ],
    [
      "0.953092879",
      "0.799110224"
    ],
    [
      "1.434034249",
      "1.195884951"
    ],
    [
      "0.946582769",
      "0.527076512"
    ],
    [
      "0.147811056",
      "0.622712312"
    ],
    [
      "1.429762155",
      "0.484761742"
    ],
    [
      "1.321122779",
      "1.402729194"
    ],
    [
      "1.666600362",
      "1.053801397"
    ],
    [
      "0.874089513",
      "0.199102212"
    ],
    [
      "0.211824231",
      "0.502656101"
    ],
    [
      "1.072514458",
      "0.864299692"
    ],
    [
      "0.733780045",
      "1.484832419"
    ],
    [
      "1.830354274",
      "0.576817090"
    ],
    [
      "1.922998320",
      "0.870980246"
    ],
    [
      "0.609658553",
      "1.845943807"
    ],
    [
      "0.856456678",
      "1.686038744"
    ],
    [
      "1.188680982",
      "0.793472304"
    ],
    [
      "0.347513630",
      "0.492677756"
    ],
    [
      "0.956347933",
      "0.935127079"
    ],
    [
      "0.830416245",
      "0.597903900"
    ],
    [
      "1.759891539",
      "0.460429018"
    ],
    [
      "0.714249720",
      "0.668731287"
    ],
    [
      "1.461392760",
      "1.762103952"
    ],
    [
      "1.071997662",
      "1.929186920"
    ],
    [
      "1.764922436",
      "0.696106045"
    ],
    [
      "1.069259403",
      "0.728282836"
    ],
    [
      "1.921035260",
      "1.142358862"
    ],
    [
      "0.737035099",
      "1.620849274"
    ],
    [
      "0.947261221",
      "0.384320791"
    ],
    [
      "0.606007593",
      "1.709114829"
    ],
    [
      "1.010132998",
      "0.196623902"
    ],
    [
      "0.846691516",
      "1.277988177"
    ],
    [
      "0.097008036",
      "0.774963700"
    ],
    [
      "0.424317543",
      "0.267086837"
    ],
    [
      "1.550200774",
      "1.125057564"
    ],
    [
      "0.334302336",
      "1.017072182"
    ],
    [
      "1.311357616",
      "0.994678628"
    ],
    [
      "1.546945719",
      "0.989040708"
    ],
    [
      "1.755949728",
      "1.295169806"
    ],
    [
      "0.070784922",
      "1.071633334"
    ],
    [
      "0.959602987",
      "1.071143934"
    ],
    [
      "0.724014883",
      "1.076781853"
    ],
    [
      "0.852833576",
      "1.822046293"
    ],
    [
      "1.890493514",
      "1.274942353"
    ],
    [
      "1.182170874",
      "0.521438594"
    ],
    [
      "1.521686691",
      "0.384456907"
    ],
    [
      "1.532549877",
      "0.573901321"
    ],
    [
      "1.390068846",
      "0.349990646"
    ],
    [
      "1.088789728",
      "1.544383968"
    ],
    [
      "0.843436462",
      "1.141971321"
    ],
    [
      "1.678911574",
      "0.918303742"
    ],
    [
      "1.314612670",
      "1.130695483"
    ],
    [
      "1.195191091",
      "1.065506015"
    ],
    [
      "1.425958954",
      "0.218753912"
    ],
    [
      "0.481916669",
      "0.810386062"
    ],
    [
      "0.585030630",
      "0.455465650"
    ],
    [
      "0.720759827",
      "0.940764998"
    ],
    [
      "1.195083098",
      "0.088674273"
    ],
    [
      "0.789829287",
      "0.092035096"
    ],
    [
      "0.121030878",
      "1.198071161"
    ],
    [
      "0.849946570",
      "1.414005032"
    ],
    [
      "0.730524991",
      "1.348815564"
    ],
    [
      "0.836926353",
      "0.869937610"
    ],
    [
      "1.421014031",
      "0.651817528"
    ],
    [
      "0.962858042",
      "1.207160790"
    ],
    [
      "1.325691070",
      "0.126789073"
    ],
    [
      "0.833671299",
      "0.733920755"
    ],
    [
      "0.935943104",
      "1.929768095"
    ],
    [
      "1.590975855",
      "1.720638283"
    ],
    [
      "0.378464640",
      "1.561336326"
    ],
    [
      "1.840973392",
      "1.401666218"
    ],
    [
      "0.727269937",
      "1.212798709"
    ],
    [
      "1.553905863",
      "1.261792968"
    ],
    [
      "1.437289303",
      "1.331901806"
    ],
    [
      "1.221324258",
      "1.629128079"
    ],
    [
      "0.378770361",
      "1.425280871"
    ],
    [
      "0.477997634",
      "0.673269206"
    ],
    [
      "0.598083194",
      "0.739558675"
    ],
    [
      "1.062749294",
      "0.456249125"
    ],
    [
      "1.060317476",
      "0.069981829"
    ],
    [
      "0.129871711",
      "1.333839419"
    ],
    [
      "0.601338248",
      "0.875575530"
    ],
    [
      "0.659871418",
      "0.132310802"
    ],
    [
      "1.665378680",
      "0.603357887"
    ],
    [
      "0.582275395",
      "0.319437752"
    ],
    [
      "0.827161189",
      "0.461887045"
    ],
    [
      "1.773530276",
      "1.519829691"
    ],
    [
      "1.317867724",
      "1.266712338"
    ],
    [
      "0.966113095",
      "1.343177645"
    ],
    [
      "1.308102561",
      "0.858661772"
    ],
    [
      "0.717504774",
      "0.804748142"
    ],
    [
      "1.066004349",
      "0.592265981"
    ],
    [
      "1.589824572",
      "1.534417211"
    ],
    [
      "1.689601528",
      "1.626914450"
    ],
    [
      "1.790562212",
      "0.834088791"
    ],
    [
      "1.289800962",
      "0.258025807"
    ],
    [
      "0.704984376",
      "0.260669679"
    ],
    [
      "0.594828139",
      "0.603541820"
    ],
    [
      "0.993306129",
      "1.785153985"
    ],
    [
      "0.497886219",
      "1.626525794"
    ],
    [
      "1.427524140",
      "0.923851240"
    ],
    [
      "0.924266356",
      "0.071110119"
    ],
    [
      "1.079024566",
      "1.136333402"
    ],
    [
      "1.204956253",
      "1.473556581"
    ],
    [
      "1.201701200",
      "1.337539726"
    ],
    [
      "1.324377832",
      "1.538746049"
    ],
    [
      "0.840181407",
      "1.005954465"
    ],
    [
      "1.424269086",
      "0.787834385"
    ],
    [
      "0.323741709",
      "0.358714791"
    ],
    [
      "0.733411997",
      "1.756856824"
    ],
    [
      "1.543690665",
      "0.853023853"
    ],
    [
      "1.136040943",
      "1.809146767"
    ],
    [
      "0.365750144",
      "0.881213449"
    ],
    [
      "1.931947718",
      "1.006741391"
    ],
    [
      "1.092044783",
      "1.680400823"
    ],
    [
      "1.185425928",
      "0.657455449"
    ],
    [
      "0.239775000",
      "1.122282049"
    ],
    [
      "1.894377797",
      "0.737968795"
    ],
    [
      "0.488426778",
      "1.082419773"
    ],
    [
      "1.491198898",
      "1.628141044"
    ],
    [
      "1.235817901",
      "1.901644006"
    ]
  ],
  "radius": "0.068027899"
}

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?