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 = 162

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 = 162
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[0.0713767094599775854524583379665, 0.0713767094599775854524583379675]见来源署名packomania.com ↗
本站纪录0.071376709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 = 0.071376709

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 = 162 的当前纪录构型,0.071376709
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
    0.071376708→0.071376709
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": [
    [
      "1.483028898",
      "0.359834960"
    ],
    [
      "0.662207247",
      "1.865007093"
    ],
    [
      "1.619834256",
      "1.691481533"
    ],
    [
      "1.096934920",
      "0.101956008"
    ],
    [
      "1.150051826",
      "0.993133923"
    ],
    [
      "0.510359525",
      "1.136589433"
    ],
    [
      "0.362855743",
      "0.892137814"
    ],
    [
      "0.473669897",
      "0.403021598"
    ],
    [
      "0.424524078",
      "0.763391726"
    ],
    [
      "1.493105237",
      "1.625767803"
    ],
    [
      "1.764872247",
      "0.539142831"
    ],
    [
      "0.940740119",
      "1.618503832"
    ],
    [
      "0.712216753",
      "0.506828483"
    ],
    [
      "0.474095754",
      "1.400376069"
    ],
    [
      "0.441367802",
      "0.621635501"
    ],
    [
      "1.374361856",
      "0.150179774"
    ],
    [
      "1.642969129",
      "0.613427207"
    ],
    [
      "1.724482866",
      "1.248828582"
    ],
    [
      "0.957283217",
      "0.072359718"
    ],
    [
      "1.506526275",
      "1.778313785"
    ],
    [
      "1.016058722",
      "1.497236992"
    ],
    [
      "1.005020116",
      "1.801527846"
    ],
    [
      "0.588644520",
      "1.742667142"
    ],
    [
      "0.400245490",
      "1.278209473"
    ],
    [
      "1.578244435",
      "0.985522279"
    ],
    [
      "1.372558179",
      "1.361333586"
    ],
    [
      "1.713695053",
      "0.405878306"
    ],
    [
      "1.207771349",
      "1.621753850"
    ],
    [
      "1.858775179",
      "0.646663889"
    ],
    [
      "0.998531787",
      "0.501236901"
    ],
    [
      "1.154446411",
      "1.240351041"
    ],
    [
      "1.381248247",
      "1.846753205"
    ],
    [
      "1.426653020",
      "0.490984871"
    ],
    [
      "1.049556302",
      "0.367913844"
    ],
    [
      "1.147504999",
      "1.792775899"
    ],
    [
      "1.366345423",
      "1.112936660"
    ],
    [
      "1.288388111",
      "0.743379590"
    ],
    [
      "0.323959460",
      "0.363360067"
    ],
    [
      "1.648972781",
      "0.861522119"
    ],
    [
      "1.575287405",
      "0.476115035"
    ],
    [
      "0.909878654",
      "0.338440242"
    ],
    [
      "0.436796798",
      "1.014249481"
    ],
    [
      "0.368103424",
      "0.499116647"
    ],
    [
      "0.979027453",
      "0.213549766"
    ],
    [
      "0.151025986",
      "1.016922591"
    ],
    [
      "0.798060652",
      "1.613909428"
    ],
    [
      "1.073641240",
      "1.670617462"
    ],
    [
      "1.905179304",
      "1.207344264"
    ],
    [
      "1.301228997",
      "1.485236457"
    ],
    [
      "0.815607511",
      "0.089867798"
    ],
    [
      "1.902784257",
      "0.782464255"
    ],
    [
      "1.575410054",
      "0.739182167"
    ],
    [
      "0.933758229",
      "0.873331185"
    ],
    [
      "1.011715541",
      "1.242888255"
    ],
    [
      "1.214825384",
      "0.621039638"
    ],
    [
      "1.509076293",
      "1.110399445"
    ],
    [
      "0.721859217",
      "1.000745566"
    ],
    [
      "1.800175611",
      "1.471232647"
    ],
    [
      "0.874054856",
      "1.744724331"
    ],
    [
      "0.293779333",
      "1.017065747"
    ],
    [
      "0.543263897",
      "1.275498903"
    ],
    [
      "0.873379256",
      "1.492642588"
    ],
    [
      "0.643048611",
      "0.631705650"
    ],
    [
      "1.072094514",
      "0.623576853"
    ],
    [
      "1.361950838",
      "0.865719542"
    ],
    [
      "1.839122601",
      "1.333894849"
    ],
    [
      "0.804886713",
      "1.869601497"
    ],
    [
      "0.795421944",
      "1.123085518"
    ],
    [
      "0.868984672",
      "1.245425470"
    ],
    [
      "0.938152814",
      "1.120548304"
    ],
    [
      "1.297177281",
      "1.237813826"
    ],
    [
      "0.724497925",
      "1.491569477"
    ],
    [
      "1.720634270",
      "0.995704297"
    ],
    [
      "0.246209467",
      "1.542347719"
    ],
    [
      "0.139041763",
      "0.765890611"
    ],
    [
      "0.168906053",
      "1.414117599"
    ],
    [
      "1.500191338",
      "0.217599948"
    ],
    [
      "1.340369405",
      "0.365012562"
    ],
    [
      "1.145657242",
      "0.745916804"
    ],
    [
      "1.791362616",
      "0.871704137"
    ],
    [
      "0.257630950",
      "1.284504831"
    ],
    [
      "0.077373428",
      "0.894636699"
    ],
    [
      "0.220102396",
      "0.891994659"
    ],
    [
      "0.717464633",
      "0.753528448"
    ],
    [
      "0.547240875",
      "0.525878354"
    ],
    [
      "1.223614553",
      "1.115473874"
    ],
    [
      "0.221184376",
      "0.494263847"
    ],
    [
      "1.581751996",
      "1.251365796"
    ],
    [
      "0.678289301",
      "0.128883795"
    ],
    [
      "0.655329783",
      "1.616446643"
    ],
    [
      "0.799816529",
      "1.370302636"
    ],
    [
      "0.726253802",
      "1.247962685"
    ],
    [
      "0.445913650",
      "1.745204356"
    ],
    [
      "1.239685588",
      "0.102842247"
    ],
    [
      "0.581767056",
      "1.494106691"
    ],
    [
      "0.225016022",
      "1.145527117"
    ],
    [
      "1.500215747",
      "0.613324822"
    ],
    [
      "1.357556254",
      "0.618502424"
    ],
    [
      "0.676752709",
      "0.271628944"
    ],
    [
      "1.926263749",
      "1.066156500"
    ],
    [
      "1.246960696",
      "1.895182456"
    ],
    [
      "0.942547399",
      "1.367765422"
    ],
    [
      "1.158498128",
      "1.487773671"
    ],
    [
      "0.429525933",
      "0.267265017"
    ],
    [
      "0.548573644",
      "0.188485699"
    ],
    [
      "0.935851973",
      "1.926405012"
    ],
    [
      "1.141262657",
      "0.498699686"
    ],
    [
      "0.579128348",
      "1.003282781"
    ],
    [
      "1.591765517",
      "1.522594707"
    ],
    [
      "0.151060995",
      "0.623644076"
    ],
    [
      "1.431232822",
      "0.740773851"
    ],
    [
      "1.439908150",
      "1.235276611"
    ],
    [
      "0.860195502",
      "0.750991234"
    ],
    [
      "0.404927612",
      "1.525253236"
    ],
    [
      "1.292782696",
      "0.990596708"
    ],
    [
      "1.614200479",
      "0.303509521"
    ],
    [
      "0.648154471",
      "0.878327057"
    ],
    [
      "1.734474146",
      "0.734317244"
    ],
    [
      "1.219219969",
      "0.868256756"
    ],
    [
      "0.791027360",
      "0.875868400"
    ],
    [
      "0.786632775",
      "0.628651282"
    ],
    [
      "0.731375389",
      "1.740129927"
    ],
    [
      "1.192287171",
      "0.365376629"
    ],
    [
      "1.650920139",
      "1.126488630"
    ],
    [
      "0.082305044",
      "1.142046412"
    ],
    [
      "1.443832015",
      "1.491787625"
    ],
    [
      "0.854758596",
      "0.497085649"
    ],
    [
      "1.085278268",
      "1.365228207"
    ],
    [
      "0.114919972",
      "1.281024126"
    ],
    [
      "0.864590087",
      "0.998208352"
    ],
    [
      "0.650935198",
      "1.369229525"
    ],
    [
      "1.118724444",
      "0.243036678"
    ],
    [
      "1.080883684",
      "1.118011089"
    ],
    [
      "1.283993527",
      "0.496162472"
    ],
    [
      "1.435513566",
      "0.988059493"
    ],
    [
      "0.293789963",
      "0.621002036"
    ],
    [
      "1.925459062",
      "0.923405349"
    ],
    [
      "0.616409018",
      "0.401001188"
    ],
    [
      "0.505187293",
      "0.881171115"
    ],
    [
      "1.002926372",
      "0.748454019"
    ],
    [
      "1.266028221",
      "0.243144079"
    ],
    [
      "1.281792550",
      "1.744346648"
    ],
    [
      "0.512598913",
      "1.618983857"
    ],
    [
      "1.087315461",
      "1.922983833"
    ],
    [
      "1.076489099",
      "0.870793971"
    ],
    [
      "0.281770730",
      "0.763248571"
    ],
    [
      "1.542492295",
      "1.388614529"
    ],
    [
      "0.566855628",
      "0.752425027"
    ],
    [
      "1.228009138",
      "1.362690992"
    ],
    [
      "1.350502219",
      "1.619216635"
    ],
    [
      "0.367630562",
      "1.139231759"
    ],
    [
      "0.652691075",
      "1.125622733"
    ],
    [
      "0.772560445",
      "0.377456239"
    ],
    [
      "0.814070918",
      "0.232612947"
    ],
    [
      "0.929363645",
      "0.626114067"
    ],
    [
      "0.338242349",
      "1.651473734"
    ],
    [
      "1.007320957",
      "0.995671137"
    ],
    [
      "1.685247912",
      "1.386550233"
    ],
    [
      "1.718494536",
      "1.588308437"
    ],
    [
      "1.793449336",
      "1.118490717"
    ],
    [
      "0.312894730",
      "1.416127220"
    ],
    [
      "1.504681708",
      "0.863182327"
    ]
  ],
  "radius": "0.071376709"
}
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": 162
}

当前第一名的答案

{
  "centers": [
    [
      "1.483028898",
      "0.359834960"
    ],
    [
      "0.662207247",
      "1.865007093"
    ],
    [
      "1.619834256",
      "1.691481533"
    ],
    [
      "1.096934920",
      "0.101956008"
    ],
    [
      "1.150051826",
      "0.993133923"
    ],
    [
      "0.510359525",
      "1.136589433"
    ],
    [
      "0.362855743",
      "0.892137814"
    ],
    [
      "0.473669897",
      "0.403021598"
    ],
    [
      "0.424524078",
      "0.763391726"
    ],
    [
      "1.493105237",
      "1.625767803"
    ],
    [
      "1.764872247",
      "0.539142831"
    ],
    [
      "0.940740119",
      "1.618503832"
    ],
    [
      "0.712216753",
      "0.506828483"
    ],
    [
      "0.474095754",
      "1.400376069"
    ],
    [
      "0.441367802",
      "0.621635501"
    ],
    [
      "1.374361856",
      "0.150179774"
    ],
    [
      "1.642969129",
      "0.613427207"
    ],
    [
      "1.724482866",
      "1.248828582"
    ],
    [
      "0.957283217",
      "0.072359718"
    ],
    [
      "1.506526275",
      "1.778313785"
    ],
    [
      "1.016058722",
      "1.497236992"
    ],
    [
      "1.005020116",
      "1.801527846"
    ],
    [
      "0.588644520",
      "1.742667142"
    ],
    [
      "0.400245490",
      "1.278209473"
    ],
    [
      "1.578244435",
      "0.985522279"
    ],
    [
      "1.372558179",
      "1.361333586"
    ],
    [
      "1.713695053",
      "0.405878306"
    ],
    [
      "1.207771349",
      "1.621753850"
    ],
    [
      "1.858775179",
      "0.646663889"
    ],
    [
      "0.998531787",
      "0.501236901"
    ],
    [
      "1.154446411",
      "1.240351041"
    ],
    [
      "1.381248247",
      "1.846753205"
    ],
    [
      "1.426653020",
      "0.490984871"
    ],
    [
      "1.049556302",
      "0.367913844"
    ],
    [
      "1.147504999",
      "1.792775899"
    ],
    [
      "1.366345423",
      "1.112936660"
    ],
    [
      "1.288388111",
      "0.743379590"
    ],
    [
      "0.323959460",
      "0.363360067"
    ],
    [
      "1.648972781",
      "0.861522119"
    ],
    [
      "1.575287405",
      "0.476115035"
    ],
    [
      "0.909878654",
      "0.338440242"
    ],
    [
      "0.436796798",
      "1.014249481"
    ],
    [
      "0.368103424",
      "0.499116647"
    ],
    [
      "0.979027453",
      "0.213549766"
    ],
    [
      "0.151025986",
      "1.016922591"
    ],
    [
      "0.798060652",
      "1.613909428"
    ],
    [
      "1.073641240",
      "1.670617462"
    ],
    [
      "1.905179304",
      "1.207344264"
    ],
    [
      "1.301228997",
      "1.485236457"
    ],
    [
      "0.815607511",
      "0.089867798"
    ],
    [
      "1.902784257",
      "0.782464255"
    ],
    [
      "1.575410054",
      "0.739182167"
    ],
    [
      "0.933758229",
      "0.873331185"
    ],
    [
      "1.011715541",
      "1.242888255"
    ],
    [
      "1.214825384",
      "0.621039638"
    ],
    [
      "1.509076293",
      "1.110399445"
    ],
    [
      "0.721859217",
      "1.000745566"
    ],
    [
      "1.800175611",
      "1.471232647"
    ],
    [
      "0.874054856",
      "1.744724331"
    ],
    [
      "0.293779333",
      "1.017065747"
    ],
    [
      "0.543263897",
      "1.275498903"
    ],
    [
      "0.873379256",
      "1.492642588"
    ],
    [
      "0.643048611",
      "0.631705650"
    ],
    [
      "1.072094514",
      "0.623576853"
    ],
    [
      "1.361950838",
      "0.865719542"
    ],
    [
      "1.839122601",
      "1.333894849"
    ],
    [
      "0.804886713",
      "1.869601497"
    ],
    [
      "0.795421944",
      "1.123085518"
    ],
    [
      "0.868984672",
      "1.245425470"
    ],
    [
      "0.938152814",
      "1.120548304"
    ],
    [
      "1.297177281",
      "1.237813826"
    ],
    [
      "0.724497925",
      "1.491569477"
    ],
    [
      "1.720634270",
      "0.995704297"
    ],
    [
      "0.246209467",
      "1.542347719"
    ],
    [
      "0.139041763",
      "0.765890611"
    ],
    [
      "0.168906053",
      "1.414117599"
    ],
    [
      "1.500191338",
      "0.217599948"
    ],
    [
      "1.340369405",
      "0.365012562"
    ],
    [
      "1.145657242",
      "0.745916804"
    ],
    [
      "1.791362616",
      "0.871704137"
    ],
    [
      "0.257630950",
      "1.284504831"
    ],
    [
      "0.077373428",
      "0.894636699"
    ],
    [
      "0.220102396",
      "0.891994659"
    ],
    [
      "0.717464633",
      "0.753528448"
    ],
    [
      "0.547240875",
      "0.525878354"
    ],
    [
      "1.223614553",
      "1.115473874"
    ],
    [
      "0.221184376",
      "0.494263847"
    ],
    [
      "1.581751996",
      "1.251365796"
    ],
    [
      "0.678289301",
      "0.128883795"
    ],
    [
      "0.655329783",
      "1.616446643"
    ],
    [
      "0.799816529",
      "1.370302636"
    ],
    [
      "0.726253802",
      "1.247962685"
    ],
    [
      "0.445913650",
      "1.745204356"
    ],
    [
      "1.239685588",
      "0.102842247"
    ],
    [
      "0.581767056",
      "1.494106691"
    ],
    [
      "0.225016022",
      "1.145527117"
    ],
    [
      "1.500215747",
      "0.613324822"
    ],
    [
      "1.357556254",
      "0.618502424"
    ],
    [
      "0.676752709",
      "0.271628944"
    ],
    [
      "1.926263749",
      "1.066156500"
    ],
    [
      "1.246960696",
      "1.895182456"
    ],
    [
      "0.942547399",
      "1.367765422"
    ],
    [
      "1.158498128",
      "1.487773671"
    ],
    [
      "0.429525933",
      "0.267265017"
    ],
    [
      "0.548573644",
      "0.188485699"
    ],
    [
      "0.935851973",
      "1.926405012"
    ],
    [
      "1.141262657",
      "0.498699686"
    ],
    [
      "0.579128348",
      "1.003282781"
    ],
    [
      "1.591765517",
      "1.522594707"
    ],
    [
      "0.151060995",
      "0.623644076"
    ],
    [
      "1.431232822",
      "0.740773851"
    ],
    [
      "1.439908150",
      "1.235276611"
    ],
    [
      "0.860195502",
      "0.750991234"
    ],
    [
      "0.404927612",
      "1.525253236"
    ],
    [
      "1.292782696",
      "0.990596708"
    ],
    [
      "1.614200479",
      "0.303509521"
    ],
    [
      "0.648154471",
      "0.878327057"
    ],
    [
      "1.734474146",
      "0.734317244"
    ],
    [
      "1.219219969",
      "0.868256756"
    ],
    [
      "0.791027360",
      "0.875868400"
    ],
    [
      "0.786632775",
      "0.628651282"
    ],
    [
      "0.731375389",
      "1.740129927"
    ],
    [
      "1.192287171",
      "0.365376629"
    ],
    [
      "1.650920139",
      "1.126488630"
    ],
    [
      "0.082305044",
      "1.142046412"
    ],
    [
      "1.443832015",
      "1.491787625"
    ],
    [
      "0.854758596",
      "0.497085649"
    ],
    [
      "1.085278268",
      "1.365228207"
    ],
    [
      "0.114919972",
      "1.281024126"
    ],
    [
      "0.864590087",
      "0.998208352"
    ],
    [
      "0.650935198",
      "1.369229525"
    ],
    [
      "1.118724444",
      "0.243036678"
    ],
    [
      "1.080883684",
      "1.118011089"
    ],
    [
      "1.283993527",
      "0.496162472"
    ],
    [
      "1.435513566",
      "0.988059493"
    ],
    [
      "0.293789963",
      "0.621002036"
    ],
    [
      "1.925459062",
      "0.923405349"
    ],
    [
      "0.616409018",
      "0.401001188"
    ],
    [
      "0.505187293",
      "0.881171115"
    ],
    [
      "1.002926372",
      "0.748454019"
    ],
    [
      "1.266028221",
      "0.243144079"
    ],
    [
      "1.281792550",
      "1.744346648"
    ],
    [
      "0.512598913",
      "1.618983857"
    ],
    [
      "1.087315461",
      "1.922983833"
    ],
    [
      "1.076489099",
      "0.870793971"
    ],
    [
      "0.281770730",
      "0.763248571"
    ],
    [
      "1.542492295",
      "1.388614529"
    ],
    [
      "0.566855628",
      "0.752425027"
    ],
    [
      "1.228009138",
      "1.362690992"
    ],
    [
      "1.350502219",
      "1.619216635"
    ],
    [
      "0.367630562",
      "1.139231759"
    ],
    [
      "0.652691075",
      "1.125622733"
    ],
    [
      "0.772560445",
      "0.377456239"
    ],
    [
      "0.814070918",
      "0.232612947"
    ],
    [
      "0.929363645",
      "0.626114067"
    ],
    [
      "0.338242349",
      "1.651473734"
    ],
    [
      "1.007320957",
      "0.995671137"
    ],
    [
      "1.685247912",
      "1.386550233"
    ],
    [
      "1.718494536",
      "1.588308437"
    ],
    [
      "1.793449336",
      "1.118490717"
    ],
    [
      "0.312894730",
      "1.416127220"
    ],
    [
      "1.504681708",
      "0.863182327"
    ]
  ],
  "radius": "0.071376709"
}

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?