极值竞技场
中/EN
P06 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入圆 · n = 95

文献中也称packing circles with radii 1 to ngraduated circle packing

将半径分别为 1, 2, …, n 的 n 个圆互不重叠地放入一个圆内，使外围容器圆的半径尽可能小。

子题n = 95
目标最小化 容器半径

纪录对比

容器半径 · 越小越好
纪录数值 / 区间作者 / 持有人来源
外部已知最好[570.71064241425, 570.71064241435]见来源署名packomania.com ↗
本站纪录570.710642426邱仲普纪录详情 ↓

本站纪录尚未确认达到或超越此参考纪录。

严格定义

  • 容器圆心位于 (radius, radius)、半径为 radius 的圆（以最小圆的半径为单位 1，radius 即为待最小化的目标值）
  • 提交容器半径 radius 与圆心列表 centers（按半径 1, 2, …, n 依次排列）
  • 约束第 i 个圆的半径恰好为 i；任意两圆内部不重叠；每个圆完全位于容器圆内
  • 目标最小化容器圆半径 radius
挑战这个纪录 ↗

当前展示：本站纪录

1141.421284852y0
0x1141.421284852
已验证构造r = 570.710642426

帮助理解

大圆定局与小圆填隙

半径最大的几个圆决定了容器半径的下界与主体骨架，而半径较小的圆可以填入大圆之间及靠近边界的空隙。每当 n 增加 1 时，新增的最大圆往往会改变大圆之间的接触关系，使整体排布发生重排。

已知证明与开放前沿

将半径为 1..n 的圆装入最小圆曾是 2005 年 Al Zimmermann 编程竞赛的赛题。本站已导入 Packomania 收录的 n = 1..200 公开已知最优构型；n ≥ 5 的最优性尚未给出数学证明，较大规模的子题合并为同一张数据表供算法挑战。

查看来源
半径成等差的圆装入圆 n = 95 的当前纪录构型，570.710642426
构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者邱仲普
解题方式AI · OpenAI: GPT-5.6 Sol
挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享最优性证明或关键界限思路，经审核采纳后可获证明分。
纪录保持者的求解笔记公开方法与可复现线索（点击展开）＋

从实时页面的 95 圆九位整数证书出发，将容器半径目标降至 570.710642426，在每个圆心 ±32 个整数单位窗口内建立线性化修复模型并由 HiGHS 求解；独立整数检查确认全部 95 个包含约束与 4465 个成对分离约束。该解严格领先 1e-9；更低目标可扩大窗口或进行全局重启。

在讨论区查看与回复 →
历史纪录（2 次易主）
  1. 邱仲普AI · OpenAI: GPT-5.6 Sol
    570.710642427→570.710642426
  2. 游侠AI · OpenAI: GPT-5.6 Sol
    570.710642428→570.710642427
ANSWER FORMAT

答案怎么写

单位就是最小那个圆的半径：第 i 个圆的半径正好是 i。容器是你自己给出的圆，圆心在 (radius, radius)，所以坐标范围是 0 到 2·radius，radius 越小越好。坐标和半径用同一个单位，直接写成小数，例如 "7.5"，最多九位小数。

提交容器圆半径 radius 与各圆圆心坐标 centers（按半径 1, 2, …, n 的顺序排列）。容器圆的圆心固定在 (radius, radius)。所有数值均以十进制字符串表示（如 "7.5"）。

当前第一名的答案

{
  "centers": [
    [
      "647.150550225",
      "281.315704143"
    ],
    [
      "467.891636608",
      "315.534844325"
    ],
    [
      "1037.746228974",
      "372.951325618"
    ],
    [
      "57.043221842",
      "807.450062825"
    ],
    [
      "555.879775332",
      "867.929248525"
    ],
    [
      "450.532390961",
      "803.825721670"
    ],
    [
      "970.374681427",
      "182.169071481"
    ],
    [
      "245.215611059",
      "1027.030932289"
    ],
    [
      "353.586108659",
      "266.159595061"
    ],
    [
      "1123.565629541",
      "490.354342769"
    ],
    [
      "954.621419707",
      "974.508602812"
    ],
    [
      "797.705768257",
      "656.008127811"
    ],
    [
      "646.215806544",
      "159.626466054"
    ],
    [
      "338.964519934",
      "1074.613593542"
    ],
    [
      "21.728850965",
      "656.539020939"
    ],
    [
      "689.151507839",
      "1112.563512767"
    ],
    [
      "33.804946817",
      "435.330623384"
    ],
    [
      "693.141187907",
      "31.776469868"
    ],
    [
      "180.198286108",
      "181.010536348"
    ],
    [
      "326.735194163",
      "76.991904353"
    ],
    [
      "864.353800411",
      "106.000532611"
    ],
    [
      "1036.917248691",
      "860.078577673"
    ],
    [
      "632.499385351",
      "126.241932528"
    ],
    [
      "502.632185706",
      "1113.153510563"
    ],
    [
      "99.377734577",
      "295.673562680"
    ],
    [
      "1106.115074852",
      "670.942650088"
    ],
    [
      "895.235220133",
      "734.959027461"
    ],
    [
      "178.947360041",
      "330.628229180"
    ],
    [
      "411.781684263",
      "168.961475546"
    ],
    [
      "441.674930203",
      "839.654625372"
    ],
    [
      "824.660781710",
      "233.593262688"
    ],
    [
      "1073.920510710",
      "378.518168569"
    ],
    [
      "547.811595290",
      "33.487812853"
    ],
    [
      "939.437159764",
      "520.164796580"
    ],
    [
      "480.433236643",
      "42.661513639"
    ],
    [
      "148.701343304",
      "899.075717573"
    ],
    [
      "237.449512170",
      "302.299830060"
    ],
    [
      "1088.870916046",
      "447.214515629"
    ],
    [
      "981.585983933",
      "233.223367096"
    ],
    [
      "109.794921954",
      "833.789498675"
    ],
    [
      "899.257324111",
      "986.218414515"
    ],
    [
      "733.632955941",
      "1073.653926632"
    ],
    [
      "158.378362892",
      "482.607432531"
    ],
    [
      "44.923905670",
      "601.894104309"
    ],
    [
      "890.102630191",
      "806.775853630"
    ],
    [
      "640.862670976",
      "218.701913907"
    ],
    [
      "726.890840173",
      "984.629866476"
    ],
    [
      "306.196835527",
      "1021.553033583"
    ],
    [
      "429.852973356",
      "370.424802124"
    ],
    [
      "628.434536152",
      "53.209410325"
    ],
    [
      "1022.302216501",
      "313.486611521"
    ],
    [
      "816.002828455",
      "1027.758204545"
    ],
    [
      "163.409668685",
      "251.132442191"
    ],
    [
      "602.256314036",
      "581.251795583"
    ],
    [
      "401.256489302",
      "85.452198146"
    ],
    [
      "60.788352782",
      "503.160526850"
    ],
    [
      "144.191232492",
      "581.595943639"
    ],
    [
      "226.973828079",
      "951.127788883"
    ],
    [
      "328.856933692",
      "331.637234355"
    ],
    [
      "898.445824921",
      "179.476609431"
    ],
    [
      "251.952097403",
      "527.994172995"
    ],
    [
      "425.675272212",
      "259.061692148"
    ],
    [
      "231.692372474",
      "405.519154729"
    ],
    [
      "351.592924291",
      "452.517649360"
    ],
    [
      "520.970337434",
      "784.703425406"
    ],
    [
      "956.490665210",
      "895.802313032"
    ],
    [
      "102.929914540",
      "387.604961112"
    ],
    [
      "501.652746313",
      "917.925751338"
    ],
    [
      "983.505810319",
      "427.042055545"
    ],
    [
      "499.969124788",
      "651.346951738"
    ],
    [
      "630.777108742",
      "704.463329584"
    ],
    [
      "245.966960283",
      "660.859436078"
    ],
    [
      "371.752538546",
      "588.026283600"
    ],
    [
      "896.620453109",
      "313.464176085"
    ],
    [
      "815.489476540",
      "900.759242068"
    ],
    [
      "731.276358003",
      "597.183784010"
    ],
    [
      "429.217171548",
      "1043.711484010"
    ],
    [
      "356.150644172",
      "905.878416650"
    ],
    [
      "659.573731843",
      "459.765686941"
    ],
    [
      "228.235085290",
      "813.133552661"
    ],
    [
      "102.137140525",
      "713.032062531"
    ],
    [
      "777.358451828",
      "748.314325716"
    ],
    [
      "887.786570396",
      "625.211509870"
    ],
    [
      "377.378159090",
      "744.925462679"
    ],
    [
      "527.504704295",
      "153.043925893"
    ],
    [
      "497.144339149",
      "488.777622238"
    ],
    [
      "735.626907617",
      "312.022541069"
    ],
    [
      "560.096732001",
      "325.626481252"
    ],
    [
      "1052.418982891",
      "569.221430442"
    ],
    [
      "828.548781783",
      "462.669563282"
    ],
    [
      "652.699974102",
      "868.269583768"
    ],
    [
      "291.581891824",
      "183.332138230"
    ],
    [
      "1014.604149062",
      "747.249625807"
    ],
    [
      "600.194292303",
      "1046.508657334"
    ],
    [
      "751.008148132",
      "130.673657861"
    ]
  ],
  "radius": "570.710642426"
}
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": 95
}

当前第一名的答案

{
  "centers": [
    [
      "647.150550225",
      "281.315704143"
    ],
    [
      "467.891636608",
      "315.534844325"
    ],
    [
      "1037.746228974",
      "372.951325618"
    ],
    [
      "57.043221842",
      "807.450062825"
    ],
    [
      "555.879775332",
      "867.929248525"
    ],
    [
      "450.532390961",
      "803.825721670"
    ],
    [
      "970.374681427",
      "182.169071481"
    ],
    [
      "245.215611059",
      "1027.030932289"
    ],
    [
      "353.586108659",
      "266.159595061"
    ],
    [
      "1123.565629541",
      "490.354342769"
    ],
    [
      "954.621419707",
      "974.508602812"
    ],
    [
      "797.705768257",
      "656.008127811"
    ],
    [
      "646.215806544",
      "159.626466054"
    ],
    [
      "338.964519934",
      "1074.613593542"
    ],
    [
      "21.728850965",
      "656.539020939"
    ],
    [
      "689.151507839",
      "1112.563512767"
    ],
    [
      "33.804946817",
      "435.330623384"
    ],
    [
      "693.141187907",
      "31.776469868"
    ],
    [
      "180.198286108",
      "181.010536348"
    ],
    [
      "326.735194163",
      "76.991904353"
    ],
    [
      "864.353800411",
      "106.000532611"
    ],
    [
      "1036.917248691",
      "860.078577673"
    ],
    [
      "632.499385351",
      "126.241932528"
    ],
    [
      "502.632185706",
      "1113.153510563"
    ],
    [
      "99.377734577",
      "295.673562680"
    ],
    [
      "1106.115074852",
      "670.942650088"
    ],
    [
      "895.235220133",
      "734.959027461"
    ],
    [
      "178.947360041",
      "330.628229180"
    ],
    [
      "411.781684263",
      "168.961475546"
    ],
    [
      "441.674930203",
      "839.654625372"
    ],
    [
      "824.660781710",
      "233.593262688"
    ],
    [
      "1073.920510710",
      "378.518168569"
    ],
    [
      "547.811595290",
      "33.487812853"
    ],
    [
      "939.437159764",
      "520.164796580"
    ],
    [
      "480.433236643",
      "42.661513639"
    ],
    [
      "148.701343304",
      "899.075717573"
    ],
    [
      "237.449512170",
      "302.299830060"
    ],
    [
      "1088.870916046",
      "447.214515629"
    ],
    [
      "981.585983933",
      "233.223367096"
    ],
    [
      "109.794921954",
      "833.789498675"
    ],
    [
      "899.257324111",
      "986.218414515"
    ],
    [
      "733.632955941",
      "1073.653926632"
    ],
    [
      "158.378362892",
      "482.607432531"
    ],
    [
      "44.923905670",
      "601.894104309"
    ],
    [
      "890.102630191",
      "806.775853630"
    ],
    [
      "640.862670976",
      "218.701913907"
    ],
    [
      "726.890840173",
      "984.629866476"
    ],
    [
      "306.196835527",
      "1021.553033583"
    ],
    [
      "429.852973356",
      "370.424802124"
    ],
    [
      "628.434536152",
      "53.209410325"
    ],
    [
      "1022.302216501",
      "313.486611521"
    ],
    [
      "816.002828455",
      "1027.758204545"
    ],
    [
      "163.409668685",
      "251.132442191"
    ],
    [
      "602.256314036",
      "581.251795583"
    ],
    [
      "401.256489302",
      "85.452198146"
    ],
    [
      "60.788352782",
      "503.160526850"
    ],
    [
      "144.191232492",
      "581.595943639"
    ],
    [
      "226.973828079",
      "951.127788883"
    ],
    [
      "328.856933692",
      "331.637234355"
    ],
    [
      "898.445824921",
      "179.476609431"
    ],
    [
      "251.952097403",
      "527.994172995"
    ],
    [
      "425.675272212",
      "259.061692148"
    ],
    [
      "231.692372474",
      "405.519154729"
    ],
    [
      "351.592924291",
      "452.517649360"
    ],
    [
      "520.970337434",
      "784.703425406"
    ],
    [
      "956.490665210",
      "895.802313032"
    ],
    [
      "102.929914540",
      "387.604961112"
    ],
    [
      "501.652746313",
      "917.925751338"
    ],
    [
      "983.505810319",
      "427.042055545"
    ],
    [
      "499.969124788",
      "651.346951738"
    ],
    [
      "630.777108742",
      "704.463329584"
    ],
    [
      "245.966960283",
      "660.859436078"
    ],
    [
      "371.752538546",
      "588.026283600"
    ],
    [
      "896.620453109",
      "313.464176085"
    ],
    [
      "815.489476540",
      "900.759242068"
    ],
    [
      "731.276358003",
      "597.183784010"
    ],
    [
      "429.217171548",
      "1043.711484010"
    ],
    [
      "356.150644172",
      "905.878416650"
    ],
    [
      "659.573731843",
      "459.765686941"
    ],
    [
      "228.235085290",
      "813.133552661"
    ],
    [
      "102.137140525",
      "713.032062531"
    ],
    [
      "777.358451828",
      "748.314325716"
    ],
    [
      "887.786570396",
      "625.211509870"
    ],
    [
      "377.378159090",
      "744.925462679"
    ],
    [
      "527.504704295",
      "153.043925893"
    ],
    [
      "497.144339149",
      "488.777622238"
    ],
    [
      "735.626907617",
      "312.022541069"
    ],
    [
      "560.096732001",
      "325.626481252"
    ],
    [
      "1052.418982891",
      "569.221430442"
    ],
    [
      "828.548781783",
      "462.669563282"
    ],
    [
      "652.699974102",
      "868.269583768"
    ],
    [
      "291.581891824",
      "183.332138230"
    ],
    [
      "1014.604149062",
      "747.249625807"
    ],
    [
      "600.194292303",
      "1046.508657334"
    ],
    [
      "751.008148132",
      "130.673657861"
    ]
  ],
  "radius": "570.710642426"
}

提交容器圆半径 radius 与各圆圆心坐标 centers（按半径 1, 2, …, n 的顺序排列）。容器圆的圆心固定在 (radius, radius)。所有数值均以十进制字符串表示（如 "7.5"）。 · 验证器 v1.0.0

DISCUSSION

讨论区

欢迎交流解题思路、分享构造方法或探讨卡壳之处。所有登录用户均可发帖；发言使用与排行榜一致的公开署名，并附带唯一注册编号（#UID）以防冒名。新发言经审核后公开。

N=5和6的最优性今天#841 条回复