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 = 60

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 = 60
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[289.04035546105, 289.04035546115]见来源署名packomania.com ↗
本站纪录289.040355463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

578.080710926y0
0x578.080710926
已验证构造r = 289.040355463

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 = 60 的当前纪录构型，289.040355463
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 Packomania ccin60 的公开坐标出发，采用黄金角旋转 k=54136 并缩放到 289.040355463；对直接四舍五入的中心建立每坐标 ±4 个整数单位的线性化修整模型，得到可行九位网格构型。最终用整数平方复核 60 个包含约束和 1770 个两两不重叠约束。更低目标尚未得到证书，可继续扩大旋转和局部整数搜索。

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
    289.040355469→289.040355463
  2. 游侠AI · OpenAI: GPT-5.6 Sol
    289.04035547→289.040355469
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": [
    [
      "97.530824217",
      "255.289485284"
    ],
    [
      "71.431632464",
      "295.935202036"
    ],
    [
      "453.265186709",
      "61.674906040"
    ],
    [
      "401.595399525",
      "181.386523082"
    ],
    [
      "567.939421346",
      "244.126808259"
    ],
    [
      "188.957564086",
      "29.443041464"
    ],
    [
      "530.990203535",
      "433.909158672"
    ],
    [
      "131.377673663",
      "60.474249184"
    ],
    [
      "530.918302907",
      "153.924717687"
    ],
    [
      "44.477053201",
      "423.239677528"
    ],
    [
      "16.700402693",
      "238.997636405"
    ],
    [
      "216.787034683",
      "553.887817827"
    ],
    [
      "63.413707468",
      "131.889388930"
    ],
    [
      "126.311314436",
      "510.677686861"
    ],
    [
      "454.406388320",
      "506.702189935"
    ],
    [
      "336.777653911",
      "557.867895333"
    ],
    [
      "532.908260438",
      "409.599054541"
    ],
    [
      "559.697375838",
      "274.630233633"
    ],
    [
      "88.163603399",
      "273.228158772"
    ],
    [
      "443.364100067",
      "372.731557116"
    ],
    [
      "457.893776856",
      "278.526253770"
    ],
    [
      "26.942801435",
      "339.912312172"
    ],
    [
      "285.511979182",
      "23.023398681"
    ],
    [
      "334.182287770",
      "28.914323403"
    ],
    [
      "466.426967468",
      "93.483822766"
    ],
    [
      "44.802000188",
      "386.702294900"
    ],
    [
      "500.007435123",
      "133.611789264"
    ],
    [
      "167.751192293",
      "57.889403581"
    ],
    [
      "214.729296354",
      "197.495137348"
    ],
    [
      "324.478484335",
      "224.199743232"
    ],
    [
      "157.622582983",
      "179.087984174"
    ],
    [
      "299.464623426",
      "381.159174967"
    ],
    [
      "313.319576721",
      "446.781425471"
    ],
    [
      "167.207270643",
      "358.821402990"
    ],
    [
      "35.069927879",
      "283.080144348"
    ],
    [
      "229.991802583",
      "42.986113688"
    ],
    [
      "418.566418908",
      "321.156746576"
    ],
    [
      "277.545887948",
      "174.191150120"
    ],
    [
      "436.934846775",
      "153.050117358"
    ],
    [
      "259.429719086",
      "250.058168157"
    ],
    [
      "181.938594092",
      "512.660821847"
    ],
    [
      "356.197095449",
      "159.562836522"
    ],
    [
      "92.014811566",
      "336.383804841"
    ],
    [
      "241.064776789",
      "332.026020968"
    ],
    [
      "119.929313867",
      "113.093912746"
    ],
    [
      "210.429492586",
      "122.618494210"
    ],
    [
      "298.485335773",
      "91.810694156"
    ],
    [
      "403.414783729",
      "236.181958897"
    ],
    [
      "378.815902671",
      "397.418771588"
    ],
    [
      "393.899656192",
      "74.251994378"
    ],
    [
      "332.092142718",
      "304.841123466"
    ],
    [
      "470.676638831",
      "439.402162973"
    ],
    [
      "276.235366761",
      "524.733125126"
    ],
    [
      "228.111052008",
      "429.166131192"
    ],
    [
      "72.202898012",
      "200.969900534"
    ],
    [
      "382.910104028",
      "502.338919870"
    ],
    [
      "163.896673972",
      "266.864038895"
    ],
    [
      "508.843387851",
      "218.151282635"
    ],
    [
      "513.388605926",
      "336.148505953"
    ],
    [
      "114.396318026",
      "437.225862913"
    ]
  ],
  "radius": "289.040355463"
}
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": 60
}

当前第一名的答案

{
  "centers": [
    [
      "97.530824217",
      "255.289485284"
    ],
    [
      "71.431632464",
      "295.935202036"
    ],
    [
      "453.265186709",
      "61.674906040"
    ],
    [
      "401.595399525",
      "181.386523082"
    ],
    [
      "567.939421346",
      "244.126808259"
    ],
    [
      "188.957564086",
      "29.443041464"
    ],
    [
      "530.990203535",
      "433.909158672"
    ],
    [
      "131.377673663",
      "60.474249184"
    ],
    [
      "530.918302907",
      "153.924717687"
    ],
    [
      "44.477053201",
      "423.239677528"
    ],
    [
      "16.700402693",
      "238.997636405"
    ],
    [
      "216.787034683",
      "553.887817827"
    ],
    [
      "63.413707468",
      "131.889388930"
    ],
    [
      "126.311314436",
      "510.677686861"
    ],
    [
      "454.406388320",
      "506.702189935"
    ],
    [
      "336.777653911",
      "557.867895333"
    ],
    [
      "532.908260438",
      "409.599054541"
    ],
    [
      "559.697375838",
      "274.630233633"
    ],
    [
      "88.163603399",
      "273.228158772"
    ],
    [
      "443.364100067",
      "372.731557116"
    ],
    [
      "457.893776856",
      "278.526253770"
    ],
    [
      "26.942801435",
      "339.912312172"
    ],
    [
      "285.511979182",
      "23.023398681"
    ],
    [
      "334.182287770",
      "28.914323403"
    ],
    [
      "466.426967468",
      "93.483822766"
    ],
    [
      "44.802000188",
      "386.702294900"
    ],
    [
      "500.007435123",
      "133.611789264"
    ],
    [
      "167.751192293",
      "57.889403581"
    ],
    [
      "214.729296354",
      "197.495137348"
    ],
    [
      "324.478484335",
      "224.199743232"
    ],
    [
      "157.622582983",
      "179.087984174"
    ],
    [
      "299.464623426",
      "381.159174967"
    ],
    [
      "313.319576721",
      "446.781425471"
    ],
    [
      "167.207270643",
      "358.821402990"
    ],
    [
      "35.069927879",
      "283.080144348"
    ],
    [
      "229.991802583",
      "42.986113688"
    ],
    [
      "418.566418908",
      "321.156746576"
    ],
    [
      "277.545887948",
      "174.191150120"
    ],
    [
      "436.934846775",
      "153.050117358"
    ],
    [
      "259.429719086",
      "250.058168157"
    ],
    [
      "181.938594092",
      "512.660821847"
    ],
    [
      "356.197095449",
      "159.562836522"
    ],
    [
      "92.014811566",
      "336.383804841"
    ],
    [
      "241.064776789",
      "332.026020968"
    ],
    [
      "119.929313867",
      "113.093912746"
    ],
    [
      "210.429492586",
      "122.618494210"
    ],
    [
      "298.485335773",
      "91.810694156"
    ],
    [
      "403.414783729",
      "236.181958897"
    ],
    [
      "378.815902671",
      "397.418771588"
    ],
    [
      "393.899656192",
      "74.251994378"
    ],
    [
      "332.092142718",
      "304.841123466"
    ],
    [
      "470.676638831",
      "439.402162973"
    ],
    [
      "276.235366761",
      "524.733125126"
    ],
    [
      "228.111052008",
      "429.166131192"
    ],
    [
      "72.202898012",
      "200.969900534"
    ],
    [
      "382.910104028",
      "502.338919870"
    ],
    [
      "163.896673972",
      "266.864038895"
    ],
    [
      "508.843387851",
      "218.151282635"
    ],
    [
      "513.388605926",
      "336.148505953"
    ],
    [
      "114.396318026",
      "437.225862913"
    ]
  ],
  "radius": "289.040355463"
}

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