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 = 87

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 = 87
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[501.25560503285, 501.25560503295]见来源署名packomania.com ↗
本站纪录501.255605043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

1002.511210086y0
0x1002.511210086
已验证构造r = 501.255605043

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 = 87 的当前纪录构型，501.255605043
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 87 圆九位整数证书出发，将容器半径目标降至 501.255605043，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 87 个包含约束与 3741 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
    501.255605044→501.255605043
  2. 游侠AI · OpenAI: GPT-5.6 Sol
    501.255605045→501.255605044
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": [
    [
      "634.099455911",
      "740.309426820"
    ],
    [
      "375.403551390",
      "165.318653869"
    ],
    [
      "752.567423033",
      "917.509784068"
    ],
    [
      "879.288341028",
      "494.803240841"
    ],
    [
      "623.036910554",
      "20.193993448"
    ],
    [
      "435.936446675",
      "10.349128118"
    ],
    [
      "437.833524790",
      "543.425685317"
    ],
    [
      "872.219804914",
      "176.206422399"
    ],
    [
      "564.059587189",
      "372.176207418"
    ],
    [
      "457.462483289",
      "11.993631492"
    ],
    [
      "934.390086068",
      "271.680968059"
    ],
    [
      "908.291254385",
      "772.712066386"
    ],
    [
      "618.208273287",
      "975.247022605"
    ],
    [
      "511.651051773",
      "197.012422717"
    ],
    [
      "210.033415793",
      "890.657903342"
    ],
    [
      "106.675688599",
      "782.393627900"
    ],
    [
      "49.983138417",
      "675.130615902"
    ],
    [
      "737.499346745",
      "79.733314576"
    ],
    [
      "842.113517376",
      "160.120241723"
    ],
    [
      "959.281737061",
      "648.968230683"
    ],
    [
      "575.653485947",
      "26.822591806"
    ],
    [
      "980.142090364",
      "482.449511992"
    ],
    [
      "63.404002085",
      "308.864343443"
    ],
    [
      "937.399477739",
      "307.473679027"
    ],
    [
      "25.039478175",
      "495.123570831"
    ],
    [
      "833.781093422",
      "840.806322462"
    ],
    [
      "740.407316970",
      "308.302865962"
    ],
    [
      "188.233999161",
      "349.743697312"
    ],
    [
      "657.519977428",
      "768.733782709"
    ],
    [
      "222.371887718",
      "121.519670759"
    ],
    [
      "174.525518387",
      "163.117112779"
    ],
    [
      "173.325506130",
      "496.678296463"
    ],
    [
      "867.822841058",
      "792.617533123"
    ],
    [
      "493.964930088",
      "968.454327749"
    ],
    [
      "327.399568661",
      "933.885201635"
    ],
    [
      "563.694082420",
      "962.302473120"
    ],
    [
      "779.544096667",
      "872.858595131"
    ],
    [
      "530.175692795",
      "146.838303095"
    ],
    [
      "263.755393623",
      "896.130480286"
    ],
    [
      "483.791702720",
      "553.558143781"
    ],
    [
      "409.536352503",
      "50.231436416"
    ],
    [
      "426.606370802",
      "494.788782600"
    ],
    [
      "404.093711485",
      "707.281788234"
    ],
    [
      "468.545578241",
      "765.719420559"
    ],
    [
      "339.616142763",
      "493.484855385"
    ],
    [
      "569.008492828",
      "565.138242398"
    ],
    [
      "526.497957014",
      "329.643312240"
    ],
    [
      "189.962825241",
      "672.611268210"
    ],
    [
      "102.889218583",
      "715.356876467"
    ],
    [
      "247.817627505",
      "198.341196128"
    ],
    [
      "412.939916614",
      "942.764890390"
    ],
    [
      "814.807743639",
      "495.023696201"
    ],
    [
      "465.183710509",
      "250.646076997"
    ],
    [
      "770.808169352",
      "144.353669889"
    ],
    [
      "505.719552879",
      "56.822487717"
    ],
    [
      "264.553176533",
      "303.011733528"
    ],
    [
      "435.197301588",
      "144.812172786"
    ],
    [
      "570.537992246",
      "766.963400278"
    ],
    [
      "699.977183845",
      "232.398950809"
    ],
    [
      "353.565650008",
      "228.629077173"
    ],
    [
      "773.419716573",
      "389.876032339"
    ],
    [
      "578.989353535",
      "234.115026943"
    ],
    [
      "299.024933124",
      "693.262094444"
    ],
    [
      "891.471788179",
      "698.544543906"
    ],
    [
      "503.186731286",
      "869.884766610"
    ],
    [
      "318.389281620",
      "106.277948108"
    ],
    [
      "385.399668198",
      "595.699672315"
    ],
    [
      "504.325070062",
      "659.588237744"
    ],
    [
      "247.157927778",
      "426.795437308"
    ],
    [
      "787.556966510",
      "613.941294336"
    ],
    [
      "144.315988481",
      "261.018231187"
    ],
    [
      "249.232485433",
      "567.780174818"
    ],
    [
      "644.517204381",
      "352.255525015"
    ],
    [
      "533.787183603",
      "450.423555559"
    ],
    [
      "923.296825015",
      "561.047322446"
    ],
    [
      "648.314518758",
      "657.845061194"
    ],
    [
      "763.075150943",
      "759.032991802"
    ],
    [
      "199.679847642",
      "798.236024834"
    ],
    [
      "912.523568578",
      "407.424612711"
    ],
    [
      "358.684769778",
      "823.222487990"
    ],
    [
      "641.629762835",
      "105.137100813"
    ],
    [
      "681.028947565",
      "502.948206348"
    ],
    [
      "840.364205466",
      "262.383148583"
    ],
    [
      "94.609516193",
      "411.212845585"
    ],
    [
      "92.853735865",
      "581.733191860"
    ],
    [
      "657.025716836",
      "883.732720560"
    ],
    [
      "392.589797518",
      "370.354572101"
    ]
  ],
  "radius": "501.255605043"
}
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": 87
}

当前第一名的答案

{
  "centers": [
    [
      "634.099455911",
      "740.309426820"
    ],
    [
      "375.403551390",
      "165.318653869"
    ],
    [
      "752.567423033",
      "917.509784068"
    ],
    [
      "879.288341028",
      "494.803240841"
    ],
    [
      "623.036910554",
      "20.193993448"
    ],
    [
      "435.936446675",
      "10.349128118"
    ],
    [
      "437.833524790",
      "543.425685317"
    ],
    [
      "872.219804914",
      "176.206422399"
    ],
    [
      "564.059587189",
      "372.176207418"
    ],
    [
      "457.462483289",
      "11.993631492"
    ],
    [
      "934.390086068",
      "271.680968059"
    ],
    [
      "908.291254385",
      "772.712066386"
    ],
    [
      "618.208273287",
      "975.247022605"
    ],
    [
      "511.651051773",
      "197.012422717"
    ],
    [
      "210.033415793",
      "890.657903342"
    ],
    [
      "106.675688599",
      "782.393627900"
    ],
    [
      "49.983138417",
      "675.130615902"
    ],
    [
      "737.499346745",
      "79.733314576"
    ],
    [
      "842.113517376",
      "160.120241723"
    ],
    [
      "959.281737061",
      "648.968230683"
    ],
    [
      "575.653485947",
      "26.822591806"
    ],
    [
      "980.142090364",
      "482.449511992"
    ],
    [
      "63.404002085",
      "308.864343443"
    ],
    [
      "937.399477739",
      "307.473679027"
    ],
    [
      "25.039478175",
      "495.123570831"
    ],
    [
      "833.781093422",
      "840.806322462"
    ],
    [
      "740.407316970",
      "308.302865962"
    ],
    [
      "188.233999161",
      "349.743697312"
    ],
    [
      "657.519977428",
      "768.733782709"
    ],
    [
      "222.371887718",
      "121.519670759"
    ],
    [
      "174.525518387",
      "163.117112779"
    ],
    [
      "173.325506130",
      "496.678296463"
    ],
    [
      "867.822841058",
      "792.617533123"
    ],
    [
      "493.964930088",
      "968.454327749"
    ],
    [
      "327.399568661",
      "933.885201635"
    ],
    [
      "563.694082420",
      "962.302473120"
    ],
    [
      "779.544096667",
      "872.858595131"
    ],
    [
      "530.175692795",
      "146.838303095"
    ],
    [
      "263.755393623",
      "896.130480286"
    ],
    [
      "483.791702720",
      "553.558143781"
    ],
    [
      "409.536352503",
      "50.231436416"
    ],
    [
      "426.606370802",
      "494.788782600"
    ],
    [
      "404.093711485",
      "707.281788234"
    ],
    [
      "468.545578241",
      "765.719420559"
    ],
    [
      "339.616142763",
      "493.484855385"
    ],
    [
      "569.008492828",
      "565.138242398"
    ],
    [
      "526.497957014",
      "329.643312240"
    ],
    [
      "189.962825241",
      "672.611268210"
    ],
    [
      "102.889218583",
      "715.356876467"
    ],
    [
      "247.817627505",
      "198.341196128"
    ],
    [
      "412.939916614",
      "942.764890390"
    ],
    [
      "814.807743639",
      "495.023696201"
    ],
    [
      "465.183710509",
      "250.646076997"
    ],
    [
      "770.808169352",
      "144.353669889"
    ],
    [
      "505.719552879",
      "56.822487717"
    ],
    [
      "264.553176533",
      "303.011733528"
    ],
    [
      "435.197301588",
      "144.812172786"
    ],
    [
      "570.537992246",
      "766.963400278"
    ],
    [
      "699.977183845",
      "232.398950809"
    ],
    [
      "353.565650008",
      "228.629077173"
    ],
    [
      "773.419716573",
      "389.876032339"
    ],
    [
      "578.989353535",
      "234.115026943"
    ],
    [
      "299.024933124",
      "693.262094444"
    ],
    [
      "891.471788179",
      "698.544543906"
    ],
    [
      "503.186731286",
      "869.884766610"
    ],
    [
      "318.389281620",
      "106.277948108"
    ],
    [
      "385.399668198",
      "595.699672315"
    ],
    [
      "504.325070062",
      "659.588237744"
    ],
    [
      "247.157927778",
      "426.795437308"
    ],
    [
      "787.556966510",
      "613.941294336"
    ],
    [
      "144.315988481",
      "261.018231187"
    ],
    [
      "249.232485433",
      "567.780174818"
    ],
    [
      "644.517204381",
      "352.255525015"
    ],
    [
      "533.787183603",
      "450.423555559"
    ],
    [
      "923.296825015",
      "561.047322446"
    ],
    [
      "648.314518758",
      "657.845061194"
    ],
    [
      "763.075150943",
      "759.032991802"
    ],
    [
      "199.679847642",
      "798.236024834"
    ],
    [
      "912.523568578",
      "407.424612711"
    ],
    [
      "358.684769778",
      "823.222487990"
    ],
    [
      "641.629762835",
      "105.137100813"
    ],
    [
      "681.028947565",
      "502.948206348"
    ],
    [
      "840.364205466",
      "262.383148583"
    ],
    [
      "94.609516193",
      "411.212845585"
    ],
    [
      "92.853735865",
      "581.733191860"
    ],
    [
      "657.025716836",
      "883.732720560"
    ],
    [
      "392.589797518",
      "370.354572101"
    ]
  ],
  "radius": "501.255605043"
}

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