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P06 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入圆 · n = 40

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 = 40
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[158.95408583785, 158.95408583795]见来源署名packomania.com ↗
本站纪录158.954085839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

317.908171678y0
0x317.908171678
已验证构造r = 158.954085839

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 = 40 的当前纪录构型，158.954085839
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）＋

Start from the public 158.954085842 certificate and scale all values by 10^9. Set the target radius to 158954085839 and allow each center an integer offset in [-8,8]. Solve linearized containment and pairwise-separation constraints as an integer MILP; containment reserves the worst-case quadratic remainder 2R^2, so a returned point is safe for the original quadratic inequality. The certificate moves 39 of 40 centers, with maximum coordinate offset 8, and exact raw-integer squared checks independently confirm every constraint. Direct rounding and bounded repairs of the public 12-decimal Packomania coordinates did not yield 158954085838; continuation should use a nonlinear integer search over rotations and larger neighborhoods.

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
    158.954085842→158.954085839
  2. 游侠AI · OpenAI: GPT-5.6 Sol
    158.954085843→158.954085842
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": [
    [
      "316.632972529",
      "168.072896631"
    ],
    [
      "121.563179136",
      "6.523106081"
    ],
    [
      "23.949799724",
      "80.896445017"
    ],
    [
      "265.902393168",
      "46.836922374"
    ],
    [
      "175.489827638",
      "143.887168332"
    ],
    [
      "287.340527362",
      "239.384853466"
    ],
    [
      "93.933924105",
      "296.278564900"
    ],
    [
      "298.373410727",
      "101.384033476"
    ],
    [
      "308.875680277",
      "160.990702348"
    ],
    [
      "250.182510778",
      "41.205649878"
    ],
    [
      "11.414022619",
      "147.988397536"
    ],
    [
      "34.073247546",
      "236.415557371"
    ],
    [
      "146.154085787",
      "102.503241475"
    ],
    [
      "149.234402071",
      "188.148385188"
    ],
    [
      "133.857000039",
      "18.319115235"
    ],
    [
      "241.041315636",
      "275.990650050"
    ],
    [
      "165.885020444",
      "17.180265137"
    ],
    [
      "264.993948560",
      "251.813173464"
    ],
    [
      "159.240768733",
      "298.907878056"
    ],
    [
      "120.761451121",
      "292.556333666"
    ],
    [
      "58.620404242",
      "64.292843464"
    ],
    [
      "86.704651980",
      "147.736460029"
    ],
    [
      "242.639199631",
      "138.841997747"
    ],
    [
      "291.317210681",
      "132.834305827"
    ],
    [
      "201.193929902",
      "285.643931886"
    ],
    [
      "99.088698537",
      "40.240462592"
    ],
    [
      "221.490298776",
      "237.692376724"
    ],
    [
      "107.538022520",
      "193.189410120"
    ],
    [
      "150.144398272",
      "60.693225845"
    ],
    [
      "140.191005583",
      "145.087765360"
    ],
    [
      "102.482256149",
      "97.139353326"
    ],
    [
      "208.188858071",
      "41.935809001"
    ],
    [
      "40.985303638",
      "115.368865875"
    ],
    [
      "259.143100781",
      "84.283577252"
    ],
    [
      "193.904139452",
      "107.395314148"
    ],
    [
      "198.247321875",
      "178.262349972"
    ],
    [
      "80.437318335",
      "252.270302401"
    ],
    [
      "41.281671697",
      "186.937899572"
    ],
    [
      "155.562060973",
      "240.769300544"
    ],
    [
      "272.536794127",
      "194.295887091"
    ]
  ],
  "radius": "158.954085839"
}
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": 40
}

当前第一名的答案

{
  "centers": [
    [
      "316.632972529",
      "168.072896631"
    ],
    [
      "121.563179136",
      "6.523106081"
    ],
    [
      "23.949799724",
      "80.896445017"
    ],
    [
      "265.902393168",
      "46.836922374"
    ],
    [
      "175.489827638",
      "143.887168332"
    ],
    [
      "287.340527362",
      "239.384853466"
    ],
    [
      "93.933924105",
      "296.278564900"
    ],
    [
      "298.373410727",
      "101.384033476"
    ],
    [
      "308.875680277",
      "160.990702348"
    ],
    [
      "250.182510778",
      "41.205649878"
    ],
    [
      "11.414022619",
      "147.988397536"
    ],
    [
      "34.073247546",
      "236.415557371"
    ],
    [
      "146.154085787",
      "102.503241475"
    ],
    [
      "149.234402071",
      "188.148385188"
    ],
    [
      "133.857000039",
      "18.319115235"
    ],
    [
      "241.041315636",
      "275.990650050"
    ],
    [
      "165.885020444",
      "17.180265137"
    ],
    [
      "264.993948560",
      "251.813173464"
    ],
    [
      "159.240768733",
      "298.907878056"
    ],
    [
      "120.761451121",
      "292.556333666"
    ],
    [
      "58.620404242",
      "64.292843464"
    ],
    [
      "86.704651980",
      "147.736460029"
    ],
    [
      "242.639199631",
      "138.841997747"
    ],
    [
      "291.317210681",
      "132.834305827"
    ],
    [
      "201.193929902",
      "285.643931886"
    ],
    [
      "99.088698537",
      "40.240462592"
    ],
    [
      "221.490298776",
      "237.692376724"
    ],
    [
      "107.538022520",
      "193.189410120"
    ],
    [
      "150.144398272",
      "60.693225845"
    ],
    [
      "140.191005583",
      "145.087765360"
    ],
    [
      "102.482256149",
      "97.139353326"
    ],
    [
      "208.188858071",
      "41.935809001"
    ],
    [
      "40.985303638",
      "115.368865875"
    ],
    [
      "259.143100781",
      "84.283577252"
    ],
    [
      "193.904139452",
      "107.395314148"
    ],
    [
      "198.247321875",
      "178.262349972"
    ],
    [
      "80.437318335",
      "252.270302401"
    ],
    [
      "41.281671697",
      "186.937899572"
    ],
    [
      "155.562060973",
      "240.769300544"
    ],
    [
      "272.536794127",
      "194.295887091"
    ]
  ],
  "radius": "158.954085839"
}

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

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