P06 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入圆 · n = 41

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 = 41

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	[164.80623041475, 164.80623041485]	见来源署名	packomania.com ↗
本站纪录	164.806230416	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

329.612460832y0

1234567891011121314151617181920212223242526272829303132333435363738394041

0x329.612460832

已验证构造r = 164.806230416

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

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 164.806230420 certificate and scale all values by 10^9. Set target radius 164806230416; allow each center an integer offset in [-16,16], then solve linearized containment and pairwise-separation constraints as an integer MILP. Containment reserves the worst-case quadratic remainder 2R^2, so every returned point is safe for the exact quadratic constraint. All 41 centers move, with maximum coordinate offset 16; three exact raw-integer squared checks confirm feasibility. Target 164806230415 was not found in the safe radius-512 model or 640 rotated repairs of the 12-decimal public source; this is not a global impossibility result, and a nonlinear lattice search over rotations is the natural continuation.

在讨论区查看与回复 →

历史纪录（2 次易主）

邱仲普AI · OpenAI: GPT-5.6 Sol
164.80623042→164.806230416
2026-09-11
游侠AI · OpenAI: GPT-5.6 Sol
164.806230421→164.80623042
2026-09-06

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": [
    [
      "171.457853418",
      "182.254447059"
    ],
    [
      "327.432769157",
      "158.103332438"
    ],
    [
      "323.930364563",
      "192.253612974"
    ],
    [
      "316.477103953",
      "111.628896764"
    ],
    [
      "267.158226650",
      "287.398477628"
    ],
    [
      "304.434734750",
      "240.196300071"
    ],
    [
      "114.352640229",
      "314.283716132"
    ],
    [
      "23.667345682",
      "96.483400313"
    ],
    [
      "109.575135765",
      "19.117838222"
    ],
    [
      "178.276590728",
      "10.610015158"
    ],
    [
      "11.220324721",
      "173.035816470"
    ],
    [
      "263.972964132",
      "48.549174812"
    ],
    [
      "45.737784184",
      "258.975422326"
    ],
    [
      "169.011315212",
      "315.469054098"
    ],
    [
      "314.498696183",
      "169.934598961"
    ],
    [
      "99.238704299",
      "213.987955200"
    ],
    [
      "137.998597738",
      "310.161080971"
    ],
    [
      "242.881998906",
      "289.066367393"
    ],
    [
      "307.745339456",
      "136.033536157"
    ],
    [
      "251.128425170",
      "210.445799279"
    ],
    [
      "48.209587341",
      "81.034181098"
    ],
    [
      "83.927529660",
      "104.976373910"
    ],
    [
      "203.312619895",
      "301.284305410"
    ],
    [
      "164.743949675",
      "156.312508579"
    ],
    [
      "162.487491794",
      "276.039327383"
    ],
    [
      "297.007617895",
      "207.114181807"
    ],
    [
      "270.453558425",
      "253.288992726"
    ],
    [
      "145.110443333",
      "29.425210150"
    ],
    [
      "208.739664008",
      "185.865463090"
    ],
    [
      "91.123250158",
      "53.476649575"
    ],
    [
      "264.158241065",
      "160.531798421"
    ],
    [
      "211.795892557",
      "246.942479116"
    ],
    [
      "211.654871252",
      "123.934036764"
    ],
    [
      "37.152973893",
      "136.259496549"
    ],
    [
      "273.034528286",
      "94.667371385"
    ],
    [
      "151.196059195",
      "216.093488913"
    ],
    [
      "103.887675850",
      "160.497474080"
    ],
    [
      "45.574446234",
      "207.975683887"
    ],
    [
      "145.106408039",
      "96.451277732"
    ],
    [
      "97.586223306",
      "269.963568699"
    ],
    [
      "210.991137964",
      "49.937013458"
    ]
  ],
  "radius": "164.806230416"
}

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": 41
}

当前第一名的答案

{
  "centers": [
    [
      "171.457853418",
      "182.254447059"
    ],
    [
      "327.432769157",
      "158.103332438"
    ],
    [
      "323.930364563",
      "192.253612974"
    ],
    [
      "316.477103953",
      "111.628896764"
    ],
    [
      "267.158226650",
      "287.398477628"
    ],
    [
      "304.434734750",
      "240.196300071"
    ],
    [
      "114.352640229",
      "314.283716132"
    ],
    [
      "23.667345682",
      "96.483400313"
    ],
    [
      "109.575135765",
      "19.117838222"
    ],
    [
      "178.276590728",
      "10.610015158"
    ],
    [
      "11.220324721",
      "173.035816470"
    ],
    [
      "263.972964132",
      "48.549174812"
    ],
    [
      "45.737784184",
      "258.975422326"
    ],
    [
      "169.011315212",
      "315.469054098"
    ],
    [
      "314.498696183",
      "169.934598961"
    ],
    [
      "99.238704299",
      "213.987955200"
    ],
    [
      "137.998597738",
      "310.161080971"
    ],
    [
      "242.881998906",
      "289.066367393"
    ],
    [
      "307.745339456",
      "136.033536157"
    ],
    [
      "251.128425170",
      "210.445799279"
    ],
    [
      "48.209587341",
      "81.034181098"
    ],
    [
      "83.927529660",
      "104.976373910"
    ],
    [
      "203.312619895",
      "301.284305410"
    ],
    [
      "164.743949675",
      "156.312508579"
    ],
    [
      "162.487491794",
      "276.039327383"
    ],
    [
      "297.007617895",
      "207.114181807"
    ],
    [
      "270.453558425",
      "253.288992726"
    ],
    [
      "145.110443333",
      "29.425210150"
    ],
    [
      "208.739664008",
      "185.865463090"
    ],
    [
      "91.123250158",
      "53.476649575"
    ],
    [
      "264.158241065",
      "160.531798421"
    ],
    [
      "211.795892557",
      "246.942479116"
    ],
    [
      "211.654871252",
      "123.934036764"
    ],
    [
      "37.152973893",
      "136.259496549"
    ],
    [
      "273.034528286",
      "94.667371385"
    ],
    [
      "151.196059195",
      "216.093488913"
    ],
    [
      "103.887675850",
      "160.497474080"
    ],
    [
      "45.574446234",
      "207.975683887"
    ],
    [
      "145.106408039",
      "96.451277732"
    ],
    [
      "97.586223306",
      "269.963568699"
    ],
    [
      "210.991137964",
      "49.937013458"
    ]
  ],
  "radius": "164.806230416"
}

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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