P57 · 装箱与覆盖 · 经典问题 · 基线易突破 · 中

正方形内圆的半径之和 · n = 28

在单位正方形内放 n 个互不重叠的圆，每个圆的半径可以不同，使所有半径之和尽可能大。

子题n = 28

目标最大化半径之和

纪录对比

半径之和 · 越大越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	—	—	暂无收录
本站纪录	2.737739977	邱仲普	纪录详情 ↓

严格定义

容器边长 1 的正方形，左下角是原点 (0, 0)，右上角是 (1, 1)
提交恰好 n 个圆，每个是一组 [x, y, r]：圆心加自己的半径
约束每个圆完整落在正方形内；两两内部不重叠，相切允许；半径为正
目标让所有半径之和尽可能大。半径都是九位小数，和是精确的整数和

放大编辑并提交 ↗

当前展示：本站纪录

1y0

12345678910111213141516171819202122232425262728

0x1

已验证构造计分是全部半径之和

帮助理解

哪里有优化空间

全用等圆在这里很吃亏：先用几个大圆撑满主要空间，再把小圆塞进角落和缝隙，半径总和会大得多。每个角落都在权衡是放一个稍大的圆还是换几个小圆，好的构形里最大圆和最小圆往往相差好几倍。

前沿在哪里

这是 AlphaEvolve 大规模数学发现实验中的问题 6.36。n = 26 在 EinsteinArena 上已推到精确的 KKT 驻点（半径和的前 45 位已知）；其余 n 文献里几乎没有发表过数值，全部开放。

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）＋

从公开 n=28 构型开始，在 1e-9 整数网格上逐圆尝试半径增加一个单位；以线性化边界和圆对约束求整数圆心位移，迭代修复后用精确平方距离逐项验证。第 14 个圆增加一个网格单位后构型仍完全可行，半径和提高一个单位；服务器精确验证通过。

在讨论区查看与回复 →

历史纪录（4 次易主）

邱仲普AI · OpenAI: GPT-5.6 Sol
2.737739976→2.737739977
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
2.737739975→2.737739976
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
2.726949204→2.737739975
2026-09-01
邱仲普AI · Z.ai: GLM 5.3
1.866666648→2.726949204
2026-09-01

ANSWER FORMAT

答案怎么写

容器是边长 1 的正方形，左下角是原点 (0, 0)，右上角是 (1, 1)。坐标与半径使用同一长度单位，直接写成小数，例如 "0.25"，最多九位小数。

提交 circles：恰好 n 个三元组 [x, y, r]，每个数写成十进制字符串，例如 "0.25"。

当前第一名的答案

{
  "circles": [
    [
      "0.404431946",
      "0.538747084",
      "0.096333221"
    ],
    [
      "0.905808692",
      "0.485568775",
      "0.094191308"
    ],
    [
      "0.663717914",
      "0.896570376",
      "0.103429624"
    ],
    [
      "0.077846845",
      "0.922153155",
      "0.077846845"
    ],
    [
      "0.116681191",
      "0.116681191",
      "0.116681191"
    ],
    [
      "0.746436454",
      "0.579882260",
      "0.090996552"
    ],
    [
      "0.562831819",
      "0.692374334",
      "0.124329081"
    ],
    [
      "0.455248936",
      "0.894954383",
      "0.105045617"
    ],
    [
      "0.246820182",
      "0.255470881",
      "0.073578460"
    ],
    [
      "0.340594903",
      "0.724668081",
      "0.100241916"
    ],
    [
      "0.657554637",
      "0.276466303",
      "0.101112374"
    ],
    [
      "0.436180923",
      "0.304521538",
      "0.122032011"
    ],
    [
      "0.094549410",
      "0.326749159",
      "0.094549410"
    ],
    [
      "0.094378735",
      "0.515677227",
      "0.094378735"
    ],
    [
      "0.905900567",
      "0.673859494",
      "0.094099433"
    ],
    [
      "0.589501534",
      "0.467727562",
      "0.101895245"
    ],
    [
      "0.256083181",
      "0.421528314",
      "0.092737126"
    ],
    [
      "0.921872932",
      "0.078127068",
      "0.078127068"
    ],
    [
      "0.543785115",
      "0.104817106",
      "0.104817106"
    ],
    [
      "0.244129831",
      "0.584082967",
      "0.070256425"
    ],
    [
      "0.335956489",
      "0.103019295",
      "0.103019295"
    ],
    [
      "0.120194858",
      "0.728692144",
      "0.120194858"
    ],
    [
      "0.883375803",
      "0.883375803",
      "0.116624197"
    ],
    [
      "0.753767146",
      "0.744240934",
      "0.073525520"
    ],
    [
      "0.746693162",
      "0.098198846",
      "0.098198846"
    ],
    [
      "0.755067459",
      "0.418194484",
      "0.070921424"
    ],
    [
      "0.252438145",
      "0.902108679",
      "0.097891321"
    ],
    [
      "0.879314232",
      "0.272331344",
      "0.120685768"
    ]
  ]
}

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": 28
}

当前第一名的答案

{
  "circles": [
    [
      "0.404431946",
      "0.538747084",
      "0.096333221"
    ],
    [
      "0.905808692",
      "0.485568775",
      "0.094191308"
    ],
    [
      "0.663717914",
      "0.896570376",
      "0.103429624"
    ],
    [
      "0.077846845",
      "0.922153155",
      "0.077846845"
    ],
    [
      "0.116681191",
      "0.116681191",
      "0.116681191"
    ],
    [
      "0.746436454",
      "0.579882260",
      "0.090996552"
    ],
    [
      "0.562831819",
      "0.692374334",
      "0.124329081"
    ],
    [
      "0.455248936",
      "0.894954383",
      "0.105045617"
    ],
    [
      "0.246820182",
      "0.255470881",
      "0.073578460"
    ],
    [
      "0.340594903",
      "0.724668081",
      "0.100241916"
    ],
    [
      "0.657554637",
      "0.276466303",
      "0.101112374"
    ],
    [
      "0.436180923",
      "0.304521538",
      "0.122032011"
    ],
    [
      "0.094549410",
      "0.326749159",
      "0.094549410"
    ],
    [
      "0.094378735",
      "0.515677227",
      "0.094378735"
    ],
    [
      "0.905900567",
      "0.673859494",
      "0.094099433"
    ],
    [
      "0.589501534",
      "0.467727562",
      "0.101895245"
    ],
    [
      "0.256083181",
      "0.421528314",
      "0.092737126"
    ],
    [
      "0.921872932",
      "0.078127068",
      "0.078127068"
    ],
    [
      "0.543785115",
      "0.104817106",
      "0.104817106"
    ],
    [
      "0.244129831",
      "0.584082967",
      "0.070256425"
    ],
    [
      "0.335956489",
      "0.103019295",
      "0.103019295"
    ],
    [
      "0.120194858",
      "0.728692144",
      "0.120194858"
    ],
    [
      "0.883375803",
      "0.883375803",
      "0.116624197"
    ],
    [
      "0.753767146",
      "0.744240934",
      "0.073525520"
    ],
    [
      "0.746693162",
      "0.098198846",
      "0.098198846"
    ],
    [
      "0.755067459",
      "0.418194484",
      "0.070921424"
    ],
    [
      "0.252438145",
      "0.902108679",
      "0.097891321"
    ],
    [
      "0.879314232",
      "0.272331344",
      "0.120685768"
    ]
  ]
}

提交 circles：恰好 n 个三元组 [x, y, r]，每个数写成十进制字符串，例如 "0.25"。 · 验证器 v1.0.0

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问题等价NUE_13#310 条回复32 天前