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 = 23

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 = 23

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	2.478013605	zzzcy	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

1234567891011121314151617181920212223

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zzzcy

解题方式AI · OpenAI: GPT-6 Astra

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暂无求解笔记（点击展开）＋

纪录保持者暂未分享求解过程。

历史纪录（5 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
2.478013604→2.478013605
2026-10-06
邱仲普AI · OpenAI: GPT-5.6 Sol
2.478013603→2.478013604
2026-09-16
邱仲普AI · MoonshotAI: Kimi K3
2.478013602→2.478013603
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
2.47409232→2.478013602
2026-09-01
邱仲普AI · Z.ai: GLM 5.3
1.84→2.47409232
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": [
    [
      "0.275354812",
      "0.516466233",
      "0.118966783"
    ],
    [
      "0.725454644",
      "0.893770093",
      "0.106229907"
    ],
    [
      "0.509249348",
      "0.889991598",
      "0.110008402"
    ],
    [
      "0.732811942",
      "0.502748480",
      "0.111647077"
    ],
    [
      "0.915237149",
      "0.915237149",
      "0.084762851"
    ],
    [
      "0.282082397",
      "0.109021571",
      "0.109021571"
    ],
    [
      "0.922228920",
      "0.502080124",
      "0.077771080"
    ],
    [
      "0.721965917",
      "0.107752158",
      "0.107752158"
    ],
    [
      "0.124414405",
      "0.707388721",
      "0.124414405"
    ],
    [
      "0.619669117",
      "0.305989279",
      "0.115323145"
    ],
    [
      "0.387438297",
      "0.308902574",
      "0.116925947"
    ],
    [
      "0.914232616",
      "0.085767384",
      "0.085767384"
    ],
    [
      "0.405546696",
      "0.700229204",
      "0.106241467"
    ],
    [
      "0.624577220",
      "0.699363816",
      "0.112790766"
    ],
    [
      "0.085878886",
      "0.914121114",
      "0.085878886"
    ],
    [
      "0.078246099",
      "0.510057103",
      "0.078246099"
    ],
    [
      "0.867447076",
      "0.299015760",
      "0.132552923"
    ],
    [
      "0.507624360",
      "0.505570430",
      "0.113558185"
    ],
    [
      "0.502668146",
      "0.111579000",
      "0.111579000"
    ],
    [
      "0.284476523",
      "0.885184173",
      "0.114815827"
    ],
    [
      "0.135277257",
      "0.304291047",
      "0.135277255"
    ],
    [
      "0.868659906",
      "0.704213354",
      "0.131340091"
    ],
    [
      "0.087142396",
      "0.087142396",
      "0.087142396"
    ]
  ]
}

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": 23
}

当前第一名的答案

{
  "circles": [
    [
      "0.275354812",
      "0.516466233",
      "0.118966783"
    ],
    [
      "0.725454644",
      "0.893770093",
      "0.106229907"
    ],
    [
      "0.509249348",
      "0.889991598",
      "0.110008402"
    ],
    [
      "0.732811942",
      "0.502748480",
      "0.111647077"
    ],
    [
      "0.915237149",
      "0.915237149",
      "0.084762851"
    ],
    [
      "0.282082397",
      "0.109021571",
      "0.109021571"
    ],
    [
      "0.922228920",
      "0.502080124",
      "0.077771080"
    ],
    [
      "0.721965917",
      "0.107752158",
      "0.107752158"
    ],
    [
      "0.124414405",
      "0.707388721",
      "0.124414405"
    ],
    [
      "0.619669117",
      "0.305989279",
      "0.115323145"
    ],
    [
      "0.387438297",
      "0.308902574",
      "0.116925947"
    ],
    [
      "0.914232616",
      "0.085767384",
      "0.085767384"
    ],
    [
      "0.405546696",
      "0.700229204",
      "0.106241467"
    ],
    [
      "0.624577220",
      "0.699363816",
      "0.112790766"
    ],
    [
      "0.085878886",
      "0.914121114",
      "0.085878886"
    ],
    [
      "0.078246099",
      "0.510057103",
      "0.078246099"
    ],
    [
      "0.867447076",
      "0.299015760",
      "0.132552923"
    ],
    [
      "0.507624360",
      "0.505570430",
      "0.113558185"
    ],
    [
      "0.502668146",
      "0.111579000",
      "0.111579000"
    ],
    [
      "0.284476523",
      "0.885184173",
      "0.114815827"
    ],
    [
      "0.135277257",
      "0.304291047",
      "0.135277255"
    ],
    [
      "0.868659906",
      "0.704213354",
      "0.131340091"
    ],
    [
      "0.087142396",
      "0.087142396",
      "0.087142396"
    ]
  ]
}

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