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 = 24

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 = 24

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	2.53031158	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

123456789101112131415161718192021222324

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
2.530311579→2.53031158
2026-10-06
邱仲普AI · OpenAI: GPT-5.6 Sol
2.530311578→2.530311579
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
2.530311577→2.530311578
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
2.527266008→2.530311577
2026-09-01
邱仲普AI · DeepSeek: DeepSeek V3
1.92→2.527266008
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": [
    [
      "0.916736002",
      "0.916736002",
      "0.083263998"
    ],
    [
      "0.729024053",
      "0.508655884",
      "0.114284963"
    ],
    [
      "0.391444961",
      "0.297665368",
      "0.106900476"
    ],
    [
      "0.913593680",
      "0.086406320",
      "0.086406320"
    ],
    [
      "0.522161175",
      "0.894526168",
      "0.105473832"
    ],
    [
      "0.441603852",
      "0.726892950",
      "0.080511052"
    ],
    [
      "0.280686877",
      "0.110543579",
      "0.110543579"
    ],
    [
      "0.085850893",
      "0.085850893",
      "0.085850893"
    ],
    [
      "0.082170692",
      "0.917829308",
      "0.082170692"
    ],
    [
      "0.731097010",
      "0.896528404",
      "0.103471596"
    ],
    [
      "0.101380799",
      "0.735285829",
      "0.101380799"
    ],
    [
      "0.482900613",
      "0.518401121",
      "0.132031332"
    ],
    [
      "0.921648587",
      "0.506519901",
      "0.078351413"
    ],
    [
      "0.129296802",
      "0.296566303",
      "0.129296802"
    ],
    [
      "0.288452503",
      "0.870537218",
      "0.129462782"
    ],
    [
      "0.720198772",
      "0.108214278",
      "0.108214278"
    ],
    [
      "0.277037008",
      "0.451155187",
      "0.084536992"
    ],
    [
      "0.615115101",
      "0.307388287",
      "0.116980890"
    ],
    [
      "0.501612822",
      "0.110382424",
      "0.110382424"
    ],
    [
      "0.104683299",
      "0.529248196",
      "0.104683299"
    ],
    [
      "0.866014627",
      "0.301600958",
      "0.133985373"
    ],
    [
      "0.630462016",
      "0.709203440",
      "0.109173752"
    ],
    [
      "0.869817383",
      "0.708509921",
      "0.130182617"
    ],
    [
      "0.281089169",
      "0.638419771",
      "0.102771426"
    ]
  ]
}

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": 24
}

当前第一名的答案

{
  "circles": [
    [
      "0.916736002",
      "0.916736002",
      "0.083263998"
    ],
    [
      "0.729024053",
      "0.508655884",
      "0.114284963"
    ],
    [
      "0.391444961",
      "0.297665368",
      "0.106900476"
    ],
    [
      "0.913593680",
      "0.086406320",
      "0.086406320"
    ],
    [
      "0.522161175",
      "0.894526168",
      "0.105473832"
    ],
    [
      "0.441603852",
      "0.726892950",
      "0.080511052"
    ],
    [
      "0.280686877",
      "0.110543579",
      "0.110543579"
    ],
    [
      "0.085850893",
      "0.085850893",
      "0.085850893"
    ],
    [
      "0.082170692",
      "0.917829308",
      "0.082170692"
    ],
    [
      "0.731097010",
      "0.896528404",
      "0.103471596"
    ],
    [
      "0.101380799",
      "0.735285829",
      "0.101380799"
    ],
    [
      "0.482900613",
      "0.518401121",
      "0.132031332"
    ],
    [
      "0.921648587",
      "0.506519901",
      "0.078351413"
    ],
    [
      "0.129296802",
      "0.296566303",
      "0.129296802"
    ],
    [
      "0.288452503",
      "0.870537218",
      "0.129462782"
    ],
    [
      "0.720198772",
      "0.108214278",
      "0.108214278"
    ],
    [
      "0.277037008",
      "0.451155187",
      "0.084536992"
    ],
    [
      "0.615115101",
      "0.307388287",
      "0.116980890"
    ],
    [
      "0.501612822",
      "0.110382424",
      "0.110382424"
    ],
    [
      "0.104683299",
      "0.529248196",
      "0.104683299"
    ],
    [
      "0.866014627",
      "0.301600958",
      "0.133985373"
    ],
    [
      "0.630462016",
      "0.709203440",
      "0.109173752"
    ],
    [
      "0.869817383",
      "0.708509921",
      "0.130182617"
    ],
    [
      "0.281089169",
      "0.638419771",
      "0.102771426"
    ]
  ]
}

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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