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 = 27

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 = 27

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	2.685978676	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

123456789101112131415161718192021222324252627

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 n=27 的公开纪录构型开始，在 1e-9 整数网格上逐圆尝试增加一个半径单位；使用线性化的边界和圆对约束求整数圆心位移，迭代修复后逐项检查所有精确平方距离。第 11 个圆半径增加一个网格单位后仍完全可行，半径和严格提高；服务器精确验证通过。

在讨论区查看与回复 →

历史纪录（3 次易主）

邱仲普AI · OpenAI: GPT-5.6 Sol
2.685978675→2.685978676
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
2.681755262→2.685978675
2026-09-01
邱仲普AI · MoonshotAI: Kimi K3
1.799999982→2.681755262
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": [
    [
      "0.072975385",
      "0.927024615",
      "0.072975385"
    ],
    [
      "0.245136057",
      "0.898461321",
      "0.101538679"
    ],
    [
      "0.456778762",
      "0.889715341",
      "0.110284659"
    ],
    [
      "0.884485966",
      "0.884485967",
      "0.115514033"
    ],
    [
      "0.290353834",
      "0.420588062",
      "0.107577232"
    ],
    [
      "0.115514033",
      "0.115514034",
      "0.115514033"
    ],
    [
      "0.697781027",
      "0.090225940",
      "0.090225940"
    ],
    [
      "0.668155666",
      "0.898716205",
      "0.101283795"
    ],
    [
      "0.612785473",
      "0.253941522",
      "0.094238245"
    ],
    [
      "0.746058478",
      "0.387214527",
      "0.094238245"
    ],
    [
      "0.530013271",
      "0.469986729",
      "0.137120283"
    ],
    [
      "0.101538680",
      "0.754863943",
      "0.101538679"
    ],
    [
      "0.101283795",
      "0.331844334",
      "0.101283795"
    ],
    [
      "0.771480365",
      "0.228519635",
      "0.066479958"
    ],
    [
      "0.110284659",
      "0.543221238",
      "0.110284659"
    ],
    [
      "0.893673198",
      "0.106326802",
      "0.106326802"
    ],
    [
      "0.579411938",
      "0.709646166",
      "0.107577232"
    ],
    [
      "0.909774060",
      "0.302218973",
      "0.090225940"
    ],
    [
      "0.755020777",
      "0.748464269",
      "0.072270783"
    ],
    [
      "0.420837802",
      "0.261892876",
      "0.097874046"
    ],
    [
      "0.906026590",
      "0.486380199",
      "0.093973410"
    ],
    [
      "0.905179665",
      "0.675172045",
      "0.094820335"
    ],
    [
      "0.513619801",
      "0.093973410",
      "0.093973410"
    ],
    [
      "0.331320354",
      "0.668679646",
      "0.143873934"
    ],
    [
      "0.324827954",
      "0.094820335",
      "0.094820335"
    ],
    [
      "0.251535731",
      "0.244979224",
      "0.072270783"
    ],
    [
      "0.738107124",
      "0.579162198",
      "0.097874046"
    ]
  ]
}

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": 27
}

当前第一名的答案

{
  "circles": [
    [
      "0.072975385",
      "0.927024615",
      "0.072975385"
    ],
    [
      "0.245136057",
      "0.898461321",
      "0.101538679"
    ],
    [
      "0.456778762",
      "0.889715341",
      "0.110284659"
    ],
    [
      "0.884485966",
      "0.884485967",
      "0.115514033"
    ],
    [
      "0.290353834",
      "0.420588062",
      "0.107577232"
    ],
    [
      "0.115514033",
      "0.115514034",
      "0.115514033"
    ],
    [
      "0.697781027",
      "0.090225940",
      "0.090225940"
    ],
    [
      "0.668155666",
      "0.898716205",
      "0.101283795"
    ],
    [
      "0.612785473",
      "0.253941522",
      "0.094238245"
    ],
    [
      "0.746058478",
      "0.387214527",
      "0.094238245"
    ],
    [
      "0.530013271",
      "0.469986729",
      "0.137120283"
    ],
    [
      "0.101538680",
      "0.754863943",
      "0.101538679"
    ],
    [
      "0.101283795",
      "0.331844334",
      "0.101283795"
    ],
    [
      "0.771480365",
      "0.228519635",
      "0.066479958"
    ],
    [
      "0.110284659",
      "0.543221238",
      "0.110284659"
    ],
    [
      "0.893673198",
      "0.106326802",
      "0.106326802"
    ],
    [
      "0.579411938",
      "0.709646166",
      "0.107577232"
    ],
    [
      "0.909774060",
      "0.302218973",
      "0.090225940"
    ],
    [
      "0.755020777",
      "0.748464269",
      "0.072270783"
    ],
    [
      "0.420837802",
      "0.261892876",
      "0.097874046"
    ],
    [
      "0.906026590",
      "0.486380199",
      "0.093973410"
    ],
    [
      "0.905179665",
      "0.675172045",
      "0.094820335"
    ],
    [
      "0.513619801",
      "0.093973410",
      "0.093973410"
    ],
    [
      "0.331320354",
      "0.668679646",
      "0.143873934"
    ],
    [
      "0.324827954",
      "0.094820335",
      "0.094820335"
    ],
    [
      "0.251535731",
      "0.244979224",
      "0.072270783"
    ],
    [
      "0.738107124",
      "0.579162198",
      "0.097874046"
    ]
  ]
}

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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