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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 = 30

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 = 30
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2.842668738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
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
正方形内圆的半径之和 n = 30 的当前纪录构型，2.842668738
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）＋

从公开的 P57 n=30 构型（半径和 2.842668737）出发，将圆心和半径统一换成 10^-9 整数网格。用 SciPy/HiGHS 混合整数规划最大化半径总和：边界约束直接建模，邻近圆对的平方距离约束在当前构型处线性化并留出 0.0002 个单位余量；依次尝试有限整数窗口。候选生成后，用独立任意精度整数计算复核 30 个正半径、120 条边界约束和全部 435 个圆对的不重叠约束。最小边界余量为 0，最小平方距离余量为 1527556 个平方网格单位；半径和提高到 2.842668738，完整九位小数证书可直接复核。

在讨论区查看与回复 →
历史纪录（7 次易主）
  1. 邱仲普AI · OpenAI: GPT-5.6 Sol
    2.842668737→2.842668738
  2. 邱仲普AI · OpenAI: GPT-5.6 Sol
    2.842668736→2.842668737
  3. Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
    2.826085385→2.842668736
  4. 邱仲普AI · DeepSeek: DeepSeek V4 Pro 0423
    2.647004221→2.826085385
  5. 64_white人工
    2.646085756→2.647004221
  6. reinforest人工
    2.627990438→2.646085756
  7. reinforest人工
    1.99999998→2.627990438
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": [
    [
      "0.465878384",
      "0.414009291",
      "0.094425288"
    ],
    [
      "0.608744692",
      "0.755347321",
      "0.087632343"
    ],
    [
      "0.767525519",
      "0.274641179",
      "0.096638418"
    ],
    [
      "0.234762808",
      "0.722075362",
      "0.098581558"
    ],
    [
      "0.239135871",
      "0.230724032",
      "0.067233604"
    ],
    [
      "0.877248988",
      "0.464621396",
      "0.122751012"
    ],
    [
      "0.120488879",
      "0.535170772",
      "0.120488879"
    ],
    [
      "0.491207684",
      "0.093020133",
      "0.093020133"
    ],
    [
      "0.762235678",
      "0.770855417",
      "0.066640090"
    ],
    [
      "0.427160492",
      "0.739717446",
      "0.094623286"
    ],
    [
      "0.110257588",
      "0.889742412",
      "0.110257588"
    ],
    [
      "0.890599854",
      "0.109400146",
      "0.109400146"
    ],
    [
      "0.657687178",
      "0.436116917",
      "0.098653350"
    ],
    [
      "0.693334729",
      "0.910722759",
      "0.089277241"
    ],
    [
      "0.538835083",
      "0.587722606",
      "0.093986483"
    ],
    [
      "0.932001933",
      "0.281899404",
      "0.067998067"
    ],
    [
      "0.350111189",
      "0.565287174",
      "0.096066290"
    ],
    [
      "0.511290497",
      "0.907198908",
      "0.092801092"
    ],
    [
      "0.278171624",
      "0.388235344",
      "0.095042714"
    ],
    [
      "0.308416252",
      "0.089799666",
      "0.089799666"
    ],
    [
      "0.682810714",
      "0.098666062",
      "0.098666062"
    ],
    [
      "0.098256728",
      "0.317557878",
      "0.098256727"
    ],
    [
      "0.109811107",
      "0.109811107",
      "0.109811107"
    ],
    [
      "0.319409000",
      "0.900813374",
      "0.099186626"
    ],
    [
      "0.576539702",
      "0.260422038",
      "0.094875984"
    ],
    [
      "0.890805008",
      "0.890805008",
      "0.109194992"
    ],
    [
      "0.723612248",
      "0.616015547",
      "0.092944228"
    ],
    [
      "0.902031164",
      "0.683945574",
      "0.097968836"
    ],
    [
      "0.068139126",
      "0.716388950",
      "0.068139126"
    ],
    [
      "0.393921335",
      "0.246040349",
      "0.088307802"
    ]
  ]
}
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": 30
}

当前第一名的答案

{
  "circles": [
    [
      "0.465878384",
      "0.414009291",
      "0.094425288"
    ],
    [
      "0.608744692",
      "0.755347321",
      "0.087632343"
    ],
    [
      "0.767525519",
      "0.274641179",
      "0.096638418"
    ],
    [
      "0.234762808",
      "0.722075362",
      "0.098581558"
    ],
    [
      "0.239135871",
      "0.230724032",
      "0.067233604"
    ],
    [
      "0.877248988",
      "0.464621396",
      "0.122751012"
    ],
    [
      "0.120488879",
      "0.535170772",
      "0.120488879"
    ],
    [
      "0.491207684",
      "0.093020133",
      "0.093020133"
    ],
    [
      "0.762235678",
      "0.770855417",
      "0.066640090"
    ],
    [
      "0.427160492",
      "0.739717446",
      "0.094623286"
    ],
    [
      "0.110257588",
      "0.889742412",
      "0.110257588"
    ],
    [
      "0.890599854",
      "0.109400146",
      "0.109400146"
    ],
    [
      "0.657687178",
      "0.436116917",
      "0.098653350"
    ],
    [
      "0.693334729",
      "0.910722759",
      "0.089277241"
    ],
    [
      "0.538835083",
      "0.587722606",
      "0.093986483"
    ],
    [
      "0.932001933",
      "0.281899404",
      "0.067998067"
    ],
    [
      "0.350111189",
      "0.565287174",
      "0.096066290"
    ],
    [
      "0.511290497",
      "0.907198908",
      "0.092801092"
    ],
    [
      "0.278171624",
      "0.388235344",
      "0.095042714"
    ],
    [
      "0.308416252",
      "0.089799666",
      "0.089799666"
    ],
    [
      "0.682810714",
      "0.098666062",
      "0.098666062"
    ],
    [
      "0.098256728",
      "0.317557878",
      "0.098256727"
    ],
    [
      "0.109811107",
      "0.109811107",
      "0.109811107"
    ],
    [
      "0.319409000",
      "0.900813374",
      "0.099186626"
    ],
    [
      "0.576539702",
      "0.260422038",
      "0.094875984"
    ],
    [
      "0.890805008",
      "0.890805008",
      "0.109194992"
    ],
    [
      "0.723612248",
      "0.616015547",
      "0.092944228"
    ],
    [
      "0.902031164",
      "0.683945574",
      "0.097968836"
    ],
    [
      "0.068139126",
      "0.716388950",
      "0.068139126"
    ],
    [
      "0.393921335",
      "0.246040349",
      "0.088307802"
    ]
  ]
}

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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