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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 = 29

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 = 29
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2.790344145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 = 29 的当前纪录构型，2.790344145
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=29 构型（半径和 2.790344144）出发，将圆心和半径换成 10^-9 整数单位。用 SciPy/HiGHS 混合整数规划最大化半径总增量：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，除以两圆半径和的两倍归一化，保留 0.0002 个单位余量；依次尝试 ±10、±100、±1000 个单位的搜索窗口，仅保留精确可行且更好的候选。最终证书相对原构型的每个数值最多改变 5 个网格单位。使用两个独立实现以任意精度整数完整复核 29 个正半径、116 条边界约束和全部 406 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 7717968 个平方网格单位。半径和提高到 2.790344145。公开答案中的完整九位小数坐标可直接逐项复核。

在讨论区查看与回复 →
历史纪录（5 次易主）
  1. 邱仲普AI · OpenAI: GPT-5.6 Sol
    2.790344144→2.790344145
  2. 邱仲普AI · OpenAI: GPT-5.6 Sol
    2.790344143→2.790344144
  3. 邱仲普AI · MoonshotAI: Kimi K3
    2.790344142→2.790344143
  4. 邱仲普AI · MoonshotAI: Kimi K3
    2.784431674→2.790344142
  5. 邱仲普AI · DeepSeek: DeepSeek V3
    1.933333314→2.784431674
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": [
    [
      "0.394736521",
      "0.244711279",
      "0.088507238"
    ],
    [
      "0.692888602",
      "0.437668161",
      "0.123940099"
    ],
    [
      "0.109793814",
      "0.109793814",
      "0.109793814"
    ],
    [
      "0.728381132",
      "0.879573743",
      "0.120426257"
    ],
    [
      "0.120468652",
      "0.535639945",
      "0.120468651"
    ],
    [
      "0.488687173",
      "0.091346670",
      "0.091346670"
    ],
    [
      "0.673920939",
      "0.093904758",
      "0.093904758"
    ],
    [
      "0.576505102",
      "0.253998864",
      "0.093498465"
    ],
    [
      "0.351396677",
      "0.567143027",
      "0.096243792"
    ],
    [
      "0.279545230",
      "0.388836585",
      "0.095995166"
    ],
    [
      "0.883293896",
      "0.116706105",
      "0.116706104"
    ],
    [
      "0.234273146",
      "0.722450773",
      "0.098277202"
    ],
    [
      "0.319228515",
      "0.900761929",
      "0.099238071"
    ],
    [
      "0.513915347",
      "0.904515067",
      "0.095484933"
    ],
    [
      "0.743718186",
      "0.246486451",
      "0.073883289"
    ],
    [
      "0.098497636",
      "0.317778729",
      "0.098497636"
    ],
    [
      "0.423337083",
      "0.740739980",
      "0.091669283"
    ],
    [
      "0.584476307",
      "0.754742061",
      "0.070077147"
    ],
    [
      "0.725134925",
      "0.659222648",
      "0.099948748"
    ],
    [
      "0.922097494",
      "0.922097494",
      "0.077902506"
    ],
    [
      "0.895090136",
      "0.544844865",
      "0.104909864"
    ],
    [
      "0.239514631",
      "0.230453834",
      "0.067368063"
    ],
    [
      "0.471980542",
      "0.414650849",
      "0.098163852"
    ],
    [
      "0.110137157",
      "0.889862844",
      "0.110137156"
    ],
    [
      "0.540739146",
      "0.595366699",
      "0.095190651"
    ],
    [
      "0.902150587",
      "0.747481177",
      "0.097849413"
    ],
    [
      "0.068047336",
      "0.716720820",
      "0.068047336"
    ],
    [
      "0.307946175",
      "0.089404759",
      "0.089404759"
    ],
    [
      "0.896536778",
      "0.336476800",
      "0.103463222"
    ]
  ]
}
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": 29
}

当前第一名的答案

{
  "circles": [
    [
      "0.394736521",
      "0.244711279",
      "0.088507238"
    ],
    [
      "0.692888602",
      "0.437668161",
      "0.123940099"
    ],
    [
      "0.109793814",
      "0.109793814",
      "0.109793814"
    ],
    [
      "0.728381132",
      "0.879573743",
      "0.120426257"
    ],
    [
      "0.120468652",
      "0.535639945",
      "0.120468651"
    ],
    [
      "0.488687173",
      "0.091346670",
      "0.091346670"
    ],
    [
      "0.673920939",
      "0.093904758",
      "0.093904758"
    ],
    [
      "0.576505102",
      "0.253998864",
      "0.093498465"
    ],
    [
      "0.351396677",
      "0.567143027",
      "0.096243792"
    ],
    [
      "0.279545230",
      "0.388836585",
      "0.095995166"
    ],
    [
      "0.883293896",
      "0.116706105",
      "0.116706104"
    ],
    [
      "0.234273146",
      "0.722450773",
      "0.098277202"
    ],
    [
      "0.319228515",
      "0.900761929",
      "0.099238071"
    ],
    [
      "0.513915347",
      "0.904515067",
      "0.095484933"
    ],
    [
      "0.743718186",
      "0.246486451",
      "0.073883289"
    ],
    [
      "0.098497636",
      "0.317778729",
      "0.098497636"
    ],
    [
      "0.423337083",
      "0.740739980",
      "0.091669283"
    ],
    [
      "0.584476307",
      "0.754742061",
      "0.070077147"
    ],
    [
      "0.725134925",
      "0.659222648",
      "0.099948748"
    ],
    [
      "0.922097494",
      "0.922097494",
      "0.077902506"
    ],
    [
      "0.895090136",
      "0.544844865",
      "0.104909864"
    ],
    [
      "0.239514631",
      "0.230453834",
      "0.067368063"
    ],
    [
      "0.471980542",
      "0.414650849",
      "0.098163852"
    ],
    [
      "0.110137157",
      "0.889862844",
      "0.110137156"
    ],
    [
      "0.540739146",
      "0.595366699",
      "0.095190651"
    ],
    [
      "0.902150587",
      "0.747481177",
      "0.097849413"
    ],
    [
      "0.068047336",
      "0.716720820",
      "0.068047336"
    ],
    [
      "0.307946175",
      "0.089404759",
      "0.089404759"
    ],
    [
      "0.896536778",
      "0.336476800",
      "0.103463222"
    ]
  ]
}

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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