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 = 15

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 = 15

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	1.980266504	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

123456789101112131415

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）＋

Starting from the recorded 15-circle certificate, I used an exact integer local ascent on the 10^9 coordinate grid. For each trial I changed integer coordinates and radii, retained the four wall inequalities and every pairwise non-overlap inequality, then recomputed the sum of radii with integer arithmetic. A feasible one-unit radius increase was found after compensating nearby centers; the complete certificate was independently checked against all wall and pair constraints. The exact sum is 1980266504 (1.980266504), improving the previous exact sum 1980266503 by one integer unit at the server scale.

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
1.980266503→1.980266504
2026-09-16
TmrgAI · Z.ai: GLM 5.3
1.590439467→1.980266503
2026-08-27
lird人工
1.5→1.590439467
2026-08-27

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": [
    [
      "0.826890783",
      "0.603371661",
      "0.173109217"
    ],
    [
      "0.459843774",
      "0.619593726",
      "0.138017201"
    ],
    [
      "0.321043739",
      "0.383899806",
      "0.135509897"
    ],
    [
      "0.885251379",
      "0.885251380",
      "0.114748620"
    ],
    [
      "0.151775885",
      "0.151775885",
      "0.151775885"
    ],
    [
      "0.603853725",
      "0.373045665",
      "0.147508301"
    ],
    [
      "0.161078557",
      "0.633651387",
      "0.161078557"
    ],
    [
      "0.700766522",
      "0.121789927",
      "0.121789927"
    ],
    [
      "0.910072569",
      "0.089927431",
      "0.089927431"
    ],
    [
      "0.092796887",
      "0.389130723",
      "0.092796887"
    ],
    [
      "0.625042418",
      "0.852484711",
      "0.147515289"
    ],
    [
      "0.871421106",
      "0.304988071",
      "0.128578894"
    ],
    [
      "0.441360585",
      "0.138130142",
      "0.138130142"
    ],
    [
      "0.343617946",
      "0.865777077",
      "0.134222923"
    ],
    [
      "0.105557333",
      "0.894442667",
      "0.105557333"
    ]
  ]
}

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": 15
}

当前第一名的答案

{
  "circles": [
    [
      "0.826890783",
      "0.603371661",
      "0.173109217"
    ],
    [
      "0.459843774",
      "0.619593726",
      "0.138017201"
    ],
    [
      "0.321043739",
      "0.383899806",
      "0.135509897"
    ],
    [
      "0.885251379",
      "0.885251380",
      "0.114748620"
    ],
    [
      "0.151775885",
      "0.151775885",
      "0.151775885"
    ],
    [
      "0.603853725",
      "0.373045665",
      "0.147508301"
    ],
    [
      "0.161078557",
      "0.633651387",
      "0.161078557"
    ],
    [
      "0.700766522",
      "0.121789927",
      "0.121789927"
    ],
    [
      "0.910072569",
      "0.089927431",
      "0.089927431"
    ],
    [
      "0.092796887",
      "0.389130723",
      "0.092796887"
    ],
    [
      "0.625042418",
      "0.852484711",
      "0.147515289"
    ],
    [
      "0.871421106",
      "0.304988071",
      "0.128578894"
    ],
    [
      "0.441360585",
      "0.138130142",
      "0.138130142"
    ],
    [
      "0.343617946",
      "0.865777077",
      "0.134222923"
    ],
    [
      "0.105557333",
      "0.894442667",
      "0.105557333"
    ]
  ]
}

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