P51 · 极值构型 · 经典问题 · 难

单位正方形里的照明 · n = 35

在单位正方形内放置 n 个单位亮度的点光源，任一点的光强为各光源到该点距离平方的倒数之和；使正方形内最暗点的光强尽可能大。

子题n = 35

目标最大化最小光强

纪录对比

最小光强 · 越大越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	[359.463, 359.464)	Mishari Alhussain2026-09	Erich Friedman ↗
本站纪录	359.463099	qiboqi	纪录详情 ↓

在公布精度内匹配已知最好；不代表精确相等或已证明最优。

严格定义

容器单位正方形 [0, 1]²，左下角为原点 (0, 0)，右上角为 (1, 1)
提交n 个光源坐标 lights，以及可被验证器严格证实的最小光强下界 intensity
约束所有光源位于正方形内；允许多个光源重合，重合时亮度相加
点光强P = {p₁,…,pₙ} 为光源位置，每个单位亮度光源按距离平方的倒数贡献光强 $I_{P} (x) = \sum_{i = 1}^{n} \frac{1}{{‖ x - p_{i} ‖}^{2}}$
最暗光强L(P) 是整个闭正方形 [0, 1]² 上的连续最小光强 $L (P) = \min_{x \in {[0, 1]}^{2}} I_{P} (x)$
目标在所有合法光源布局 P 中最大化 L(P)；提交的 intensity 必须是验证器可严格证实的 L(P) 下界 $\max_{P} L (P)$

放大编辑并提交 ↗

当前展示：本站纪录

1y0

1234567891011121314151617181920212223242526272829303132333435

0x1

已验证构造最暗处不低于 359.463099

帮助理解

难点在哪里

最暗点通常出现在光源之间的鞍点或正方形四角。把光源向某处集中以照亮暗区时，其他区域的光强又会下降。均匀网格排布会在边角留下明显暗区，最优构形往往需要向边界和四角倾斜配置。

研究进展

Friedman 的公开表格整理了各个 n 的已知最好构形与数值。n = 2 的对称构型 24/5 已由 NUE_13 #31 证明为连续最优（取等坐标为无理数，不宣称九位小数网格最优），其余 n 均无最优性证明。纪录对比表按来源公布的有效位数展示对应的数值区间，达到该区间仅代表在公布精度内持平。

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qiboqi

解题方式AI · OpenAI: GPT-6 Luna

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（16 次易主）

qiboqiAI · OpenAI: GPT-6 Luna
359.463081→359.463099
2026-10-09
qiboqiAI · OpenAI: GPT-6 Luna
359.463074→359.463081
2026-10-09
qiboqiAI · OpenAI: GPT-6 Luna
359.463033→359.463074
2026-10-09
qiboqiAI · OpenAI: GPT-6 Luna
359.461645→359.463033
2026-10-09
qiboqiAI · OpenAI: GPT-6 Luna
359.425036→359.461645
2026-10-08
qiboqiAI · OpenAI: GPT-6 Luna
359.25441→359.425036
2026-10-08
qiboqiAI · OpenAI: GPT-6 Luna
357.338768→359.25441
2026-10-08
qiboqi人工
357.338621→357.338768
2026-10-08
qiboqi人工
357.334366→357.338621
2026-10-08
今天人工
357.333806→357.334366
2026-09-06
今天人工
357.330575→357.333806
2026-09-04
今天人工
357.330378→357.330575
2026-09-04
😰人工
357.329982→357.330378
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
357.183674→357.329982
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
347.195206→357.183674
2026-09-02
lird人工
347.191734→347.195206
2026-08-29

ANSWER FORMAT

答案怎么写

容器是边长为 1 的正方形，左下角为原点 (0, 0)，右上角为 (1, 1)。光源坐标写成十进制小数，例如 "0.25"，最多九位小数；最小光强下界 intensity 最多六位小数。

提交 lights 与 intensity。lights 为 n 个光源坐标，写成十进制字符串；intensity 为你声称的正方形内最小光强下界，最多六位小数。验证器会严格验证该下界，建议填得比实际最小光强略小一点。某点的光强等于各光源到该点距离平方的倒数之和。

当前第一名的答案

{
  "intensity": "359.463099",
  "lights": [
    [
      "0.046212948",
      "0.044461349"
    ],
    [
      "0.953786303",
      "0.044460575"
    ],
    [
      "0.046213121",
      "0.955538838"
    ],
    [
      "0.953786990",
      "0.955538700"
    ],
    [
      "0.199816244",
      "0.052354685"
    ],
    [
      "0.800183755",
      "0.052356514"
    ],
    [
      "0.199816244",
      "0.947644981"
    ],
    [
      "0.800183755",
      "0.947645081"
    ],
    [
      "0.044844775",
      "0.201875064"
    ],
    [
      "0.955157455",
      "0.201874983"
    ],
    [
      "0.044843235",
      "0.798124383"
    ],
    [
      "0.955154661",
      "0.798125257"
    ],
    [
      "0.240194130",
      "0.275667243"
    ],
    [
      "0.759804378",
      "0.275666420"
    ],
    [
      "0.240194953",
      "0.724334592"
    ],
    [
      "0.759804737",
      "0.724331767"
    ],
    [
      "0.412115114",
      "0.291119753"
    ],
    [
      "0.587885573",
      "0.291119753"
    ],
    [
      "0.412115347",
      "0.708880247"
    ],
    [
      "0.587885168",
      "0.708879274"
    ],
    [
      "0.060183758",
      "0.359432077"
    ],
    [
      "0.939814037",
      "0.359431313"
    ],
    [
      "0.060186152",
      "0.640568371"
    ],
    [
      "0.939814023",
      "0.640570296"
    ],
    [
      "0.352223208",
      "0.058781830"
    ],
    [
      "0.647776684",
      "0.058780091"
    ],
    [
      "0.352223876",
      "0.941219367"
    ],
    [
      "0.647776949",
      "0.941218131"
    ],
    [
      "0.499999285",
      "0.062212903"
    ],
    [
      "0.500000871",
      "0.937787099"
    ],
    [
      "0.280985224",
      "0.499999555"
    ],
    [
      "0.719014776",
      "0.499998646"
    ],
    [
      "0.070731137",
      "0.500001128"
    ],
    [
      "0.929271055",
      "0.500000770"
    ],
    [
      "0.499999284",
      "0.500000650"
    ]
  ]
}

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": 35
}

当前第一名的答案

{
  "intensity": "359.463099",
  "lights": [
    [
      "0.046212948",
      "0.044461349"
    ],
    [
      "0.953786303",
      "0.044460575"
    ],
    [
      "0.046213121",
      "0.955538838"
    ],
    [
      "0.953786990",
      "0.955538700"
    ],
    [
      "0.199816244",
      "0.052354685"
    ],
    [
      "0.800183755",
      "0.052356514"
    ],
    [
      "0.199816244",
      "0.947644981"
    ],
    [
      "0.800183755",
      "0.947645081"
    ],
    [
      "0.044844775",
      "0.201875064"
    ],
    [
      "0.955157455",
      "0.201874983"
    ],
    [
      "0.044843235",
      "0.798124383"
    ],
    [
      "0.955154661",
      "0.798125257"
    ],
    [
      "0.240194130",
      "0.275667243"
    ],
    [
      "0.759804378",
      "0.275666420"
    ],
    [
      "0.240194953",
      "0.724334592"
    ],
    [
      "0.759804737",
      "0.724331767"
    ],
    [
      "0.412115114",
      "0.291119753"
    ],
    [
      "0.587885573",
      "0.291119753"
    ],
    [
      "0.412115347",
      "0.708880247"
    ],
    [
      "0.587885168",
      "0.708879274"
    ],
    [
      "0.060183758",
      "0.359432077"
    ],
    [
      "0.939814037",
      "0.359431313"
    ],
    [
      "0.060186152",
      "0.640568371"
    ],
    [
      "0.939814023",
      "0.640570296"
    ],
    [
      "0.352223208",
      "0.058781830"
    ],
    [
      "0.647776684",
      "0.058780091"
    ],
    [
      "0.352223876",
      "0.941219367"
    ],
    [
      "0.647776949",
      "0.941218131"
    ],
    [
      "0.499999285",
      "0.062212903"
    ],
    [
      "0.500000871",
      "0.937787099"
    ],
    [
      "0.280985224",
      "0.499999555"
    ],
    [
      "0.719014776",
      "0.499998646"
    ],
    [
      "0.070731137",
      "0.500001128"
    ],
    [
      "0.929271055",
      "0.500000770"
    ],
    [
      "0.499999284",
      "0.500000650"
    ]
  ]
}

提交 lights 与 intensity。lights 为 n 个光源坐标，写成十进制字符串；intensity 为你声称的正方形内最小光强下界，最多六位小数。验证器会严格验证该下界，建议填得比实际最小光强略小一点。某点的光强等于各光源到该点距离平方的倒数之和。 · 验证器 v1.0.0

DISCUSSION

讨论区

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