P51 · 极值构型 · 经典问题 · 难

单位正方形里的照明 · n = 34

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 = 34

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	[346.523, 346.524)	Mishari Alhussain2026-10	Erich Friedman ↗
本站纪录	346.523376	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

12345678910111213141516171819202122232425262728293031323334

0x1

已验证构造最暗处不低于 346.523376

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（25 次易主）

qiboqiAI · OpenAI: GPT-6 Luna
346.523375→346.523376
2026-10-11
qiboqiAI · OpenAI: GPT-6 Luna
346.523367→346.523375
2026-10-10
今天AI · DeepSeek: DeepSeek V4.1 Flash
346.523359→346.523367
2026-10-10
qiboqiAI · OpenAI: GPT-6 Luna
346.514936→346.523359
2026-10-10
qiboqiAI · OpenAI: GPT-6 Luna
346.514539→346.514936
2026-10-10
今天AI · DeepSeek: DeepSeek V4.1 Flash
346.512336→346.514539
2026-10-09
qiboqiAI · OpenAI: GPT-6 Luna
346.511819→346.512336
2026-10-09
qiboqiAI · OpenAI: GPT-6 Luna
346.5118→346.511819
2026-10-09
qiboqiAI · OpenAI: GPT-6 Luna
346.511329→346.5118
2026-10-09
NUE_13人工
346.511324→346.511329
2026-10-09
qiboqiAI · OpenAI: GPT-6 Luna
346.510705→346.511324
2026-10-09
NUE_13人工
346.510699→346.510705
2026-10-09
qiboqiAI · OpenAI: GPT-6 Luna
346.50704→346.510699
2026-10-09
今天AI · DeepSeek: DeepSeek V4.1 Flash
344.583457→346.50704
2026-10-09
qiboqiAI · OpenAI: GPT-6 Luna
344.265655→344.583457
2026-10-08
NUE_13人工
344.245377→344.265655
2026-10-05
游侠AI · OpenAI: GPT-5.6 Sol
344.242054→344.245377
2026-09-12
今天人工
344.242053→344.242054
2026-09-08
游侠AI · OpenAI: GPT-5.6 Sol
344.236429→344.242053
2026-09-07
今天人工
344.23623→344.236429
2026-09-06
今天人工
344.235704→344.23623
2026-09-06
游侠AI · OpenAI: GPT-5.6 Sol
344.088144→344.235704
2026-09-06
今天人工
344.088056→344.088144
2026-09-05
hadsajnc人工
344.088047→344.088056
2026-09-05
今天人工
344.078227→344.088047
2026-09-05

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": "346.523376",
  "lights": [
    [
      "0.496697195",
      "0.947206088"
    ],
    [
      "0.814277849",
      "0.946690849"
    ],
    [
      "0.962670640",
      "0.946323936"
    ],
    [
      "0.035933575",
      "0.945254457"
    ],
    [
      "0.336090666",
      "0.945093264"
    ],
    [
      "0.656956102",
      "0.944624743"
    ],
    [
      "0.180886986",
      "0.943241326"
    ],
    [
      "0.965279227",
      "0.774279448"
    ],
    [
      "0.029709561",
      "0.770023715"
    ],
    [
      "0.770771927",
      "0.728848500"
    ],
    [
      "0.581218158",
      "0.724877171"
    ],
    [
      "0.395322014",
      "0.721979336"
    ],
    [
      "0.219861895",
      "0.721096091"
    ],
    [
      "0.967948864",
      "0.592964400"
    ],
    [
      "0.037798810",
      "0.589206669"
    ],
    [
      "0.239595164",
      "0.500000451"
    ],
    [
      "0.435059571",
      "0.499999462"
    ],
    [
      "0.631402420",
      "0.499999789"
    ],
    [
      "0.799729873",
      "0.500000023"
    ],
    [
      "0.037798672",
      "0.410793774"
    ],
    [
      "0.967948740",
      "0.407035382"
    ],
    [
      "0.219861148",
      "0.278904487"
    ],
    [
      "0.395320112",
      "0.278019783"
    ],
    [
      "0.581217492",
      "0.275122425"
    ],
    [
      "0.770771649",
      "0.271151552"
    ],
    [
      "0.029709100",
      "0.229976459"
    ],
    [
      "0.965279097",
      "0.225720447"
    ],
    [
      "0.180886802",
      "0.056758987"
    ],
    [
      "0.656956119",
      "0.055375269"
    ],
    [
      "0.336090489",
      "0.054906405"
    ],
    [
      "0.035933604",
      "0.054745530"
    ],
    [
      "0.962670603",
      "0.053676037"
    ],
    [
      "0.814277821",
      "0.053309208"
    ],
    [
      "0.496697239",
      "0.052793953"
    ]
  ]
}

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": 34
}

当前第一名的答案

{
  "intensity": "346.523376",
  "lights": [
    [
      "0.496697195",
      "0.947206088"
    ],
    [
      "0.814277849",
      "0.946690849"
    ],
    [
      "0.962670640",
      "0.946323936"
    ],
    [
      "0.035933575",
      "0.945254457"
    ],
    [
      "0.336090666",
      "0.945093264"
    ],
    [
      "0.656956102",
      "0.944624743"
    ],
    [
      "0.180886986",
      "0.943241326"
    ],
    [
      "0.965279227",
      "0.774279448"
    ],
    [
      "0.029709561",
      "0.770023715"
    ],
    [
      "0.770771927",
      "0.728848500"
    ],
    [
      "0.581218158",
      "0.724877171"
    ],
    [
      "0.395322014",
      "0.721979336"
    ],
    [
      "0.219861895",
      "0.721096091"
    ],
    [
      "0.967948864",
      "0.592964400"
    ],
    [
      "0.037798810",
      "0.589206669"
    ],
    [
      "0.239595164",
      "0.500000451"
    ],
    [
      "0.435059571",
      "0.499999462"
    ],
    [
      "0.631402420",
      "0.499999789"
    ],
    [
      "0.799729873",
      "0.500000023"
    ],
    [
      "0.037798672",
      "0.410793774"
    ],
    [
      "0.967948740",
      "0.407035382"
    ],
    [
      "0.219861148",
      "0.278904487"
    ],
    [
      "0.395320112",
      "0.278019783"
    ],
    [
      "0.581217492",
      "0.275122425"
    ],
    [
      "0.770771649",
      "0.271151552"
    ],
    [
      "0.029709100",
      "0.229976459"
    ],
    [
      "0.965279097",
      "0.225720447"
    ],
    [
      "0.180886802",
      "0.056758987"
    ],
    [
      "0.656956119",
      "0.055375269"
    ],
    [
      "0.336090489",
      "0.054906405"
    ],
    [
      "0.035933604",
      "0.054745530"
    ],
    [
      "0.962670603",
      "0.053676037"
    ],
    [
      "0.814277821",
      "0.053309208"
    ],
    [
      "0.496697239",
      "0.052793953"
    ]
  ]
}

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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本题n=3（之后关于这道的讨论都会发在此，也欢迎各位参与）NUE_13#311 条回复2 天前 外部记录大幅更新！NUE_13#317 条回复9 天前 [证明] p51-n2-v1 最优值 24/5：完整严格证明NUE_13#315 条回复10 天前 这个外部记录是不是有点水（NUE_13#315 条回复30 天前 对称构型的最优性：本题n=2时的最终证明思路与通用思想NUE_13#310 条回复39 天前 [证明思路 · p51-n2-v1] 本题n=2时证明思路NUE_13#3112 条回复44 天前 n=10似乎可以参考我这里的摆法hadsajnc#1460 条回复58 天前 P51 求解日记 2026 9 6hadsajnc#1460 条回复58 天前 经过我不懈的努力，终于...hadsajnc#1461 条回复58 天前 还在争！！！hadsajnc#1460 条回复59 天前