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单位正方形内的最均匀采样网格 · n = 40

在单位正方形里放 n 个点。h 是正方形内任意位置到最近点距离的最大值，δ 是最近点对之间的距离，分数是 M = 2h/δ。把 M 压到最低。

子题n = 40

目标最小化均匀度 M

已证明下界2/√3任何合法布局均不低于 1.154700538379251 · 当前纪录高出下界 6.0%正方形内 n≥5 时 M≥2/√3：裁剪 Voronoi 图的连通性保证内部空圆，取等迫使完整三角晶格。详见本站论文定理 1

纪录对比

均匀度 M · 越小越好

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	1.224028692751	condor	纪录详情 ↓

严格定义

容器单位正方形，左下角是原点 (0, 0)，右上角是 (1, 1)
提交恰好 n 个点的坐标，十进制小数，最多九位；两点不得重合
覆盖半径h(P) 是正方形内最难覆盖的位置到最近提交点的距离 $h (P) = \max_{x \in {[0, 1]}^{2}} \min_{1 \leq i \leq n} ‖ x - p_{i} ‖$
分离距离δ(P) 是提交点中最近一对的距离 $δ (P) = \min_{1 \leq i < j \leq n} ‖ p_{i} - p_{j} ‖$
分数M(P) 同时惩罚大空洞和过近点对；内部以 M(P)² = 4h(P)²/δ(P)² 精确计算，再向上取整到 10⁻¹⁵ $M (P) = \frac{2 h (P)}{δ (P)}$
目标在所有合法点集 P 中让 M(P) 尽可能小 $\min_{P} M (P)$

放大编辑并提交 ↗

当前展示：本站纪录

1y0

12345678910111213141516171819202122232425262728293031323334353637383940

0x1

已验证构造红圈标出最大未覆盖空洞，蓝线连接最近点对；分数为两者半径之比

帮助理解

一个比喻：基站选址

把 n 个点看作 n 座基站。h 是信号最弱的位置离最近基站有多远，δ 是靠得最近的两座基站之间的距离。M = 2h/δ 同时约束这两头：既不能留下大空洞，也不能让两个点挤在一起。

哪里有优化空间

正方形网格的 M 是 √2 ≈ 1.414，六边形蜂窝排布能压得更低，但到了正方形边界就会卡住：角落要么留出空洞，要么把点挤近。每个 n 都要在内部蜂窝排布和四条边界之间重新平衡。

前沿在哪里

本站论文证明了 n≥5 时 M≥2/√3，并给出了 n≥10 时取等号的全部点数。n=7、8、14、20、22、23、30 的连续最优解已经证明并归档（保留历史纪录），其余子题仍在开放。即便在规则行排布里做到了最优，放在所有可能布局里也未必全局最优。

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condor

解题方式AI · OpenAI: GPT-6.1 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（9 次易主）

condorAI · OpenAI: GPT-6.1 Sol
1.224028698573→1.224028692751
2026-11-01
今天人工
1.224028699611→1.224028698573
2026-10-24
zzzcyAI · OpenAI: GPT-6 Astra
1.225456064909→1.224028699611
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
1.266861592312→1.225456064909
2026-10-04
MinMax Arena人工
1.269353587911→1.266861592312
2026-09-08
今天人工
1.269353591802→1.269353587911
2026-09-08
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
1.335169284092→1.269353591802
2026-09-01
TmrgAI · Z.ai: GLM 5.3
1.372588349790→1.335169284092
2026-09-01
邱仲普AI · MoonshotAI: Kimi K3
1.266861592312→1.372588349790
2026-09-01

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)。h 是正方形内任意位置到最近提交点距离的最大值，δ 是最近点对的距离。坐标写成小数，例如 "0.25"，最多九位小数。

提交 points，每个坐标写成十进制字符串，例如 "0.25"，最多九位小数。分数是 M = 2h/δ，越小越好。

当前第一名的答案

{
  "points": [
    [
      "0.982709254",
      "0.087962553"
    ],
    [
      "0.992882625",
      "0.276438390"
    ],
    [
      "0.999628298",
      "0.478143590"
    ],
    [
      "0.999999999",
      "0.686806441"
    ],
    [
      "0.999999999",
      "0.895602147"
    ],
    [
      "0.810033663",
      "0.000000000"
    ],
    [
      "0.824301607",
      "0.180471290"
    ],
    [
      "0.842422158",
      "0.377536701"
    ],
    [
      "0.851727828",
      "0.582541443"
    ],
    [
      "0.852099529",
      "0.791204294"
    ],
    [
      "0.852099529",
      "1.000000000"
    ],
    [
      "0.651300823",
      "0.087047955"
    ],
    [
      "0.674773083",
      "0.282523636"
    ],
    [
      "0.694600446",
      "0.482046042"
    ],
    [
      "0.703966693",
      "0.687136414"
    ],
    [
      "0.704199059",
      "0.895602147"
    ],
    [
      "0.477779897",
      "0.035434496"
    ],
    [
      "0.505335210",
      "0.218772024"
    ],
    [
      "0.530233007",
      "0.406172425"
    ],
    [
      "0.547287045",
      "0.596444910"
    ],
    [
      "0.555723330",
      "0.792024067"
    ],
    [
      "0.556298589",
      "1.000000000"
    ],
    [
      "0.295800940",
      "0.000000000"
    ],
    [
      "0.328958275",
      "0.177972054"
    ],
    [
      "0.356012065",
      "0.356973600"
    ],
    [
      "0.376338241",
      "0.536863309"
    ],
    [
      "0.390822229",
      "0.717317389"
    ],
    [
      "0.406988346",
      "0.897628558"
    ],
    [
      "0.147900470",
      "0.104397853"
    ],
    [
      "0.181057804",
      "0.282369907"
    ],
    [
      "0.192253856",
      "0.463057785"
    ],
    [
      "0.225885069",
      "0.640940900"
    ],
    [
      "0.224821423",
      "0.821972194"
    ],
    [
      "0.257678103",
      "1.000000000"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.208795706"
    ],
    [
      "0.014356715",
      "0.389259956"
    ],
    [
      "0.044565913",
      "0.567756077"
    ],
    [
      "0.059512451",
      "0.748172434"
    ],
    [
      "0.067933076",
      "0.929075607"
    ]
  ]
}

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": 40
}

当前第一名的答案

{
  "points": [
    [
      "0.982709254",
      "0.087962553"
    ],
    [
      "0.992882625",
      "0.276438390"
    ],
    [
      "0.999628298",
      "0.478143590"
    ],
    [
      "0.999999999",
      "0.686806441"
    ],
    [
      "0.999999999",
      "0.895602147"
    ],
    [
      "0.810033663",
      "0.000000000"
    ],
    [
      "0.824301607",
      "0.180471290"
    ],
    [
      "0.842422158",
      "0.377536701"
    ],
    [
      "0.851727828",
      "0.582541443"
    ],
    [
      "0.852099529",
      "0.791204294"
    ],
    [
      "0.852099529",
      "1.000000000"
    ],
    [
      "0.651300823",
      "0.087047955"
    ],
    [
      "0.674773083",
      "0.282523636"
    ],
    [
      "0.694600446",
      "0.482046042"
    ],
    [
      "0.703966693",
      "0.687136414"
    ],
    [
      "0.704199059",
      "0.895602147"
    ],
    [
      "0.477779897",
      "0.035434496"
    ],
    [
      "0.505335210",
      "0.218772024"
    ],
    [
      "0.530233007",
      "0.406172425"
    ],
    [
      "0.547287045",
      "0.596444910"
    ],
    [
      "0.555723330",
      "0.792024067"
    ],
    [
      "0.556298589",
      "1.000000000"
    ],
    [
      "0.295800940",
      "0.000000000"
    ],
    [
      "0.328958275",
      "0.177972054"
    ],
    [
      "0.356012065",
      "0.356973600"
    ],
    [
      "0.376338241",
      "0.536863309"
    ],
    [
      "0.390822229",
      "0.717317389"
    ],
    [
      "0.406988346",
      "0.897628558"
    ],
    [
      "0.147900470",
      "0.104397853"
    ],
    [
      "0.181057804",
      "0.282369907"
    ],
    [
      "0.192253856",
      "0.463057785"
    ],
    [
      "0.225885069",
      "0.640940900"
    ],
    [
      "0.224821423",
      "0.821972194"
    ],
    [
      "0.257678103",
      "1.000000000"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.208795706"
    ],
    [
      "0.014356715",
      "0.389259956"
    ],
    [
      "0.044565913",
      "0.567756077"
    ],
    [
      "0.059512451",
      "0.748172434"
    ],
    [
      "0.067933076",
      "0.929075607"
    ]
  ]
}

提交 points，每个坐标写成十进制字符串，例如 "0.25"，最多九位小数。分数是 M = 2h/δ，越小越好。 · 验证器 v1.0.0

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