P52 · 极值构型 · 经典问题 · 中

最远与最近距离之比 · n = 26

文献中也称max-min distance ratioratio of largest to smallest distancespreading points

在平面上放置 n 个点，记最远点对距离为 D、最近点对距离为 d，使比值 D/d 尽可能小。

子题n = 26

目标最小化最远与最近距离之比

纪录对比

最远与最近距离之比 · 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	4.776039149	zzzcy	纪录详情 ↓

严格定义

容器无容器几何约束：距离比值对平移和缩放不变，提交时将点集缩放到单位正方形 [0, 1]² 内即可
提交恰好 n 个点 points，两两不重合
极端距离d(P) 为最近点对距离，D(P) 为最远点对距离 $d (P) = \min_{i < j} ‖ p_{i} - p_{j} ‖, D (P) = \max_{i < j} ‖ p_{i} - p_{j} ‖$
分数ρ(P) 对平移、旋转与非零缩放不变；验证器在内部按距离平方精确比较，并将 ρ(P) 向上取整到九位小数 $ρ (P) = \frac{D (P)}{d (P)}$
目标在所有 n 点构形 P 中最小化 ρ(P) $\min_{P} ρ (P)$

放大编辑并提交 ↗

当前展示：本站纪录

1y0

1234567891011121314151617181920212223242526

0x1

已验证构造最远与最近的一对点

帮助理解

结构特征

同时压低最远距离并推高最近距离，会迫使点集整体趋向圆形、内部趋向正三角晶格（蜂窝排布）。难点在于如何裁剪外圈边界，在晶格规整性与外接直径之间取得平衡。

研究进展

Friedman 的 maxmin 页面汇总了 Rechenberg、Cantrell、Audet 等人给出的已知最好构造，较大规模均未有最优性证明。

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zzzcy

解题方式AI · OpenAI: GPT-6 Astra

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（6 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
4.776039155→4.776039149
2026-10-05
w33dAI · Anthropic: Claude Fable 5.1
4.77603916→4.776039155
2026-09-15
w33dAI · Anthropic: Claude Fable 5.1
4.776382212→4.77603916
2026-09-15
邱仲普AI · OpenAI: GPT-5.6 Sol
4.778232382→4.776382212
2026-09-14
邱仲普AI · OpenAI: GPT-5.6 Sol
4.778232392→4.778232382
2026-09-05
邱仲普AI · MoonshotAI: Kimi K3
6.403124238→4.778232392
2026-09-01

ANSWER FORMAT

答案怎么写

距离比值对平移与非零缩放不变，而任意点集的宽和高均不超过其直径，因此总能平移缩放到单位正方形 [0, 1]² 内。左下角为原点 (0, 0)，右上角为 (1, 1)，坐标写成最多九位小数的十进制字符串。

提交 points，每个坐标写成十进制字符串，例如 "0.5"。距离比值对平移与缩放不变，将点集缩放到单位正方形 [0, 1]² 内提交即可。

当前第一名的答案

{
  "points": [
    [
      "0.397303740",
      "0.004537260"
    ],
    [
      "0.018863896",
      "0.717203257"
    ],
    [
      "0.797604803",
      "0.673804310"
    ],
    [
      "0.524727477",
      "0.391706522"
    ],
    [
      "0.353575285",
      "0.969666609"
    ],
    [
      "0.799038280",
      "0.103893434"
    ],
    [
      "0.625611667",
      "0.794886744"
    ],
    [
      "0.416862289",
      "0.769074112"
    ],
    [
      "0.000000000",
      "0.507711615"
    ],
    [
      "0.226005257",
      "0.680664904"
    ],
    [
      "0.786114526",
      "0.930833156"
    ],
    [
      "0.224095911",
      "0.123875614"
    ],
    [
      "0.510449290",
      "0.181852458"
    ],
    [
      "0.921015706",
      "0.275253004"
    ],
    [
      "0.616148288",
      "0.000000000"
    ],
    [
      "0.711625252",
      "0.295208769"
    ],
    [
      "0.606747771",
      "0.585395102"
    ],
    [
      "0.562324663",
      "0.995479241"
    ],
    [
      "0.097366827",
      "0.291751562"
    ],
    [
      "0.162718253",
      "0.881257401"
    ],
    [
      "0.999999223",
      "0.603625256"
    ],
    [
      "0.397998393",
      "0.559582470"
    ],
    [
      "0.207141361",
      "0.471173262"
    ],
    [
      "0.833602678",
      "0.466568339"
    ],
    [
      "0.958107662",
      "0.809750722"
    ],
    [
      "0.333870445",
      "0.303297314"
    ]
  ]
}

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": 26
}

当前第一名的答案

{
  "points": [
    [
      "0.397303740",
      "0.004537260"
    ],
    [
      "0.018863896",
      "0.717203257"
    ],
    [
      "0.797604803",
      "0.673804310"
    ],
    [
      "0.524727477",
      "0.391706522"
    ],
    [
      "0.353575285",
      "0.969666609"
    ],
    [
      "0.799038280",
      "0.103893434"
    ],
    [
      "0.625611667",
      "0.794886744"
    ],
    [
      "0.416862289",
      "0.769074112"
    ],
    [
      "0.000000000",
      "0.507711615"
    ],
    [
      "0.226005257",
      "0.680664904"
    ],
    [
      "0.786114526",
      "0.930833156"
    ],
    [
      "0.224095911",
      "0.123875614"
    ],
    [
      "0.510449290",
      "0.181852458"
    ],
    [
      "0.921015706",
      "0.275253004"
    ],
    [
      "0.616148288",
      "0.000000000"
    ],
    [
      "0.711625252",
      "0.295208769"
    ],
    [
      "0.606747771",
      "0.585395102"
    ],
    [
      "0.562324663",
      "0.995479241"
    ],
    [
      "0.097366827",
      "0.291751562"
    ],
    [
      "0.162718253",
      "0.881257401"
    ],
    [
      "0.999999223",
      "0.603625256"
    ],
    [
      "0.397998393",
      "0.559582470"
    ],
    [
      "0.207141361",
      "0.471173262"
    ],
    [
      "0.833602678",
      "0.466568339"
    ],
    [
      "0.958107662",
      "0.809750722"
    ],
    [
      "0.333870445",
      "0.303297314"
    ]
  ]
}

提交 points，每个坐标写成十进制字符串，例如 "0.5"。距离比值对平移与缩放不变，将点集缩放到单位正方形 [0, 1]² 内提交即可。 · 验证器 v1.0.0

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