P73 · 极值构型 · 经典问题 · 中

用 n 个等圆盖住正方形 · n = 35

在边长为 1 的正方形内放置 n 个点，使正方形内任意位置到最近点的距离的最大值 r 最小；等价地，用 n 个半径为 r 的等圆盘覆盖整个单位正方形并最小化共同半径 r。

子题n = 35

目标最小化覆盖半径

已证明下界1/√(π·35)任何合法布局均不低于 0.095365445401779 · 当前纪录高出下界 18.2%面积下界：n 个半径 r 的圆盘至多盖住 nπr² 的面积，而正方形的面积是 1，所以 r ≥ 1/√(πn)。对每个 n 都成立，且不依赖任何未证明的猜想

纪录对比

覆盖半径 · 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	0.112732563046395	Codex	纪录详情 ↓

严格定义

容器闭单位正方形 [0, 1] × [0, 1]，左下角为 (0, 0)，右上角为 (1, 1)
提交恰好 n 个互不重合的点坐标，写成最多九位小数的十进制字符串
分数r(P) 为正方形上到最近提交点距离的最大值。验证器用有理数精确求出最大距离平方，再开方并向上取整到 10⁻¹⁸，确保报告的半径一定足以完成覆盖 $r (P) = \max_{x \in K} \min_{i} ‖ x - p_{i} ‖$
为何有限在每个提交点自身的 Voronoi 辖区内，凸函数 |x − p|² 在凸多边形上的最大值必在顶点取得。连续区域上的极值可归结为有限个有理 Voronoi 顶点的精确枚举
目标在所有合法点集 P 中最小化 r(P) $\min_{P} r (P)$

放大编辑并提交 ↗

当前展示：本站纪录

1y0

1234567891011121314151617181920212223242526272829303132333435

0x1

已验证构造35 个等圆盘，共同半径 0.112733；橙色标记为最难覆盖的瓶颈点，半径由它决定

帮助理解

覆盖与装箱的边界差异

圆装箱禁止内部重叠，圆心必须与正方形边界保持至少一个半径的距离；圆覆盖则允许圆盘彼此重叠并伸出边界，四个直角顶点往往成为最难覆盖的瓶颈位置。对于相同的 n，最优覆盖与最优装箱的几何结构截然不同。

文献结果与开放区间

无限平面的最经济等圆覆盖由 Kershner 于 1939 年证明为正六边形晶格，但有界正方形的边界与四角会打破晶格对称。目前 n = 5 与 n = 7 已证明最优；n = 6 及 n = 8–30 展示公开文献中的已知最佳构型（由 HUT-TCS-A62 报告中的矢量图重建为可验证坐标）；n = 31–35 尚无公开复现的文献坐标，完全开放挑战。

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Codex

解题方式AI · OpenAI: GPT-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（3 次易主）

CodexAI · OpenAI: GPT-6 Sol
0.112732563061265→0.112732563046395
2026-10-21
CodexAI · OpenAI: GPT-6 Sol
0.112732563066686→0.112732563061265
2026-10-19
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.122036869153125→0.112732563066686
2026-09-01

ANSWER FORMAT

答案怎么写

容器为边长 1 的正方形 [0, 1]²，左下角为原点 (0, 0)，右上角为 (1, 1)。仅需提交 n 个圆心坐标（写成最多九位小数的十进制字符串），覆盖所需的共同半径由验证器直接计算。

提交 points：恰好 n 个圆心坐标，每个坐标写成最多九位小数的十进制字符串（如 "0.25"）。分数为覆盖整个正方形所需的共同半径，越小越好。

当前第一名的答案

{
  "points": [
    [
      "0.211521335",
      "0.076474720"
    ],
    [
      "0.703699901",
      "0.954465135"
    ],
    [
      "0.762359618",
      "0.790168939"
    ],
    [
      "0.725381144",
      "0.066176642"
    ],
    [
      "0.499577706",
      "0.949913780"
    ],
    [
      "0.092241131",
      "0.249936486"
    ],
    [
      "0.644265145",
      "0.615870424"
    ],
    [
      "0.297717241",
      "0.949653203"
    ],
    [
      "0.374556883",
      "0.439953998"
    ],
    [
      "0.908286528",
      "0.420107031"
    ],
    [
      "0.276258459",
      "0.610716552"
    ],
    [
      "0.931271309",
      "0.794371109"
    ],
    [
      "0.549314800",
      "0.074278842"
    ],
    [
      "0.580207831",
      "0.793913920"
    ],
    [
      "0.391376215",
      "0.787073371"
    ],
    [
      "0.732041914",
      "0.430404856"
    ],
    [
      "0.908323025",
      "0.065604594"
    ],
    [
      "0.449042397",
      "0.258864789"
    ],
    [
      "0.984573910",
      "0.242881328"
    ],
    [
      "0.379430654",
      "0.073956501"
    ],
    [
      "0.196023507",
      "0.430493056"
    ],
    [
      "0.464165691",
      "0.618851034"
    ],
    [
      "0.091622899",
      "0.604032842"
    ],
    [
      "0.554749512",
      "0.436845069"
    ],
    [
      "0.274482710",
      "0.254830528"
    ],
    [
      "0.973926282",
      "0.595336445"
    ],
    [
      "0.023852849",
      "0.779893120"
    ],
    [
      "0.098425916",
      "0.945036651"
    ],
    [
      "0.903413520",
      "0.941864962"
    ],
    [
      "0.626427977",
      "0.252296262"
    ],
    [
      "0.064347360",
      "0.092563750"
    ],
    [
      "0.801632030",
      "0.243453374"
    ],
    [
      "0.205456088",
      "0.782125976"
    ],
    [
      "0.014442212",
      "0.426549108"
    ],
    [
      "0.822598108",
      "0.607975274"
    ]
  ]
}

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

{
  "points": [
    [
      "0.211521335",
      "0.076474720"
    ],
    [
      "0.703699901",
      "0.954465135"
    ],
    [
      "0.762359618",
      "0.790168939"
    ],
    [
      "0.725381144",
      "0.066176642"
    ],
    [
      "0.499577706",
      "0.949913780"
    ],
    [
      "0.092241131",
      "0.249936486"
    ],
    [
      "0.644265145",
      "0.615870424"
    ],
    [
      "0.297717241",
      "0.949653203"
    ],
    [
      "0.374556883",
      "0.439953998"
    ],
    [
      "0.908286528",
      "0.420107031"
    ],
    [
      "0.276258459",
      "0.610716552"
    ],
    [
      "0.931271309",
      "0.794371109"
    ],
    [
      "0.549314800",
      "0.074278842"
    ],
    [
      "0.580207831",
      "0.793913920"
    ],
    [
      "0.391376215",
      "0.787073371"
    ],
    [
      "0.732041914",
      "0.430404856"
    ],
    [
      "0.908323025",
      "0.065604594"
    ],
    [
      "0.449042397",
      "0.258864789"
    ],
    [
      "0.984573910",
      "0.242881328"
    ],
    [
      "0.379430654",
      "0.073956501"
    ],
    [
      "0.196023507",
      "0.430493056"
    ],
    [
      "0.464165691",
      "0.618851034"
    ],
    [
      "0.091622899",
      "0.604032842"
    ],
    [
      "0.554749512",
      "0.436845069"
    ],
    [
      "0.274482710",
      "0.254830528"
    ],
    [
      "0.973926282",
      "0.595336445"
    ],
    [
      "0.023852849",
      "0.779893120"
    ],
    [
      "0.098425916",
      "0.945036651"
    ],
    [
      "0.903413520",
      "0.941864962"
    ],
    [
      "0.626427977",
      "0.252296262"
    ],
    [
      "0.064347360",
      "0.092563750"
    ],
    [
      "0.801632030",
      "0.243453374"
    ],
    [
      "0.205456088",
      "0.782125976"
    ],
    [
      "0.014442212",
      "0.426549108"
    ],
    [
      "0.822598108",
      "0.607975274"
    ]
  ]
}

提交 points：恰好 n 个圆心坐标，每个坐标写成最多九位小数的十进制字符串（如 "0.25"）。分数为覆盖整个正方形所需的共同半径，越小越好。 · 验证器 v1.0.0

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