极值竞技场
中/EN
P02 · 装箱与覆盖 · 经典问题 · 基线易突破

单位圆内的等圆装箱 · n = 28

在半径 1 的圆内放置 n 个半径相同、互不重叠的小圆，使共同半径尽可能大。

子题n = 28
目标最大化 共同半径
已知最好（未证明）0.166252750值取自 Eckard Specht 的汇总表，该表未把这一项标为已证明最优

严格定义

  • 容器半径 1 的圆，圆心在 (1, 1)，两个坐标都在 0 到 2 之间
  • 提交恰好 n 个圆：一个共同半径 radius 与 n 个圆心 centers
  • 约束每个圆完整落在容器内；两两内部不重叠，相切允许
  • 目标让共同半径尽可能大
放大来摆，然后提交 ↗
2y0
0x2
已验证构造r = 0.166252749

帮助理解

哪里有优化空间

最优构形是「卡死」的接触结构：圆彼此顶住、顶住边界，常出现斜排、错位、以及不碰任何邻居的游离圆。规整的网格摆法几乎从不最优。

前沿在哪里

n = 2..14 与 19 已证明（Specht 汇总表 cci 的粗体标记）；15..18 与 20 以上全部开放，包括本站开到的 29。

查看来源
当前第一名

0.166252749

共同半径

纪录保持者创始基准
解题方式人工
挑战这个纪录 ↗
ANSWER FORMAT

答案怎么写

容器是半径 1 的圆，圆心在 (1, 1)，所以两个坐标都在 0 到 2 之间。坐标和半径用同一个单位，直接写成小数，例如 "0.5"，最多九位小数。

提交 radius 与 centers。每个数写成十进制字符串，例如 "0.2"。

当前第一名的答案

{
  "centers": [
    [
      "1.107395649",
      "0.173719509"
    ],
    [
      "0.552158256",
      "0.296741691"
    ],
    [
      "1.447841744",
      "0.296741691"
    ],
    [
      "0.865305921",
      "0.408538144"
    ],
    [
      "1.184906622",
      "0.500273087"
    ],
    [
      "0.312939573",
      "0.527683955"
    ],
    [
      "1.687060427",
      "0.527683955"
    ],
    [
      "0.625916022",
      "0.639958841"
    ],
    [
      "0.945661481",
      "0.731187942"
    ],
    [
      "1.424125305",
      "0.731215351"
    ],
    [
      "0.182996634",
      "0.83374725"
    ],
    [
      "1.817003366",
      "0.83374725"
    ],
    [
      "0.495973083",
      "0.946022136"
    ],
    [
      "1.184281458",
      "0.962748764"
    ],
    [
      "0.815718542",
      "1.037251236"
    ],
    [
      "1.504026917",
      "1.053977864"
    ],
    [
      "0.182996634",
      "1.16625275"
    ],
    [
      "1.817003366",
      "1.16625275"
    ],
    [
      "0.575874695",
      "1.268784649"
    ],
    [
      "1.054338519",
      "1.268812058"
    ],
    [
      "1.374083978",
      "1.360041159"
    ],
    [
      "0.312939573",
      "1.472316045"
    ],
    [
      "1.687060427",
      "1.472316045"
    ],
    [
      "0.815093378",
      "1.499726913"
    ],
    [
      "1.134694079",
      "1.591461856"
    ],
    [
      "0.552158256",
      "1.703258309"
    ],
    [
      "1.447841744",
      "1.703258309"
    ],
    [
      "0.892867304",
      "1.826632832"
    ]
  ],
  "radius": "0.166252749"
}
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": 28
}

当前第一名的答案

{
  "centers": [
    [
      "1.107395649",
      "0.173719509"
    ],
    [
      "0.552158256",
      "0.296741691"
    ],
    [
      "1.447841744",
      "0.296741691"
    ],
    [
      "0.865305921",
      "0.408538144"
    ],
    [
      "1.184906622",
      "0.500273087"
    ],
    [
      "0.312939573",
      "0.527683955"
    ],
    [
      "1.687060427",
      "0.527683955"
    ],
    [
      "0.625916022",
      "0.639958841"
    ],
    [
      "0.945661481",
      "0.731187942"
    ],
    [
      "1.424125305",
      "0.731215351"
    ],
    [
      "0.182996634",
      "0.83374725"
    ],
    [
      "1.817003366",
      "0.83374725"
    ],
    [
      "0.495973083",
      "0.946022136"
    ],
    [
      "1.184281458",
      "0.962748764"
    ],
    [
      "0.815718542",
      "1.037251236"
    ],
    [
      "1.504026917",
      "1.053977864"
    ],
    [
      "0.182996634",
      "1.16625275"
    ],
    [
      "1.817003366",
      "1.16625275"
    ],
    [
      "0.575874695",
      "1.268784649"
    ],
    [
      "1.054338519",
      "1.268812058"
    ],
    [
      "1.374083978",
      "1.360041159"
    ],
    [
      "0.312939573",
      "1.472316045"
    ],
    [
      "1.687060427",
      "1.472316045"
    ],
    [
      "0.815093378",
      "1.499726913"
    ],
    [
      "1.134694079",
      "1.591461856"
    ],
    [
      "0.552158256",
      "1.703258309"
    ],
    [
      "1.447841744",
      "1.703258309"
    ],
    [
      "0.892867304",
      "1.826632832"
    ]
  ],
  "radius": "0.166252749"
}

提交 radius 与 centers。每个数写成十进制字符串，例如 "0.2"。 · 验证器 v1.0.0