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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 = 26

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 = 26
目标最大化 共同半径
已知最好（未证明）0.171580252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 = 0.171580251

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

0.171580251

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": [
    [
      "0.961757873",
      "0.172463404"
    ],
    [
      "1.300400405",
      "0.22796446"
    ],
    [
      "0.629677334",
      "0.258959919"
    ],
    [
      "1.587497026",
      "0.415939624"
    ],
    [
      "0.361140715",
      "0.47261201"
    ],
    [
      "0.948934049",
      "0.515384213"
    ],
    [
      "1.285992274",
      "0.579811574"
    ],
    [
      "0.642274296",
      "0.669393719"
    ],
    [
      "1.773784629",
      "0.704134108"
    ],
    [
      "0.202226398",
      "0.776758967"
    ],
    [
      "1.164371857",
      "0.900697233"
    ],
    [
      "1.506991337",
      "0.919959202"
    ],
    [
      "0.82621613",
      "0.959091033"
    ],
    [
      "0.48335998",
      "0.973540676"
    ],
    [
      "1.827297998",
      "1.043096423"
    ],
    [
      "0.183593989",
      "1.140572058"
    ],
    [
      "1.045864507",
      "1.222745583"
    ],
    [
      "1.412851223",
      "1.255815259"
    ],
    [
      "0.610371036",
      "1.292331083"
    ],
    [
      "1.738854736",
      "1.374663793"
    ],
    [
      "0.310605044",
      "1.459362465"
    ],
    [
      "1.199358394",
      "1.529663746"
    ],
    [
      "0.856204279",
      "1.531757791"
    ],
    [
      "1.523630883",
      "1.641942347"
    ],
    [
      "0.556685433",
      "1.699779357"
    ],
    [
      "1.029594832",
      "1.82789095"
    ]
  ],
  "radius": "0.171580251"
}
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

{
  "centers": [
    [
      "0.961757873",
      "0.172463404"
    ],
    [
      "1.300400405",
      "0.22796446"
    ],
    [
      "0.629677334",
      "0.258959919"
    ],
    [
      "1.587497026",
      "0.415939624"
    ],
    [
      "0.361140715",
      "0.47261201"
    ],
    [
      "0.948934049",
      "0.515384213"
    ],
    [
      "1.285992274",
      "0.579811574"
    ],
    [
      "0.642274296",
      "0.669393719"
    ],
    [
      "1.773784629",
      "0.704134108"
    ],
    [
      "0.202226398",
      "0.776758967"
    ],
    [
      "1.164371857",
      "0.900697233"
    ],
    [
      "1.506991337",
      "0.919959202"
    ],
    [
      "0.82621613",
      "0.959091033"
    ],
    [
      "0.48335998",
      "0.973540676"
    ],
    [
      "1.827297998",
      "1.043096423"
    ],
    [
      "0.183593989",
      "1.140572058"
    ],
    [
      "1.045864507",
      "1.222745583"
    ],
    [
      "1.412851223",
      "1.255815259"
    ],
    [
      "0.610371036",
      "1.292331083"
    ],
    [
      "1.738854736",
      "1.374663793"
    ],
    [
      "0.310605044",
      "1.459362465"
    ],
    [
      "1.199358394",
      "1.529663746"
    ],
    [
      "0.856204279",
      "1.531757791"
    ],
    [
      "1.523630883",
      "1.641942347"
    ],
    [
      "0.556685433",
      "1.699779357"
    ],
    [
      "1.029594832",
      "1.82789095"
    ]
  ],
  "radius": "0.171580251"
}

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