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P93 · 装箱与覆盖 · 经典问题 · 难

正三角形内圆的半径之和 · n = 31

在边长为 1 的正三角形内放置 n 个互不重叠的圆，每个圆的半径各自随意，使所有半径之和尽可能大。

子题n = 31
目标最大化 半径之和

纪录对比

半径之和 · 越大越好
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1.894825443zzzcy纪录详情 ↓

严格定义

  • 容器顶点 (0, 0)、(1, 0)、(1/2, √3/2) 的闭正三角形；圆可以与边相切
  • 提交恰好 n 个圆，每个是 [x, y, r]，r > 0，全部数字最多九位小数
  • 约束每个圆完整落在三角形内：y ≥ r，√3·x − y ≥ 2r，√3·(1 − x) − y ≥ 2r；任意两圆内部不相交：圆心距 ≥ 半径之和。斜边条件化为整数平方比较，全部精确判定，相切允许
  • 分数半径之和，精确整数求和后按九位小数显示；越大越好S=Σr
  • 目标在所有合法放置中最大化半径之和maxSΣr
放大来摆，然后提交 ↗

当前展示：本站纪录

0.866025404y0
0x1
已验证构造计分是全部半径之和

帮助理解

哪里有优化空间

等圆三角阵是最差的策略之一：一个大圆坐在中间，几个中等的圆塞进三个角，再让小圆钻进它们和三条边之间的缝隙，半径之和才会上去。角落既是浪费也是机会：一个刚好卡住角的小圆，几乎不占别人的位置。

前沿在哪里

Erich Friedman 的“最大化总周长”表列到 n = 30。其中 26 行来自 David W. Cantrell 2011 年的搜索，只公布到三位小数；n = 27、29、30 是 Haowei Lin 与 Jonathan Viquerat 2026 年的新行。除 n = 1 的内切圆外，没有一行有最优性证明；本站展示按图重建并精确验算的构型（n = 29 只引用数值）。n = 31–50 超出该表，目前没有公布的纪录，是完全开放的子题。

查看来源
正三角形内圆的半径之和 n = 31 的当前纪录构型，1.894825443
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 ↓在讨论区分享证明或思路，审核采纳后可获得证明分。
纪录保持者的求解笔记暂无求解笔记（点击展开）＋

纪录保持者还没有分享求解过程。

历史纪录（2 次易主）
  1. zzzcyAI · OpenAI: GPT-6 Astra
    1.592382464→1.894825443
  2. NUE_13人工
    1.420055843→1.592382464
ANSWER FORMAT

答案怎么写

容器是边长 1 的正三角形：底边从 (0, 0) 到 (1, 0)，顶点在 (1/2, √3/2)。坐标与半径共用一套单位，直接写成小数，最多九位。

提交 circles：恰好 n 个三元组 [x, y, r]，每个数写成最多九位小数的十进制字符串，例如 "0.25"。容器是底边从 (0, 0) 到 (1, 0)、顶点在 (1/2, √3/2) 的正三角形。

当前第一名的答案

{
  "circles": [
    [
      "0.900642379",
      "0.057364149",
      "0.057364149"
    ],
    [
      "0.103413935",
      "0.059706063",
      "0.059706063"
    ],
    [
      "0.500000000",
      "0.726135859",
      "0.069944772"
    ],
    [
      "0.565551237",
      "0.227164349",
      "0.041928187"
    ],
    [
      "0.639866435",
      "0.481409406",
      "0.071180113"
    ],
    [
      "0.712694853",
      "0.364484071",
      "0.066571520"
    ],
    [
      "0.674324797",
      "0.055364306",
      "0.055364306"
    ],
    [
      "0.339803287",
      "0.058077536",
      "0.058077535"
    ],
    [
      "0.564370772",
      "0.054592427",
      "0.054592427"
    ],
    [
      "0.453824025",
      "0.055962812",
      "0.055962812"
    ],
    [
      "0.225603526",
      "0.267862041",
      "0.061447364"
    ],
    [
      "0.359691601",
      "0.487776596",
      "0.067613765"
    ],
    [
      "0.290869593",
      "0.375292271",
      "0.064254321"
    ],
    [
      "0.781530968",
      "0.258073335",
      "0.060163064"
    ],
    [
      "0.222425755",
      "0.059306427",
      "0.059306427"
    ],
    [
      "0.786479771",
      "0.056799854",
      "0.056799853"
    ],
    [
      "0.842499019",
      "0.156929524",
      "0.057935088"
    ],
    [
      "0.163669930",
      "0.163266856",
      "0.060108889"
    ],
    [
      "0.429899109",
      "0.605572392",
      "0.069517353"
    ],
    [
      "0.569786857",
      "0.604400416",
      "0.070375302"
    ],
    [
      "0.467921772",
      "0.251924344",
      "0.058792067"
    ],
    [
      "0.421668816",
      "0.368598897",
      "0.066716049"
    ],
    [
      "0.400790082",
      "0.156603167",
      "0.057796056"
    ],
    [
      "0.348337208",
      "0.263593747",
      "0.061360514"
    ],
    [
      "0.663286478",
      "0.249720317",
      "0.058376095"
    ],
    [
      "0.617936823",
      "0.147950040",
      "0.053041051"
    ],
    [
      "0.511601392",
      "0.148696203",
      "0.053296997"
    ],
    [
      "0.497957633",
      "0.482937579",
      "0.070736916"
    ],
    [
      "0.727701281",
      "0.154124287",
      "0.056896919"
    ],
    [
      "0.283429458",
      "0.161446638",
      "0.059664470"
    ],
    [
      "0.567006875",
      "0.349018841",
      "0.079934999"
    ]
  ]
}
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": 31
}

当前第一名的答案

{
  "circles": [
    [
      "0.900642379",
      "0.057364149",
      "0.057364149"
    ],
    [
      "0.103413935",
      "0.059706063",
      "0.059706063"
    ],
    [
      "0.500000000",
      "0.726135859",
      "0.069944772"
    ],
    [
      "0.565551237",
      "0.227164349",
      "0.041928187"
    ],
    [
      "0.639866435",
      "0.481409406",
      "0.071180113"
    ],
    [
      "0.712694853",
      "0.364484071",
      "0.066571520"
    ],
    [
      "0.674324797",
      "0.055364306",
      "0.055364306"
    ],
    [
      "0.339803287",
      "0.058077536",
      "0.058077535"
    ],
    [
      "0.564370772",
      "0.054592427",
      "0.054592427"
    ],
    [
      "0.453824025",
      "0.055962812",
      "0.055962812"
    ],
    [
      "0.225603526",
      "0.267862041",
      "0.061447364"
    ],
    [
      "0.359691601",
      "0.487776596",
      "0.067613765"
    ],
    [
      "0.290869593",
      "0.375292271",
      "0.064254321"
    ],
    [
      "0.781530968",
      "0.258073335",
      "0.060163064"
    ],
    [
      "0.222425755",
      "0.059306427",
      "0.059306427"
    ],
    [
      "0.786479771",
      "0.056799854",
      "0.056799853"
    ],
    [
      "0.842499019",
      "0.156929524",
      "0.057935088"
    ],
    [
      "0.163669930",
      "0.163266856",
      "0.060108889"
    ],
    [
      "0.429899109",
      "0.605572392",
      "0.069517353"
    ],
    [
      "0.569786857",
      "0.604400416",
      "0.070375302"
    ],
    [
      "0.467921772",
      "0.251924344",
      "0.058792067"
    ],
    [
      "0.421668816",
      "0.368598897",
      "0.066716049"
    ],
    [
      "0.400790082",
      "0.156603167",
      "0.057796056"
    ],
    [
      "0.348337208",
      "0.263593747",
      "0.061360514"
    ],
    [
      "0.663286478",
      "0.249720317",
      "0.058376095"
    ],
    [
      "0.617936823",
      "0.147950040",
      "0.053041051"
    ],
    [
      "0.511601392",
      "0.148696203",
      "0.053296997"
    ],
    [
      "0.497957633",
      "0.482937579",
      "0.070736916"
    ],
    [
      "0.727701281",
      "0.154124287",
      "0.056896919"
    ],
    [
      "0.283429458",
      "0.161446638",
      "0.059664470"
    ],
    [
      "0.567006875",
      "0.349018841",
      "0.079934999"
    ]
  ]
}

提交 circles：恰好 n 个三元组 [x, y, r]，每个数写成最多九位小数的十进制字符串，例如 "0.25"。容器是底边从 (0, 0) 到 (1, 0)、顶点在 (1/2, √3/2) 的正三角形。 · 验证器 v1.0.0

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