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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 = 40

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 = 40
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2.16253093zzzcy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 = 40 的当前纪录构型，2.16253093
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
    1.701344595→2.16253093
  2. NUE_13人工
    1.64405224→1.701344595
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": [
    [
      "0.091437280",
      "0.052791338",
      "0.052791338"
    ],
    [
      "0.196845309",
      "0.052616834",
      "0.052616834"
    ],
    [
      "0.301554603",
      "0.052093767",
      "0.052093766"
    ],
    [
      "0.404858252",
      "0.051213632",
      "0.051213631"
    ],
    [
      "0.506442065",
      "0.050373653",
      "0.050373653"
    ],
    [
      "0.606969962",
      "0.050154483",
      "0.050154482"
    ],
    [
      "0.707672179",
      "0.050548505",
      "0.050548504"
    ],
    [
      "0.809213182",
      "0.050993473",
      "0.050993473"
    ],
    [
      "0.911374476",
      "0.051167970",
      "0.051167970"
    ],
    [
      "0.144467859",
      "0.144292617",
      "0.052966527"
    ],
    [
      "0.250180237",
      "0.143486677",
      "0.052748923"
    ],
    [
      "0.354854066",
      "0.141442543",
      "0.051944863"
    ],
    [
      "0.457211557",
      "0.138370934",
      "0.050458704"
    ],
    [
      "0.557111213",
      "0.136393987",
      "0.049460511"
    ],
    [
      "0.656588618",
      "0.137176958",
      "0.050019975"
    ],
    [
      "0.757610037",
      "0.138990049",
      "0.051017711"
    ],
    [
      "0.859967247",
      "0.139857494",
      "0.051343174"
    ],
    [
      "0.198247253",
      "0.236249921",
      "0.053562196"
    ],
    [
      "0.305120975",
      "0.234230174",
      "0.053330607"
    ],
    [
      "0.410211238",
      "0.229274463",
      "0.051876438"
    ],
    [
      "0.509807334",
      "0.221620143",
      "0.048013355"
    ],
    [
      "0.605640650",
      "0.220710204",
      "0.047824280"
    ],
    [
      "0.704510643",
      "0.226340615",
      "0.051205901"
    ],
    [
      "0.807701097",
      "0.229032170",
      "0.052019649"
    ],
    [
      "0.253700798",
      "0.329522656",
      "0.054950008"
    ],
    [
      "0.363572830",
      "0.325435816",
      "0.054998005"
    ],
    [
      "0.470848968",
      "0.314625040",
      "0.052821484"
    ],
    [
      "0.558566973",
      "0.292149804",
      "0.037730073"
    ],
    [
      "0.646527673",
      "0.312345932",
      "0.052519405"
    ],
    [
      "0.752867823",
      "0.319872014",
      "0.054086736"
    ],
    [
      "0.312238775",
      "0.425637620",
      "0.057587901"
    ],
    [
      "0.429541523",
      "0.419479705",
      "0.059876366"
    ],
    [
      "0.560046878",
      "0.401703905",
      "0.071834023"
    ],
    [
      "0.625577926",
      "0.520624500",
      "0.063946777"
    ],
    [
      "0.690993520",
      "0.415533218",
      "0.059840852"
    ],
    [
      "0.374015970",
      "0.526475914",
      "0.060669374"
    ],
    [
      "0.437055110",
      "0.632184333",
      "0.062408661"
    ],
    [
      "0.498122973",
      "0.522058060",
      "0.063516235"
    ],
    [
      "0.562656287",
      "0.631107266",
      "0.063197132"
    ],
    [
      "0.500000000",
      "0.740422537",
      "0.062801433"
    ]
  ]
}
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": 40
}

当前第一名的答案

{
  "circles": [
    [
      "0.091437280",
      "0.052791338",
      "0.052791338"
    ],
    [
      "0.196845309",
      "0.052616834",
      "0.052616834"
    ],
    [
      "0.301554603",
      "0.052093767",
      "0.052093766"
    ],
    [
      "0.404858252",
      "0.051213632",
      "0.051213631"
    ],
    [
      "0.506442065",
      "0.050373653",
      "0.050373653"
    ],
    [
      "0.606969962",
      "0.050154483",
      "0.050154482"
    ],
    [
      "0.707672179",
      "0.050548505",
      "0.050548504"
    ],
    [
      "0.809213182",
      "0.050993473",
      "0.050993473"
    ],
    [
      "0.911374476",
      "0.051167970",
      "0.051167970"
    ],
    [
      "0.144467859",
      "0.144292617",
      "0.052966527"
    ],
    [
      "0.250180237",
      "0.143486677",
      "0.052748923"
    ],
    [
      "0.354854066",
      "0.141442543",
      "0.051944863"
    ],
    [
      "0.457211557",
      "0.138370934",
      "0.050458704"
    ],
    [
      "0.557111213",
      "0.136393987",
      "0.049460511"
    ],
    [
      "0.656588618",
      "0.137176958",
      "0.050019975"
    ],
    [
      "0.757610037",
      "0.138990049",
      "0.051017711"
    ],
    [
      "0.859967247",
      "0.139857494",
      "0.051343174"
    ],
    [
      "0.198247253",
      "0.236249921",
      "0.053562196"
    ],
    [
      "0.305120975",
      "0.234230174",
      "0.053330607"
    ],
    [
      "0.410211238",
      "0.229274463",
      "0.051876438"
    ],
    [
      "0.509807334",
      "0.221620143",
      "0.048013355"
    ],
    [
      "0.605640650",
      "0.220710204",
      "0.047824280"
    ],
    [
      "0.704510643",
      "0.226340615",
      "0.051205901"
    ],
    [
      "0.807701097",
      "0.229032170",
      "0.052019649"
    ],
    [
      "0.253700798",
      "0.329522656",
      "0.054950008"
    ],
    [
      "0.363572830",
      "0.325435816",
      "0.054998005"
    ],
    [
      "0.470848968",
      "0.314625040",
      "0.052821484"
    ],
    [
      "0.558566973",
      "0.292149804",
      "0.037730073"
    ],
    [
      "0.646527673",
      "0.312345932",
      "0.052519405"
    ],
    [
      "0.752867823",
      "0.319872014",
      "0.054086736"
    ],
    [
      "0.312238775",
      "0.425637620",
      "0.057587901"
    ],
    [
      "0.429541523",
      "0.419479705",
      "0.059876366"
    ],
    [
      "0.560046878",
      "0.401703905",
      "0.071834023"
    ],
    [
      "0.625577926",
      "0.520624500",
      "0.063946777"
    ],
    [
      "0.690993520",
      "0.415533218",
      "0.059840852"
    ],
    [
      "0.374015970",
      "0.526475914",
      "0.060669374"
    ],
    [
      "0.437055110",
      "0.632184333",
      "0.062408661"
    ],
    [
      "0.498122973",
      "0.522058060",
      "0.063516235"
    ],
    [
      "0.562656287",
      "0.631107266",
      "0.063197132"
    ],
    [
      "0.500000000",
      "0.740422537",
      "0.062801433"
    ]
  ]
}

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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