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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 = 44

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 = 44
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2.275662242zzzcy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 = 44 的当前纪录构型，2.275662242
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
    1.948745714→2.275662242
  2. NUE_13人工
    1.808457464→1.948745714
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": [
    [
      "0.092768706",
      "0.053560037",
      "0.053560037"
    ],
    [
      "0.200169042",
      "0.053840665",
      "0.053840664"
    ],
    [
      "0.308906875",
      "0.054902351",
      "0.054902351"
    ],
    [
      "0.422026999",
      "0.058267826",
      "0.058267825"
    ],
    [
      "0.551466423",
      "0.071886003",
      "0.071886001"
    ],
    [
      "0.681197247",
      "0.058530472",
      "0.058530471"
    ],
    [
      "0.795118052",
      "0.055432450",
      "0.055432450"
    ],
    [
      "0.905251285",
      "0.054703196",
      "0.054703196"
    ],
    [
      "0.145947579",
      "0.146226344",
      "0.053281139"
    ],
    [
      "0.252578310",
      "0.147351588",
      "0.053355529"
    ],
    [
      "0.359387081",
      "0.150845389",
      "0.053510368"
    ],
    [
      "0.464448540",
      "0.159993309",
      "0.051948602"
    ],
    [
      "0.638207434",
      "0.160427632",
      "0.052064081"
    ],
    [
      "0.743751304",
      "0.151863817",
      "0.053826651"
    ],
    [
      "0.851529986",
      "0.149187077",
      "0.053985264"
    ],
    [
      "0.198224022",
      "0.238240910",
      "0.052546583"
    ],
    [
      "0.302854592",
      "0.240050493",
      "0.052099634"
    ],
    [
      "0.405713363",
      "0.244359154",
      "0.050849338"
    ],
    [
      "0.503659643",
      "0.251206943",
      "0.047336027"
    ],
    [
      "0.598357501",
      "0.251518778",
      "0.047362343"
    ],
    [
      "0.696484734",
      "0.245380751",
      "0.050956673"
    ],
    [
      "0.799758303",
      "0.242090184",
      "0.052369304"
    ],
    [
      "0.249430936",
      "0.328891929",
      "0.051567562"
    ],
    [
      "0.351825012",
      "0.330596677",
      "0.050840703"
    ],
    [
      "0.452241354",
      "0.333399473",
      "0.049614745"
    ],
    [
      "0.550706030",
      "0.335119189",
      "0.048864946"
    ],
    [
      "0.649161417",
      "0.334101012",
      "0.049595705"
    ],
    [
      "0.749539062",
      "0.332211631",
      "0.050799719"
    ],
    [
      "0.299883893",
      "0.417908535",
      "0.050752802"
    ],
    [
      "0.400832089",
      "0.418957875",
      "0.050200847"
    ],
    [
      "0.500728314",
      "0.420075637",
      "0.049701631"
    ],
    [
      "0.600086672",
      "0.420372692",
      "0.049657170"
    ],
    [
      "0.699781018",
      "0.419918105",
      "0.050038212"
    ],
    [
      "0.350002403",
      "0.505644229",
      "0.050288857"
    ],
    [
      "0.450310838",
      "0.506114100",
      "0.050020677"
    ],
    [
      "0.550194144",
      "0.506483446",
      "0.049863311"
    ],
    [
      "0.649946548",
      "0.506532157",
      "0.049889103"
    ],
    [
      "0.400018059",
      "0.592673736",
      "0.050088933"
    ],
    [
      "0.500089884",
      "0.592848123",
      "0.049983042"
    ],
    [
      "0.600002324",
      "0.592957383",
      "0.049929457"
    ],
    [
      "0.450008856",
      "0.679403999",
      "0.050017101"
    ],
    [
      "0.550005051",
      "0.679455902",
      "0.049979106"
    ],
    [
      "0.500000000",
      "0.766029204",
      "0.049998099"
    ],
    [
      "0.551216313",
      "0.181127703",
      "0.037355983"
    ]
  ]
}
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": 44
}

当前第一名的答案

{
  "circles": [
    [
      "0.092768706",
      "0.053560037",
      "0.053560037"
    ],
    [
      "0.200169042",
      "0.053840665",
      "0.053840664"
    ],
    [
      "0.308906875",
      "0.054902351",
      "0.054902351"
    ],
    [
      "0.422026999",
      "0.058267826",
      "0.058267825"
    ],
    [
      "0.551466423",
      "0.071886003",
      "0.071886001"
    ],
    [
      "0.681197247",
      "0.058530472",
      "0.058530471"
    ],
    [
      "0.795118052",
      "0.055432450",
      "0.055432450"
    ],
    [
      "0.905251285",
      "0.054703196",
      "0.054703196"
    ],
    [
      "0.145947579",
      "0.146226344",
      "0.053281139"
    ],
    [
      "0.252578310",
      "0.147351588",
      "0.053355529"
    ],
    [
      "0.359387081",
      "0.150845389",
      "0.053510368"
    ],
    [
      "0.464448540",
      "0.159993309",
      "0.051948602"
    ],
    [
      "0.638207434",
      "0.160427632",
      "0.052064081"
    ],
    [
      "0.743751304",
      "0.151863817",
      "0.053826651"
    ],
    [
      "0.851529986",
      "0.149187077",
      "0.053985264"
    ],
    [
      "0.198224022",
      "0.238240910",
      "0.052546583"
    ],
    [
      "0.302854592",
      "0.240050493",
      "0.052099634"
    ],
    [
      "0.405713363",
      "0.244359154",
      "0.050849338"
    ],
    [
      "0.503659643",
      "0.251206943",
      "0.047336027"
    ],
    [
      "0.598357501",
      "0.251518778",
      "0.047362343"
    ],
    [
      "0.696484734",
      "0.245380751",
      "0.050956673"
    ],
    [
      "0.799758303",
      "0.242090184",
      "0.052369304"
    ],
    [
      "0.249430936",
      "0.328891929",
      "0.051567562"
    ],
    [
      "0.351825012",
      "0.330596677",
      "0.050840703"
    ],
    [
      "0.452241354",
      "0.333399473",
      "0.049614745"
    ],
    [
      "0.550706030",
      "0.335119189",
      "0.048864946"
    ],
    [
      "0.649161417",
      "0.334101012",
      "0.049595705"
    ],
    [
      "0.749539062",
      "0.332211631",
      "0.050799719"
    ],
    [
      "0.299883893",
      "0.417908535",
      "0.050752802"
    ],
    [
      "0.400832089",
      "0.418957875",
      "0.050200847"
    ],
    [
      "0.500728314",
      "0.420075637",
      "0.049701631"
    ],
    [
      "0.600086672",
      "0.420372692",
      "0.049657170"
    ],
    [
      "0.699781018",
      "0.419918105",
      "0.050038212"
    ],
    [
      "0.350002403",
      "0.505644229",
      "0.050288857"
    ],
    [
      "0.450310838",
      "0.506114100",
      "0.050020677"
    ],
    [
      "0.550194144",
      "0.506483446",
      "0.049863311"
    ],
    [
      "0.649946548",
      "0.506532157",
      "0.049889103"
    ],
    [
      "0.400018059",
      "0.592673736",
      "0.050088933"
    ],
    [
      "0.500089884",
      "0.592848123",
      "0.049983042"
    ],
    [
      "0.600002324",
      "0.592957383",
      "0.049929457"
    ],
    [
      "0.450008856",
      "0.679403999",
      "0.050017101"
    ],
    [
      "0.550005051",
      "0.679455902",
      "0.049979106"
    ],
    [
      "0.500000000",
      "0.766029204",
      "0.049998099"
    ],
    [
      "0.551216313",
      "0.181127703",
      "0.037355983"
    ]
  ]
}

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

DISCUSSION

讨论区

聊思路、贴方法、问为什么卡住。所有登录用户都可以发帖；发言公开署名，与纪录使用同一个名字，署名后的 #编号是账号注册序号，冒不了名。新发言经自动审核后公开。

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