P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 48

在半径为 1 的圆内放置 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 = 48

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	[6.408615, 6.408625]	见来源署名	Erich Friedman ↗
本站纪录	暂无选手纪录	—	—

严格定义

容器圆心 (1, 1)、半径 1 的闭圆盘；圆可以与容器边界相切
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圆盘内：(x − 1)² + (y − 1)² ≤ (1 − r)²；任意两圆内部不相交：圆心距 ≥ 半径之和。全部按整数精确判定，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 $S = Σ r$
目标在所有合法放置中最大化半径之和 $\max_{S} Σ r$

放大来摆，然后提交 ↗

当前展示：起步布局

2y0

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748

0x2

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

和正方形里一样，等圆是最差的策略之一：一个大圆压在圆心附近，或者几个大圆围成一圈，再让小圆钻进它们和圆周之间的缝隙。圆盘没有角，所以“靠墙的大圆”换成了“沿圆周的一圈”，大小的取舍换了一种几何。

前沿在哪里

Erich Friedman 的“最大化总周长”表列到 n = 50：n ≤ 4 和 6–10 是 Elser 与 Cantrell 的闭式构型，n = 5 是 Haowei Lin 2026 年 7 月的改进，n = 33–50 是 Jonathan Viquerat 2026 年 9 月新加的行。表还在被改写，没有一行有最优性证明；本站展示的是按图重建并精确验算的构型。

查看来源

构型与历史

起步布局

查看构型、求解笔记与纪录历史。

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。

ANSWER FORMAT

答案怎么写

容器是圆心 (1, 1)、半径 1 的圆盘，所以坐标都在 [0, 2] 之内，不会出现负数。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"。容器是圆心 (1, 1)、半径 1 的闭圆盘。

起步布局答案

{
  "circles": [
    [
      "1.235886324",
      "0.432198610",
      "0.169896351"
    ],
    [
      "0.240910992",
      "1.343430005",
      "0.166837176"
    ],
    [
      "1.675642639",
      "1.492076274",
      "0.164157888"
    ],
    [
      "1.065570463",
      "0.156614741",
      "0.154069636"
    ],
    [
      "0.148116721",
      "1.043624599",
      "0.147000453"
    ],
    [
      "0.403981640",
      "1.610958293",
      "0.146473245"
    ],
    [
      "0.920792883",
      "0.418853517",
      "0.145479564"
    ],
    [
      "1.825914695",
      "1.222228490",
      "0.144710233"
    ],
    [
      "1.465042472",
      "1.717906022",
      "0.144632501"
    ],
    [
      "0.432978057",
      "1.099851211",
      "0.143356933"
    ],
    [
      "0.550288570",
      "1.361126848",
      "0.143046129"
    ],
    [
      "1.044268904",
      "0.679666705",
      "0.143085587"
    ],
    [
      "0.769634940",
      "0.173887441",
      "0.142369531"
    ],
    [
      "1.716245093",
      "0.528323660",
      "0.142395427"
    ],
    [
      "1.373329755",
      "1.446849521",
      "0.141519290"
    ],
    [
      "1.540014452",
      "1.218475511",
      "0.141214641"
    ],
    [
      "1.525189604",
      "0.319654513",
      "0.140526845"
    ],
    [
      "0.713803514",
      "1.131072454",
      "0.139198730"
    ],
    [
      "0.877910525",
      "0.906954039",
      "0.138578472"
    ],
    [
      "0.764605142",
      "0.655020001",
      "0.137662128"
    ],
    [
      "1.262403084",
      "1.190497821",
      "0.137802961"
    ],
    [
      "1.151970961",
      "0.938537333",
      "0.137295823"
    ],
    [
      "0.988931064",
      "1.159623871",
      "0.137406316"
    ],
    [
      "0.603941488",
      "0.877722882",
      "0.136945561"
    ],
    [
      "0.828676848",
      "1.381222559",
      "0.136066519"
    ],
    [
      "1.098806540",
      "1.409474153",
      "0.135536504"
    ],
    [
      "1.836510571",
      "0.777683480",
      "0.134451289"
    ],
    [
      "1.421473416",
      "0.970051035",
      "0.134042870"
    ],
    [
      "0.229409815",
      "0.603544990",
      "0.133405626"
    ],
    [
      "0.494750772",
      "0.630515647",
      "0.133302527"
    ],
    [
      "0.385463896",
      "0.387996332",
      "0.132703560"
    ],
    [
      "1.316871760",
      "0.724204205",
      "0.133131548"
    ],
    [
      "0.785562049",
      "1.840479023",
      "0.132596620"
    ],
    [
      "0.940211944",
      "1.625360304",
      "0.132342340"
    ],
    [
      "1.049052810",
      "1.866442359",
      "0.132170212"
    ],
    [
      "0.339211091",
      "0.843027511",
      "0.130048729"
    ],
    [
      "0.679643993",
      "1.600875866",
      "0.129373430"
    ],
    [
      "0.646007873",
      "0.415822809",
      "0.129322157"
    ],
    [
      "1.199559495",
      "1.653569468",
      "0.128534850"
    ],
    [
      "1.682673032",
      "0.990241904",
      "0.127935966"
    ],
    [
      "1.575252130",
      "0.758478087",
      "0.127512104"
    ],
    [
      "1.327095696",
      "0.169915259",
      "0.107793593"
    ],
    [
      "1.475846212",
      "0.554474676",
      "0.099421647"
    ],
    [
      "0.117211045",
      "0.803085642",
      "0.095515835"
    ],
    [
      "0.563087076",
      "1.792573319",
      "0.094977696"
    ],
    [
      "1.905259349",
      "0.996313557",
      "0.094733144"
    ],
    [
      "1.275359428",
      "1.862970328",
      "0.094163038"
    ],
    [
      "0.538031226",
      "0.220482322",
      "0.093874755"
    ]
  ]
}

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": 48
}

起步布局答案

{
  "circles": [
    [
      "1.235886324",
      "0.432198610",
      "0.169896351"
    ],
    [
      "0.240910992",
      "1.343430005",
      "0.166837176"
    ],
    [
      "1.675642639",
      "1.492076274",
      "0.164157888"
    ],
    [
      "1.065570463",
      "0.156614741",
      "0.154069636"
    ],
    [
      "0.148116721",
      "1.043624599",
      "0.147000453"
    ],
    [
      "0.403981640",
      "1.610958293",
      "0.146473245"
    ],
    [
      "0.920792883",
      "0.418853517",
      "0.145479564"
    ],
    [
      "1.825914695",
      "1.222228490",
      "0.144710233"
    ],
    [
      "1.465042472",
      "1.717906022",
      "0.144632501"
    ],
    [
      "0.432978057",
      "1.099851211",
      "0.143356933"
    ],
    [
      "0.550288570",
      "1.361126848",
      "0.143046129"
    ],
    [
      "1.044268904",
      "0.679666705",
      "0.143085587"
    ],
    [
      "0.769634940",
      "0.173887441",
      "0.142369531"
    ],
    [
      "1.716245093",
      "0.528323660",
      "0.142395427"
    ],
    [
      "1.373329755",
      "1.446849521",
      "0.141519290"
    ],
    [
      "1.540014452",
      "1.218475511",
      "0.141214641"
    ],
    [
      "1.525189604",
      "0.319654513",
      "0.140526845"
    ],
    [
      "0.713803514",
      "1.131072454",
      "0.139198730"
    ],
    [
      "0.877910525",
      "0.906954039",
      "0.138578472"
    ],
    [
      "0.764605142",
      "0.655020001",
      "0.137662128"
    ],
    [
      "1.262403084",
      "1.190497821",
      "0.137802961"
    ],
    [
      "1.151970961",
      "0.938537333",
      "0.137295823"
    ],
    [
      "0.988931064",
      "1.159623871",
      "0.137406316"
    ],
    [
      "0.603941488",
      "0.877722882",
      "0.136945561"
    ],
    [
      "0.828676848",
      "1.381222559",
      "0.136066519"
    ],
    [
      "1.098806540",
      "1.409474153",
      "0.135536504"
    ],
    [
      "1.836510571",
      "0.777683480",
      "0.134451289"
    ],
    [
      "1.421473416",
      "0.970051035",
      "0.134042870"
    ],
    [
      "0.229409815",
      "0.603544990",
      "0.133405626"
    ],
    [
      "0.494750772",
      "0.630515647",
      "0.133302527"
    ],
    [
      "0.385463896",
      "0.387996332",
      "0.132703560"
    ],
    [
      "1.316871760",
      "0.724204205",
      "0.133131548"
    ],
    [
      "0.785562049",
      "1.840479023",
      "0.132596620"
    ],
    [
      "0.940211944",
      "1.625360304",
      "0.132342340"
    ],
    [
      "1.049052810",
      "1.866442359",
      "0.132170212"
    ],
    [
      "0.339211091",
      "0.843027511",
      "0.130048729"
    ],
    [
      "0.679643993",
      "1.600875866",
      "0.129373430"
    ],
    [
      "0.646007873",
      "0.415822809",
      "0.129322157"
    ],
    [
      "1.199559495",
      "1.653569468",
      "0.128534850"
    ],
    [
      "1.682673032",
      "0.990241904",
      "0.127935966"
    ],
    [
      "1.575252130",
      "0.758478087",
      "0.127512104"
    ],
    [
      "1.327095696",
      "0.169915259",
      "0.107793593"
    ],
    [
      "1.475846212",
      "0.554474676",
      "0.099421647"
    ],
    [
      "0.117211045",
      "0.803085642",
      "0.095515835"
    ],
    [
      "0.563087076",
      "1.792573319",
      "0.094977696"
    ],
    [
      "1.905259349",
      "0.996313557",
      "0.094733144"
    ],
    [
      "1.275359428",
      "1.862970328",
      "0.094163038"
    ],
    [
      "0.538031226",
      "0.220482322",
      "0.093874755"
    ]
  ]
}

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"。容器是圆心 (1, 1)、半径 1 的闭圆盘。 · 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。

对本周挑战的评价NUE_13#311 条回复1 小时前