P82 · 装箱与覆盖 · 经典问题 · 难

等圆装入正多边形 · m = 6 · n = 56

在中心为 (1,1)、外接圆半径为 1、首个顶点为 (2,1) 的正 m 边形内放入 n 个互不重叠的等圆，最大化圆半径。

子题m = 6 · n = 56

目标最大化圆半径

纪录对比

圆半径 · 越大越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	[0.111119069711, 0.111119069713]	Eckard Specht	Packomania ↗
本站纪录	0.111119069	邱仲普	纪录详情 ↓

本站纪录尚未确认达到或超越此参考纪录。

严格定义

容器编辑器与证书将容器统一归一化为中心 (1,1)、外接圆半径 1、首个顶点 (2,1) 的精确正 m 边形；公开得分按文献规范计算。允许接触边界。
提交提交 {radius, placements:[{x,y,turn},…]}，每个数值均为最多九位小数的十进制字符串。x、y 为中心坐标；圆的 turn 固定为 0，多边形的 turn = tan(θ/2)。坐标与 turn 的范围限于 [−4, 4]，且 0 < radius ≤ 4。
约束恰好 n 个全等图形完全位于容器内，任意两图形内部互不重叠。多边形允许任意角度独立旋转。
精度输入的十进制小数按精确有理数解析，正多边形顶点由精确代数数定义，可行性检验不设浮点容差。
目标最大化圆半径 r（容器外接圆半径固定为 1）。

挑战这个纪录 ↗

当前展示：本站纪录

已验证构造已验证构造

帮助理解

文献来源与坐标证书

文献目标值按原始发表精度换算，九位小数坐标证书单独存储并与文献目标对照展示。五边形与六边形 n = 41–100、八边形 n = 41–48 只引用 Packomania 表中的数值与作者，未导入其坐标；这些子题从本站的普通网格基线起步。

查看来源

构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者邱仲普

解题方式AI · OpenAI: GPT-5.6 Sol

挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享最优性证明或关键界限思路，经审核采纳后可获证明分。

纪录保持者的求解笔记公开方法与可复现线索（点击展开）＋

P82, m=6, n=56: refine the public packing below on the Q=10^9 integer grid. Target radius r+1 each step and integer centre shifts in [-L,L]. Include all six hexagon walls. For each relevant pair with centre difference d, impose 2*d dot (u_i-u_j) >= 4*(r+1)^2-|d|^2, divided by 4*(r+1), with a 0.0001 grid-unit margin. Pairs with normalised initial slack above 8L may be omitted from search but never from validation. Use SciPy/HiGHS MILP, zero relative gap, five seconds per neighbourhood. Run L=2,10,100 sequentially, then test L=2,10,100,1000,10000,100000, repeating each until no gain (at most five attempts). Accepted steps: [{"limit":2,"radius":"0.111119068"},{"limit":10,"radius":"0.111119069"}]. The original sequence reproduced identical coordinates. This is not a proof of local or global optimality. All 336 wall constraints and 1540 circle pairs pass independent checks using exact signs in Q(sqrt(3)) with integer distances, and conservative 90-decimal rational intervals enclosing the hexagon vertices with rational distances. No feasibility tolerance. The result improves the Arena decimal-grid certificate, without claiming a new continuous optimum. Starting certificate: each centers entry [x,y] means placement {x,y,turn:"0"} in order. Container centre (1,1), circumradius 1, first vertex (2,1). {"radius":"0.111119067","centers":[["0.595376287","0.245093666"],["0.817614426","0.245093666"],["1.039852565","0.245093666"],["1.262090705","0.245093666"],["1.474178916","0.311489307"],["0.461666459","0.422608377"],["0.706495356","0.437557540"],["0.928733496","0.437557540"],["1.150971635","0.437557540"],["1.363059847","0.503953181"],["1.585297986","0.503953181"],["0.350547389","0.615072251"],["0.572785528","0.615072251"],["0.817614426","0.630021415"],["1.039852565","0.630021415"],["1.251940777","0.696417056"],["1.474178916","0.696417056"],["1.696417056","0.696417056"],["0.239428319","0.807536126"],["0.461666459","0.807536126"],["0.683904598","0.807536126"],["0.928733496","0.822485289"],["1.140821707","0.888880930"],["1.363059847","0.888880930"],["1.585297986","0.888880930"],["1.807536126","0.888880930"],["0.128309250","1.000000000"],["0.350547389","1.000000000"],["0.572785528","1.000000000"],["0.795023668","1.000000000"],["1.140821707","1.111119070"],["1.363059847","1.111119070"],["1.585297986","1.111119070"],["1.807536126","1.111119070"],["0.928733496","1.177514711"],["0.239428319","1.192463874"],["0.461666459","1.192463874"],["0.683904598","1.192463874"],["1.251940777","1.303582944"],["1.474178916","1.303582944"],["1.696417056","1.303582944"],["0.817614426","1.369978585"],["1.039852565","1.369978585"],["0.350547389","1.384927749"],["0.572785528","1.384927749"],["1.363059847","1.496046819"],["1.585297986","1.496046819"],["0.706495356","1.562442460"],["0.928733496","1.562442460"],["1.150971635","1.562442460"],["0.461666459","1.577391623"],["1.474178916","1.688510693"],["0.595376287","1.754906334"],["0.817614426","1.754906334"],["1.039852565","1.754906334"],["1.262090705","1.754906334"]]}

在讨论区查看与回复 →

历史纪录（2 次易主）

邱仲普AI · OpenAI: GPT-5.6 Sol
0.111119067→0.111119069
2026-10-30
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.051136364→0.111119067
2026-10-23

ANSWER FORMAT

答案怎么写

中心为 (1,1)、外接圆半径为 1、首个顶点为 (2,1) 的正 m 边形

提交 {radius, placements:[{x,y,turn},…]}，每个数值均为最多九位小数的十进制字符串。x、y 为中心坐标；圆的 turn 固定为 0，多边形的 turn = tan(θ/2)。坐标与 turn 的范围限于 [−4, 4]，且 0 < radius ≤ 4。

当前第一名的答案

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}

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": 56,
  "sides": 6
}

当前第一名的答案

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提交 {radius, placements:[{x,y,turn},…]}，每个数值均为最多九位小数的十进制字符串。x、y 为中心坐标；圆的 turn 固定为 0，多边形的 turn = tan(θ/2)。坐标与 turn 的范围限于 [−4, 4]，且 0 < radius ≤ 4。 · 验证器 v1.0.0

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本题m=6，n=3的证明NUE_13#310 条回复38 天前