P82 · 装箱与覆盖 · 经典问题 · 难

等圆装入正多边形 · m = 6 · n = 53

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 = 53

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	[0.11225352737, 0.112253527372]	Eckard Specht	Packomania ↗
本站纪录	0.112253526	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=53. Refine the public packing below on the Q=10^9 grid. Target radius r+1 per step and integer centre shifts in [-L,L]. Include all six hexagon walls. For each pair with centre difference d, impose 2*d dot (u_i-u_j) >= 4*(r+1)^2-|d|^2, divided by 4*(r+1), plus a 0.0001 grid-unit margin. Pairs with normalised slack above 8L may be omitted from search, never from final verification. Use SciPy/HiGHS MILP with zero relative gap and a five-second limit. Run L=2,10,100 sequentially; then test L=2,10,100,1000,10000,100000, repeating at each L while it improves (up to five attempts). Accepted steps: [{"limit":2,"radius":"0.112253525"},{"limit":10,"radius":"0.112253526"}]. The original sequence replayed exactly; larger neighbourhoods gave no further gain. This is not an optimality proof. All 318 wall constraints and 1378 circle pairs pass two independent checks: exact signs in Q(sqrt(3)) and integer squared distances; conservative 90-decimal rational intervals enclosing the exact hexagon vertices and exact rational pair distances. No feasibility tolerance. This 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 the given order. Container centre (1,1), circumradius 1, first vertex (2,1). {"radius":"0.112253524","centers":[["0.652687175","0.246228124"],["0.877194230","0.246228124"],["1.101701284","0.246228124"],["1.326208339","0.246228124"],["1.509797227","0.375450946"],["0.481171744","0.391093146"],["0.764940702","0.440656936"],["0.989447757","0.440656936"],["1.213954812","0.440656936"],["1.397543700","0.569879759"],["1.622050754","0.569879759"],["0.368918217","0.585521959"],["0.593425272","0.585521959"],["0.877194230","0.635085749"],["1.101701284","0.635085749"],["1.285290172","0.764308572"],["1.509797227","0.764308572"],["1.734304282","0.764308572"],["0.256664690","0.779950772"],["0.481171744","0.779950772"],["0.705678799","0.779950772"],["0.986759785","0.831119483"],["1.173036645","0.958737384"],["1.397543700","0.958737384"],["1.622050754","0.958737384"],["1.846557809","0.958737384"],["0.144411162","0.974379584"],["0.368918217","0.974379584"],["0.815173405","0.975946560"],["0.591958599","1.000000000"],["1.000605116","1.102510766"],["1.285290172","1.153166197"],["1.509797227","1.153166197"],["1.734304282","1.153166197"],["0.233743422","1.180348429"],["0.458250477","1.180348429"],["0.733629862","1.208227043"],["0.989445535","1.326740294"],["1.210683341","1.364914251"],["1.435190396","1.364914251"],["1.659697450","1.364914251"],["0.345996950","1.374777242"],["0.570504005","1.374777242"],["0.788760754","1.427383408"],["1.083029897","1.550538448"],["1.322936868","1.559343064"],["1.547443923","1.559343064"],["0.458250477","1.569206054"],["0.786125186","1.651874992"],["0.586074086","1.753771876"],["0.986176286","1.753771876"],["1.210683341","1.753771876"],["1.435190396","1.753771876"]]}

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
0.112253524→0.112253526
2026-10-30
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.051136364→0.112253524
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": 53,
  "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