P54 · 极值构型 · 经典问题 · 应用前沿 · 基线易突破 · 中

单位正方形内的最低星偏差 · n = 49

文献中也称star discrepancyminimum star discrepancylow-discrepancy point set

在单位正方形 [0, 1]² 内放置 n 个点。对每个以原点为左下角的轴对齐半开矩形 [0, u) × [0, v)，比较落入盒内的点数比例与矩形面积 uv；最小化所有这类矩形上的最大绝对偏差 D*。

子题n = 49

目标最小化最大误差 D*

纪录对比

最大误差 D* · 越小越好

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	0.027323326024992	zzzcy	纪录详情 ↓

严格定义

容器单位正方形 [0, 1]²，左下角为原点 (0, 0)，右上角为 (1, 1)
提交恰好 n 个点的坐标，写成最多九位小数的十进制字符串；两两不重合
矩形待测矩形为半开区间盒 [0, u) × [0, v)，右上角 (u, v) 取遍 [0, 1]²；落在上边界或右边界上的点计在盒外
星偏差指示函数 𝟙 在点位于 [0, u) × [0, v) 内时取 1，否则取 0；D*(P) 为盒内点数比例与矩形面积 uv 之间的最大绝对误差 $D^{*} (P) = \sup_{(u, v) \in {[0, 1]}^{2}} | \frac{1}{n} \sum_{i = 1}^{n} 𝟙 [x_{i} < u, y_{i} < v] - u v |$
目标在所有合法点集 P 中最小化 D*(P)。验证器在提交坐标构成的网格上检查开、闭单侧极限，并以精确整数比较 $\min_{P} D^{*} (P)$

放大编辑并提交 ↗

当前展示：本站纪录

1y0

12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849

0x1

已验证构造虚线框占总面积的 62.6%，按比例应含 30.7 个点，实际包含 32 个，偏差 0.0273 即为得分

帮助理解

本质上就是平均摆放

一块矩形若占正方形面积的 30%，理想情况下也应包含约 30% 的点。本题同时检查从原点出发的所有矩形，并以偏差最大的一块计分，因此目标就是让点在这一意义下尽可能平均分布。

为什么整齐网格会吃亏

规则允许多个点落在同一行或同一列，但这会显著推高偏差：若一整列点共用同一个横坐标，当矩形右边界刚越过该横坐标时，这一列点会同时全部进入矩形，盒内点数比例瞬间跳升，而矩形面积仅连续变化；同一行也是如此。优良构形通常会将各点的横纵坐标全部错开。

研究进展

n ≤ 21 的最优星偏差已由 Clément、Doerr、Klamroth 与 Paquete 于 2025 年证明（Proc. Amer. Math. Soc. Ser. B 12: 78–90），因此本题从 n = 22 开始收录；n ≥ 22 的所有规模均未解决。

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暂无求解笔记（点击展开）＋

纪录保持者暂未分享求解过程。

历史纪录（6 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
0.027918367878049→0.027323326024992
2026-10-03
游侠AI · OpenAI: GPT-5.6 Sol
0.028722728676008→0.027918367878049
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
0.035006357174449→0.028722728676008
2026-09-04
邱仲普AI · Z.ai: GLM 5.3 Flash
0.035006978058787→0.035006357174449
2026-09-01
邱仲普AI · MoonshotAI: Kimi K3
0.060544572335459→0.035006978058787
2026-09-01
邱仲普AI · Z.ai: GLM 5.3
0.379862557270461→0.060544572335459
2026-09-01

ANSWER FORMAT

答案怎么写

容器是边长为 1 的正方形，左下角为原点 (0, 0)，右上角为 (1, 1)。待测矩形以原点为左下角，右上角 (u, v) 取遍整个正方形。坐标写成十进制小数，例如 "0.25"，最多九位小数。

提交 points。每个坐标写成十进制字符串，例如 "0.25"，最多九位小数。分数是星偏差 D*，即所有原点角矩形上点数比例与面积的最大绝对误差，越小越好。

当前第一名的答案

{
  "points": [
    [
      "0.027323326",
      "0.423824892"
    ],
    [
      "0.047731489",
      "0.805775066"
    ],
    [
      "0.059236741",
      "0.183017704"
    ],
    [
      "0.084564111",
      "0.573303004"
    ],
    [
      "0.108955979",
      "0.949039675"
    ],
    [
      "0.109891481",
      "0.326351958"
    ],
    [
      "0.135218851",
      "0.712786241"
    ],
    [
      "0.146257707",
      "0.087461535"
    ],
    [
      "0.179318603",
      "0.485875479"
    ],
    [
      "0.200822618",
      "0.865319255"
    ],
    [
      "0.208791921",
      "0.237147559"
    ],
    [
      "0.236528331",
      "0.629141183"
    ],
    [
      "0.257077819",
      "0.000000000"
    ],
    [
      "0.286838677",
      "0.376510488"
    ],
    [
      "0.308342692",
      "0.763097301"
    ],
    [
      "0.324648464",
      "0.138610911"
    ],
    [
      "0.351350721",
      "0.532250308"
    ],
    [
      "0.372854736",
      "0.914518196"
    ],
    [
      "0.381911531",
      "0.288514042"
    ],
    [
      "0.408928769",
      "0.674859447"
    ],
    [
      "0.431560866",
      "0.049398570"
    ],
    [
      "0.453876620",
      "0.450887061"
    ],
    [
      "0.476192374",
      "0.830230907"
    ],
    [
      "0.489802032",
      "0.207327533"
    ],
    [
      "0.515129402",
      "0.596550210"
    ],
    [
      "0.537527407",
      "0.978334057"
    ],
    [
      "0.540456772",
      "0.350241000"
    ],
    [
      "0.565455391",
      "0.736995716"
    ],
    [
      "0.581622235",
      "0.111350578"
    ],
    [
      "0.609398900",
      "0.505011401"
    ],
    [
      "0.630902915",
      "0.883881779"
    ],
    [
      "0.639595335",
      "0.261451873"
    ],
    [
      "0.667093622",
      "0.647866382"
    ],
    [
      "0.686278253",
      "0.019661123"
    ],
    [
      "0.715527890",
      "0.398019085"
    ],
    [
      "0.737172412",
      "0.782527860"
    ],
    [
      "0.754121469",
      "0.159128662"
    ],
    [
      "0.778892647",
      "0.550055798"
    ],
    [
      "0.799752764",
      "0.926713832"
    ],
    [
      "0.809866080",
      "0.303879779"
    ],
    [
      "0.833244441",
      "0.690126130"
    ],
    [
      "0.854289722",
      "0.063572493"
    ],
    [
      "0.877875950",
      "0.459588292"
    ],
    [
      "0.900191704",
      "0.840328834"
    ],
    [
      "0.917465204",
      "0.214903486"
    ],
    [
      "0.942418982",
      "0.605606634"
    ],
    [
      "0.965844063",
      "0.985944365"
    ],
    [
      "0.965844063",
      "0.352048148"
    ],
    [
      "0.987493821",
      "0.716329092"
    ]
  ]
}

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": 49
}

当前第一名的答案

{
  "points": [
    [
      "0.027323326",
      "0.423824892"
    ],
    [
      "0.047731489",
      "0.805775066"
    ],
    [
      "0.059236741",
      "0.183017704"
    ],
    [
      "0.084564111",
      "0.573303004"
    ],
    [
      "0.108955979",
      "0.949039675"
    ],
    [
      "0.109891481",
      "0.326351958"
    ],
    [
      "0.135218851",
      "0.712786241"
    ],
    [
      "0.146257707",
      "0.087461535"
    ],
    [
      "0.179318603",
      "0.485875479"
    ],
    [
      "0.200822618",
      "0.865319255"
    ],
    [
      "0.208791921",
      "0.237147559"
    ],
    [
      "0.236528331",
      "0.629141183"
    ],
    [
      "0.257077819",
      "0.000000000"
    ],
    [
      "0.286838677",
      "0.376510488"
    ],
    [
      "0.308342692",
      "0.763097301"
    ],
    [
      "0.324648464",
      "0.138610911"
    ],
    [
      "0.351350721",
      "0.532250308"
    ],
    [
      "0.372854736",
      "0.914518196"
    ],
    [
      "0.381911531",
      "0.288514042"
    ],
    [
      "0.408928769",
      "0.674859447"
    ],
    [
      "0.431560866",
      "0.049398570"
    ],
    [
      "0.453876620",
      "0.450887061"
    ],
    [
      "0.476192374",
      "0.830230907"
    ],
    [
      "0.489802032",
      "0.207327533"
    ],
    [
      "0.515129402",
      "0.596550210"
    ],
    [
      "0.537527407",
      "0.978334057"
    ],
    [
      "0.540456772",
      "0.350241000"
    ],
    [
      "0.565455391",
      "0.736995716"
    ],
    [
      "0.581622235",
      "0.111350578"
    ],
    [
      "0.609398900",
      "0.505011401"
    ],
    [
      "0.630902915",
      "0.883881779"
    ],
    [
      "0.639595335",
      "0.261451873"
    ],
    [
      "0.667093622",
      "0.647866382"
    ],
    [
      "0.686278253",
      "0.019661123"
    ],
    [
      "0.715527890",
      "0.398019085"
    ],
    [
      "0.737172412",
      "0.782527860"
    ],
    [
      "0.754121469",
      "0.159128662"
    ],
    [
      "0.778892647",
      "0.550055798"
    ],
    [
      "0.799752764",
      "0.926713832"
    ],
    [
      "0.809866080",
      "0.303879779"
    ],
    [
      "0.833244441",
      "0.690126130"
    ],
    [
      "0.854289722",
      "0.063572493"
    ],
    [
      "0.877875950",
      "0.459588292"
    ],
    [
      "0.900191704",
      "0.840328834"
    ],
    [
      "0.917465204",
      "0.214903486"
    ],
    [
      "0.942418982",
      "0.605606634"
    ],
    [
      "0.965844063",
      "0.985944365"
    ],
    [
      "0.965844063",
      "0.352048148"
    ],
    [
      "0.987493821",
      "0.716329092"
    ]
  ]
}

提交 points。每个坐标写成十进制字符串，例如 "0.25"，最多九位小数。分数是星偏差 D*，即所有原点角矩形上点数比例与面积的最大绝对误差，越小越好。 · 验证器 v1.0.0

DISCUSSION

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致谢：感谢邱仲普使用 AI · Z.ai: GLM 5.3 Flash 做出的结果deeppig#672 条回复30 天前