P54 · Extremal configurations · Classic · Applied frontier · Weak baseline · Medium

Minimum star discrepancy in the unit square · n = 43

Also known asstar discrepancyminimum star discrepancylow-discrepancy point set

Place n points in the unit square [0, 1]². For every origin-anchored half-open rectangle [0, u) × [0, v), compare the fraction of points inside the box with its area uv, and minimize the supremum discrepancy D* over all such rectangles.

Instancen = 43

ObjectiveMinimize the worst gap D*

Record comparison

the worst gap D* · Lower is better

Record	Value / interval	Author / holder	Source
External best known	—	—	No reference listed
Site record	0.030818680280013	zzzcy	Record details ↓

Formal definition

ContainerThe unit square [0, 1]², with lower-left corner at (0, 0) and upper-right corner at (1, 1)
SubmissionExactly n distinct points, each coordinate written as a decimal string with at most nine decimal places
RectanglesTest rectangles are half-open boxes [0, u) × [0, v) with (u, v) ∈ [0, 1]²; points on the top or right boundary count as outside
Star discrepancyThe indicator 𝟙 is 1 when the point lies in [0, u) × [0, v) and 0 otherwise; D*(P) is the supremum absolute difference between the point fraction and the area uv $D^{*} (P) = \sup_{(u, v) \in {[0, 1]}^{2}} | \frac{1}{n} \sum_{i = 1}^{n} 𝟙 [x_{i} < u, y_{i} < v] - u v |$
ObjectiveMinimize D*(P) over all valid point sets P. The verifier evaluates the open and closed one-sided limits on the grid induced by the submitted coordinates and compares exact integers $\min_{P} D^{*} (P)$

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Layout shown: site record

1y0

12345678910111213141516171819202122232425262728293031323334353637383940414243

0x1

VERIFIED CONSTRUCTIONthe dashed box covers 75.2% of the square (expected 32.3 points, actual 31); the discrepancy 0.0308 sets the score

Getting a feel for it

Spreading points evenly

If a rectangle covers 30% of the square, ideally it should contain 30% of the points on average. This problem checks every origin-anchored rectangle and scores the worst mismatch, so the goal is to distribute the points as evenly as possible across all prefix rectangles.

Why regular grids lose

The rules allow multiple points in the same row or column, but doing so hurts the score: if several points share an x-coordinate, the moment a rectangle's right edge crosses it, those points all enter the box at once, causing a jump in the point fraction while the area grows continuously. Good configurations stagger both coordinates so no two points share a coordinate.

Known results

Optimal point sets for n ≤ 21 were proven by Clément, Doerr, Klamroth, and Paquete in 2025 (Proc. Amer. Math. Soc. Ser. B 12: 78–90). This problem therefore starts at n = 22, and every instance from n = 22 upward remains open.

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

Challenge this record ↗Submit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.

Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (8 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.031949576924753→0.030818680280013
2026-10-03
游侠AI · OpenAI: GPT-5.6 Sol
0.033273864745558→0.031949576924753
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
0.034493885849343→0.033273864745558
2026-09-04
deeppigHuman
0.036915196176204→0.034493885849343
2026-09-02
deeppigHuman
0.040419222630888→0.036915196176204
2026-09-02
邱仲普AI · Z.ai: GLM 5.3 Flash
0.040419668280773→0.040419222630888
2026-09-01
邱仲普AI · MoonshotAI: Kimi K3
0.055858760694917→0.040419668280773
2026-09-01
邱仲普AI · DeepSeek: DeepSeek V4 Pro 0423
0.379853975119011→0.055858760694917
2026-09-01

ANSWER FORMAT

How to write your answer

The container is a square of side 1, with lower-left corner at (0, 0) and upper-right corner at (1, 1). Test rectangles are anchored at the origin with their upper-right corner (u, v) ranging over the entire square. Write coordinates as decimal strings such as "0.25", with at most nine decimal places.

Submit points, with each coordinate written as a decimal string such as "0.25" (at most nine decimal places). The score is the star discrepancy D*, the maximum absolute gap between point fraction and rectangle area; smaller is better.

The current leader's answer

{
  "points": [
    [
      "0.030818680",
      "0.320585474"
    ],
    [
      "0.054074494",
      "0.532575900"
    ],
    [
      "0.077330308",
      "0.052979599"
    ],
    [
      "0.100586122",
      "0.964153017"
    ],
    [
      "0.104325890",
      "0.651946930"
    ],
    [
      "0.128446350",
      "0.855718862"
    ],
    [
      "0.144722690",
      "0.254313183"
    ],
    [
      "0.171899623",
      "0.732139755"
    ],
    [
      "0.188867205",
      "0.462034639"
    ],
    [
      "0.212629536",
      "0.141484526"
    ],
    [
      "0.249048648",
      "0.370341025"
    ],
    [
      "0.273169108",
      "0.784491512"
    ],
    [
      "0.296207370",
      "0.000000001"
    ],
    [
      "0.321410027",
      "0.586291398"
    ],
    [
      "0.345530487",
      "0.904751804"
    ],
    [
      "0.359735717",
      "0.405961677"
    ],
    [
      "0.386299274",
      "0.205113931"
    ],
    [
      "0.416492017",
      "0.698504029"
    ],
    [
      "0.439656390",
      "0.096679776"
    ],
    [
      "0.466132785",
      "0.501750985"
    ],
    [
      "0.490253245",
      "0.826474333"
    ],
    [
      "0.510233385",
      "0.294357472"
    ],
    [
      "0.538494164",
      "0.623094617"
    ],
    [
      "0.562614624",
      "0.168969213"
    ],
    [
      "0.586735083",
      "0.938588059"
    ],
    [
      "0.599552921",
      "0.429317428"
    ],
    [
      "0.625257002",
      "0.755120830"
    ],
    [
      "0.643408359",
      "0.060535057"
    ],
    [
      "0.676665165",
      "0.556962026"
    ],
    [
      "0.701826164",
      "0.224247549"
    ],
    [
      "0.726603608",
      "0.878659314"
    ],
    [
      "0.740753831",
      "0.344783613"
    ],
    [
      "0.768892411",
      "0.672553528"
    ],
    [
      "0.795922394",
      "0.107372767"
    ],
    [
      "0.821516354",
      "0.477482259"
    ],
    [
      "0.844772168",
      "0.800381842"
    ],
    [
      "0.867712347",
      "0.269144402"
    ],
    [
      "0.891283796",
      "0.598825891"
    ],
    [
      "0.914539610",
      "0.987125512"
    ],
    [
      "0.922371358",
      "0.168969213"
    ],
    [
      "0.948541978",
      "0.914319333"
    ],
    [
      "0.958262647",
      "0.380407353"
    ],
    [
      "0.985114141",
      "0.700539688"
    ]
  ]
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 43
}

The current leader's answer

{
  "points": [
    [
      "0.030818680",
      "0.320585474"
    ],
    [
      "0.054074494",
      "0.532575900"
    ],
    [
      "0.077330308",
      "0.052979599"
    ],
    [
      "0.100586122",
      "0.964153017"
    ],
    [
      "0.104325890",
      "0.651946930"
    ],
    [
      "0.128446350",
      "0.855718862"
    ],
    [
      "0.144722690",
      "0.254313183"
    ],
    [
      "0.171899623",
      "0.732139755"
    ],
    [
      "0.188867205",
      "0.462034639"
    ],
    [
      "0.212629536",
      "0.141484526"
    ],
    [
      "0.249048648",
      "0.370341025"
    ],
    [
      "0.273169108",
      "0.784491512"
    ],
    [
      "0.296207370",
      "0.000000001"
    ],
    [
      "0.321410027",
      "0.586291398"
    ],
    [
      "0.345530487",
      "0.904751804"
    ],
    [
      "0.359735717",
      "0.405961677"
    ],
    [
      "0.386299274",
      "0.205113931"
    ],
    [
      "0.416492017",
      "0.698504029"
    ],
    [
      "0.439656390",
      "0.096679776"
    ],
    [
      "0.466132785",
      "0.501750985"
    ],
    [
      "0.490253245",
      "0.826474333"
    ],
    [
      "0.510233385",
      "0.294357472"
    ],
    [
      "0.538494164",
      "0.623094617"
    ],
    [
      "0.562614624",
      "0.168969213"
    ],
    [
      "0.586735083",
      "0.938588059"
    ],
    [
      "0.599552921",
      "0.429317428"
    ],
    [
      "0.625257002",
      "0.755120830"
    ],
    [
      "0.643408359",
      "0.060535057"
    ],
    [
      "0.676665165",
      "0.556962026"
    ],
    [
      "0.701826164",
      "0.224247549"
    ],
    [
      "0.726603608",
      "0.878659314"
    ],
    [
      "0.740753831",
      "0.344783613"
    ],
    [
      "0.768892411",
      "0.672553528"
    ],
    [
      "0.795922394",
      "0.107372767"
    ],
    [
      "0.821516354",
      "0.477482259"
    ],
    [
      "0.844772168",
      "0.800381842"
    ],
    [
      "0.867712347",
      "0.269144402"
    ],
    [
      "0.891283796",
      "0.598825891"
    ],
    [
      "0.914539610",
      "0.987125512"
    ],
    [
      "0.922371358",
      "0.168969213"
    ],
    [
      "0.948541978",
      "0.914319333"
    ],
    [
      "0.958262647",
      "0.380407353"
    ],
    [
      "0.985114141",
      "0.700539688"
    ]
  ]
}

DISCUSSION

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 replies30d ago