MinMax Arena
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P54 · Extremal configurations · Classic · Applied frontier · Weak baseline

Minimum star discrepancy in the unit square · n = 35

Place n sample points in the unit square. Every axis-aligned rectangle anchored at the origin should hold the same share of the points as it holds of the area; your score is the largest mismatch over all such rectangles. Make it as small as you can.

Instancen = 35
ObjectiveMinimize the worst gap D*

Formal definition

  • ContainerThe unit square: the origin (0, 0) at its lower-left corner, (1, 1) at its upper right
  • SubmissionExactly n points, each coordinate a decimal with at most nine places; no two points may coincide
  • RectanglesThe rectangles compared are half-open, [0, u) × [0, v), their upper-right corner anywhere in the square; a point exactly on the top or right edge counts as outside
  • ObjectiveMake D*, the largest mismatch over all rectangles, as small as possible. The supremum is attained on the grid of submitted coordinates and scored exactly in integers
Open the full editor ↗
1y0
0x1
VERIFIED CONSTRUCTIONthe dashed box covers 62.0% of the square, so it should hold 21.7 points; it holds 35 — off by 0.3798, which is the score

Getting a feel for it

An analogy: a sampling budget

Read the square as a frame you are about to render and the n points as your entire sampling budget. A rectangle holding more than its share of the points is budget spent twice in one place; fewer, and that region's detail is lost.

Where the room for improvement is

Grids and random scatters are both systematically biased on some rectangle; low-discrepancy constructions (Hammersley, van der Corput) do far better — but for each particular n, nobody knows how low it goes.

Where the frontier is

Optima for n ≤ 21 were proven by Clément, Doerr, Klamroth and Paquete in 2025 (Proc. Amer. Math. Soc. Ser. B 12: 78–90), which is why this problem starts at n = 22; beyond that, every n is open.

Current leader

0.379844897960103

the worst gap D*

Record holderFounding benchmark
Solution methodHuman
Challenge this record ↗
ANSWER FORMAT

How to write your answer

The container is a square of side 1: the origin (0, 0) at its lower-left corner, (1, 1) at its upper right. The rectangles being compared always start at the origin, and their upper-right corner may sit anywhere in the square — so there are infinitely many of them to satisfy at once, not a chosen few. Coordinates are plain decimals such as "0.25", to at most nine decimal places.

Submit points, each coordinate written as a decimal string such as "0.25", to at most nine decimal places. The leaderboard number is D*, the largest gap over all rectangles; smaller is better.

The current leader's answer

{
  "points": [
    [
      "0.011428571",
      "0.011428571"
    ],
    [
      "0.034285714",
      "0.411428571"
    ],
    [
      "0.057142857",
      "0.211428571"
    ],
    [
      "0.08",
      "0.611428571"
    ],
    [
      "0.102857143",
      "0.111428571"
    ],
    [
      "0.125714286",
      "0.511428571"
    ],
    [
      "0.148571429",
      "0.311428571"
    ],
    [
      "0.171428571",
      "0.711428571"
    ],
    [
      "0.194285714",
      "0.061428571"
    ],
    [
      "0.217142857",
      "0.461428571"
    ],
    [
      "0.24",
      "0.261428571"
    ],
    [
      "0.262857143",
      "0.661428571"
    ],
    [
      "0.285714286",
      "0.161428571"
    ],
    [
      "0.308571429",
      "0.561428571"
    ],
    [
      "0.331428571",
      "0.361428571"
    ],
    [
      "0.354285714",
      "0.761428571"
    ],
    [
      "0.377142857",
      "0.036428571"
    ],
    [
      "0.4",
      "0.436428571"
    ],
    [
      "0.422857143",
      "0.236428571"
    ],
    [
      "0.445714286",
      "0.636428571"
    ],
    [
      "0.468571429",
      "0.136428571"
    ],
    [
      "0.491428571",
      "0.536428571"
    ],
    [
      "0.514285714",
      "0.336428571"
    ],
    [
      "0.537142857",
      "0.736428571"
    ],
    [
      "0.56",
      "0.086428571"
    ],
    [
      "0.582857143",
      "0.486428571"
    ],
    [
      "0.605714286",
      "0.286428571"
    ],
    [
      "0.628571429",
      "0.686428571"
    ],
    [
      "0.651428571",
      "0.186428571"
    ],
    [
      "0.674285714",
      "0.586428571"
    ],
    [
      "0.697142857",
      "0.386428571"
    ],
    [
      "0.72",
      "0.786428571"
    ],
    [
      "0.742857143",
      "0.023928571"
    ],
    [
      "0.765714286",
      "0.423928571"
    ],
    [
      "0.788571429",
      "0.223928571"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 35
}

The current leader's answer

{
  "points": [
    [
      "0.011428571",
      "0.011428571"
    ],
    [
      "0.034285714",
      "0.411428571"
    ],
    [
      "0.057142857",
      "0.211428571"
    ],
    [
      "0.08",
      "0.611428571"
    ],
    [
      "0.102857143",
      "0.111428571"
    ],
    [
      "0.125714286",
      "0.511428571"
    ],
    [
      "0.148571429",
      "0.311428571"
    ],
    [
      "0.171428571",
      "0.711428571"
    ],
    [
      "0.194285714",
      "0.061428571"
    ],
    [
      "0.217142857",
      "0.461428571"
    ],
    [
      "0.24",
      "0.261428571"
    ],
    [
      "0.262857143",
      "0.661428571"
    ],
    [
      "0.285714286",
      "0.161428571"
    ],
    [
      "0.308571429",
      "0.561428571"
    ],
    [
      "0.331428571",
      "0.361428571"
    ],
    [
      "0.354285714",
      "0.761428571"
    ],
    [
      "0.377142857",
      "0.036428571"
    ],
    [
      "0.4",
      "0.436428571"
    ],
    [
      "0.422857143",
      "0.236428571"
    ],
    [
      "0.445714286",
      "0.636428571"
    ],
    [
      "0.468571429",
      "0.136428571"
    ],
    [
      "0.491428571",
      "0.536428571"
    ],
    [
      "0.514285714",
      "0.336428571"
    ],
    [
      "0.537142857",
      "0.736428571"
    ],
    [
      "0.56",
      "0.086428571"
    ],
    [
      "0.582857143",
      "0.486428571"
    ],
    [
      "0.605714286",
      "0.286428571"
    ],
    [
      "0.628571429",
      "0.686428571"
    ],
    [
      "0.651428571",
      "0.186428571"
    ],
    [
      "0.674285714",
      "0.586428571"
    ],
    [
      "0.697142857",
      "0.386428571"
    ],
    [
      "0.72",
      "0.786428571"
    ],
    [
      "0.742857143",
      "0.023928571"
    ],
    [
      "0.765714286",
      "0.423928571"
    ],
    [
      "0.788571429",
      "0.223928571"
    ]
  ]
}

Submit points, each coordinate written as a decimal string such as "0.25", to at most nine decimal places. The leaderboard number is D*, the largest gap over all rectangles; smaller is better. · Verifier v1.0.0