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

Minimum star discrepancy in the unit square · n = 46

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 = 46
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 28.5 points; it holds 46 — off by 0.3799, 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

0.379858223063706

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": [
    [
      "0.008695652",
      "0.008695652"
    ],
    [
      "0.026086957",
      "0.408695652"
    ],
    [
      "0.043478261",
      "0.208695652"
    ],
    [
      "0.060869565",
      "0.608695652"
    ],
    [
      "0.07826087",
      "0.108695652"
    ],
    [
      "0.095652174",
      "0.508695652"
    ],
    [
      "0.113043478",
      "0.308695652"
    ],
    [
      "0.130434783",
      "0.708695652"
    ],
    [
      "0.147826087",
      "0.058695652"
    ],
    [
      "0.165217391",
      "0.458695652"
    ],
    [
      "0.182608696",
      "0.258695652"
    ],
    [
      "0.2",
      "0.658695652"
    ],
    [
      "0.217391304",
      "0.158695652"
    ],
    [
      "0.234782609",
      "0.558695652"
    ],
    [
      "0.252173913",
      "0.358695652"
    ],
    [
      "0.269565217",
      "0.758695652"
    ],
    [
      "0.286956522",
      "0.033695652"
    ],
    [
      "0.304347826",
      "0.433695652"
    ],
    [
      "0.32173913",
      "0.233695652"
    ],
    [
      "0.339130435",
      "0.633695652"
    ],
    [
      "0.356521739",
      "0.133695652"
    ],
    [
      "0.373913043",
      "0.533695652"
    ],
    [
      "0.391304348",
      "0.333695652"
    ],
    [
      "0.408695652",
      "0.733695652"
    ],
    [
      "0.426086957",
      "0.083695652"
    ],
    [
      "0.443478261",
      "0.483695652"
    ],
    [
      "0.460869565",
      "0.283695652"
    ],
    [
      "0.47826087",
      "0.683695652"
    ],
    [
      "0.495652174",
      "0.183695652"
    ],
    [
      "0.513043478",
      "0.583695652"
    ],
    [
      "0.530434783",
      "0.383695652"
    ],
    [
      "0.547826087",
      "0.783695652"
    ],
    [
      "0.565217391",
      "0.021195652"
    ],
    [
      "0.582608696",
      "0.421195652"
    ],
    [
      "0.6",
      "0.221195652"
    ],
    [
      "0.617391304",
      "0.621195652"
    ],
    [
      "0.634782609",
      "0.121195652"
    ],
    [
      "0.652173913",
      "0.521195652"
    ],
    [
      "0.669565217",
      "0.321195652"
    ],
    [
      "0.686956522",
      "0.721195652"
    ],
    [
      "0.704347826",
      "0.071195652"
    ],
    [
      "0.72173913",
      "0.471195652"
    ],
    [
      "0.739130435",
      "0.271195652"
    ],
    [
      "0.756521739",
      "0.671195652"
    ],
    [
      "0.773913043",
      "0.171195652"
    ],
    [
      "0.791304348",
      "0.571195652"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 46
}

The current leader's answer

{
  "points": [
    [
      "0.008695652",
      "0.008695652"
    ],
    [
      "0.026086957",
      "0.408695652"
    ],
    [
      "0.043478261",
      "0.208695652"
    ],
    [
      "0.060869565",
      "0.608695652"
    ],
    [
      "0.07826087",
      "0.108695652"
    ],
    [
      "0.095652174",
      "0.508695652"
    ],
    [
      "0.113043478",
      "0.308695652"
    ],
    [
      "0.130434783",
      "0.708695652"
    ],
    [
      "0.147826087",
      "0.058695652"
    ],
    [
      "0.165217391",
      "0.458695652"
    ],
    [
      "0.182608696",
      "0.258695652"
    ],
    [
      "0.2",
      "0.658695652"
    ],
    [
      "0.217391304",
      "0.158695652"
    ],
    [
      "0.234782609",
      "0.558695652"
    ],
    [
      "0.252173913",
      "0.358695652"
    ],
    [
      "0.269565217",
      "0.758695652"
    ],
    [
      "0.286956522",
      "0.033695652"
    ],
    [
      "0.304347826",
      "0.433695652"
    ],
    [
      "0.32173913",
      "0.233695652"
    ],
    [
      "0.339130435",
      "0.633695652"
    ],
    [
      "0.356521739",
      "0.133695652"
    ],
    [
      "0.373913043",
      "0.533695652"
    ],
    [
      "0.391304348",
      "0.333695652"
    ],
    [
      "0.408695652",
      "0.733695652"
    ],
    [
      "0.426086957",
      "0.083695652"
    ],
    [
      "0.443478261",
      "0.483695652"
    ],
    [
      "0.460869565",
      "0.283695652"
    ],
    [
      "0.47826087",
      "0.683695652"
    ],
    [
      "0.495652174",
      "0.183695652"
    ],
    [
      "0.513043478",
      "0.583695652"
    ],
    [
      "0.530434783",
      "0.383695652"
    ],
    [
      "0.547826087",
      "0.783695652"
    ],
    [
      "0.565217391",
      "0.021195652"
    ],
    [
      "0.582608696",
      "0.421195652"
    ],
    [
      "0.6",
      "0.221195652"
    ],
    [
      "0.617391304",
      "0.621195652"
    ],
    [
      "0.634782609",
      "0.121195652"
    ],
    [
      "0.652173913",
      "0.521195652"
    ],
    [
      "0.669565217",
      "0.321195652"
    ],
    [
      "0.686956522",
      "0.721195652"
    ],
    [
      "0.704347826",
      "0.071195652"
    ],
    [
      "0.72173913",
      "0.471195652"
    ],
    [
      "0.739130435",
      "0.271195652"
    ],
    [
      "0.756521739",
      "0.671195652"
    ],
    [
      "0.773913043",
      "0.171195652"
    ],
    [
      "0.791304348",
      "0.571195652"
    ]
  ]
}

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