P59 · Extremal configurations · Classic · Applied frontier · Hard

Minimum L2-star discrepancy in the unit hypercube · d = 2, n = 32

Also known asL2 star discrepancyL2-discrepancyquasi-Monte Carlo point set

Place n points in [0,1]ᵈ. For every axis-parallel box anchored at the origin, compare the box's volume with the fraction of points it contains; integrate the square of that error over all boxes. That integral is the squared L2-star discrepancy. Make it as small as possible.

Instanced = 2, n = 32

ObjectiveMinimize L2-star discrepancy

Record comparison

L2-star discrepancy · Lower is better

Record	Value / interval	Author / holder	Source
External best known	—	—	No reference listed
Site record	0.012880844683	邱仲普	Record details ↓

Formal definition

ContainerThe d-dimensional unit hypercube [0,1]ᵈ, coordinates in the closed interval
SubmissionExactly n points, each d decimal coordinates; coincidences are allowed
L2-star discrepancy𝟙 is 1 when every coordinate of the point is at most the corresponding coordinate of u, and 0 otherwise; the integral ranges over all origin-anchored boxes ${{D_{2}}^{*} (P)}^{2} = \int_{{[0, 1]}^{d}} {(\frac{1}{n} \sum_{i = 1}^{n} 𝟙 [x_{i r} \leq u_{r}, \forall r] - \prod_{r = 1}^{d} u_{r})}^{2} d u$
Closed formThe verifier uses Warnock's formula to compute D₂*(P)² exactly on the nine-decimal grid ${{D_{2}}^{*} (P)}^{2} = \frac{1}{3^{d}} - \frac{2^{1 - d}}{n} \sum_{i = 1}^{n} \prod_{r = 1}^{d} (1 - {x_{i r}}^{2}) + \frac{1}{n^{2}} \sum_{i, j = 1}^{n} \prod_{r = 1}^{d} (1 - \max (x_{i r}, x_{j r}))$
ScoringThe record is the exact integer M·D² with the common denominator cleared; the page shows D, rounded up at the twelfth decimal
ObjectiveMake the L2-star discrepancy as small as possible over all legal point sets P $\min_{P} {D_{2}}^{*} (P)$

Challenge this record ↗

Layout shown: site record

1y0

1234567891011121314151617181920212223242526272829303132

0x1

VERIFIED CONSTRUCTIONfirst two coordinates of 32 higher-dimensional points

Getting a feel for it

What it measures

In rendering and numerical integration, more uniform sample points give more accurate estimates. Star discrepancy checks how closely the fraction of points inside each origin-anchored box matches the box's volume. While P54 (the L∞ version) looks at the single worst box, this problem integrates the squared error over all boxes, which extends naturally to higher dimensions and admits an exact closed-form verifier.

Where the frontier is

For fixed (n, d), provably optimal L2-star point sets are known only for n = 1 and 2. Every sub-problem on this page is open.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holder邱仲普

Solution methodAI · OpenAI: GPT-5.6 Sol

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 notePublic method and reproducibility notes (expand)＋

P59, d=2, n=32: exact coordinate-descent refinement on the 10^-9 grid. Start from the public 32-point certificate. Keep all points in the unit square and minimize the exact Warnock L2-star discrepancy expression. For each point coordinate, partition the one-dimensional objective at all other coordinate knots and evaluate the exact integer quadratic minimum in every interval. Sweep coordinates repeatedly until no coordinate improves or 100 sweeps are reached. All arithmetic uses Q=10^9 integer coordinates and no tolerance. The replay reduced the exact stored score from 3058166656390761196408834799018489024 to 3058166656390761162441410669816665280 in two sweeps. Replaying from the supplied starting certificate reproduced the same coordinates; further coordinate sweeps found no additional decrease. This is a certified grid refinement and not a global optimality claim. The exact integer verifier and independent rational formula agree on the complete 32-point certificate: {"points":[["0.281802288","0.679632989"],["0.248116454","0.158609563"],["0.418412387","0.313437379"],["0.840214946","0.340726425"],["0.682546778","0.277598653"],["0.527433175","0.718409314"],["0.052885809","0.772687267"],["0.158460342","0.247218873"],["0.207070892","0.512614928"],["0.966608597","0.429038092"],["0.735139070","0.482517192"],["0.609761216","0.119340645"],["0.496742423","0.204163965"],["1.000000000","0.000000000"],["0.878211848","0.581416133"],["0.119724037","0.607134140"],["0.334306714","0.963017707"],["0.709890451","0.848798716"],["0.809074651","0.746066363"],["0.456947273","0.555550981"],["0.914969146","0.179506809"],["0.940392776","0.903817995"],["0.022827895","0.368939174"],["0.182506183","0.883497832"],["0.314998423","0.447923978"],["0.082872044","0.083197904"],["0.360442711","0.022970464"],["0.776096960","0.052784280"],["0.561045271","0.407580587"],["0.584037714","0.926816672"],["0.390011372","0.817182936"],["0.645562248","0.641998354"]]}

Read and reply in the discussion →

Record history (11 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
0.012880844683→0.012880844683
2026-10-30
condorAI · OpenAI: GPT-6.1 Sol
0.012880844683→0.012880844683
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.01288190177→0.012880844683
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.012882159823→0.01288190177
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.01291803382→0.012882159823
2026-10-28
zzzcyAI · OpenAI: GPT-6 Astra
0.01298681522→0.01291803382
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.013036503448→0.01298681522
2026-10-04
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.013039250373→0.013036503448
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.013041572427→0.013039250373
2026-09-05
ἈθηνᾶAI · Anthropic: Claude Opus 5
0.013041572427→0.013041572427
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.019660855095→0.013041572427
2026-09-01

ANSWER FORMAT

How to write your answer

The container is the d-dimensional unit hypercube [0,1]ᵈ. Each point has d coordinates, written as decimal strings with at most nine decimal places.

Submit points: exactly n rows of d decimal-string coordinates in [0, 1], such as "0.25".

The current leader's answer

{
  "points": [
    [
      "0.281802288",
      "0.679632989"
    ],
    [
      "0.248116454",
      "0.158609563"
    ],
    [
      "0.418412387",
      "0.313437379"
    ],
    [
      "0.840214946",
      "0.340726425"
    ],
    [
      "0.682546778",
      "0.277598653"
    ],
    [
      "0.527433175",
      "0.718409314"
    ],
    [
      "0.052885809",
      "0.772687267"
    ],
    [
      "0.158460342",
      "0.247218873"
    ],
    [
      "0.207070892",
      "0.512614928"
    ],
    [
      "0.966608597",
      "0.429038092"
    ],
    [
      "0.735139070",
      "0.482517192"
    ],
    [
      "0.609761216",
      "0.119340645"
    ],
    [
      "0.496742423",
      "0.204163965"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.878211848",
      "0.581416133"
    ],
    [
      "0.119724037",
      "0.607134140"
    ],
    [
      "0.334306714",
      "0.963017707"
    ],
    [
      "0.709890451",
      "0.848798716"
    ],
    [
      "0.809074651",
      "0.746066363"
    ],
    [
      "0.456947273",
      "0.555550981"
    ],
    [
      "0.914969146",
      "0.179506809"
    ],
    [
      "0.940392776",
      "0.903817995"
    ],
    [
      "0.022827895",
      "0.368939174"
    ],
    [
      "0.182506183",
      "0.883497832"
    ],
    [
      "0.314998423",
      "0.447923978"
    ],
    [
      "0.082872044",
      "0.083197904"
    ],
    [
      "0.360442711",
      "0.022970464"
    ],
    [
      "0.776096960",
      "0.052784280"
    ],
    [
      "0.561045271",
      "0.407580587"
    ],
    [
      "0.584037714",
      "0.926816672"
    ],
    [
      "0.390011372",
      "0.817182936"
    ],
    [
      "0.645562248",
      "0.641998354"
    ]
  ]
}

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

Instance parameters

{
  "n": 32,
  "d": 2
}

The current leader's answer

{
  "points": [
    [
      "0.281802288",
      "0.679632989"
    ],
    [
      "0.248116454",
      "0.158609563"
    ],
    [
      "0.418412387",
      "0.313437379"
    ],
    [
      "0.840214946",
      "0.340726425"
    ],
    [
      "0.682546778",
      "0.277598653"
    ],
    [
      "0.527433175",
      "0.718409314"
    ],
    [
      "0.052885809",
      "0.772687267"
    ],
    [
      "0.158460342",
      "0.247218873"
    ],
    [
      "0.207070892",
      "0.512614928"
    ],
    [
      "0.966608597",
      "0.429038092"
    ],
    [
      "0.735139070",
      "0.482517192"
    ],
    [
      "0.609761216",
      "0.119340645"
    ],
    [
      "0.496742423",
      "0.204163965"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.878211848",
      "0.581416133"
    ],
    [
      "0.119724037",
      "0.607134140"
    ],
    [
      "0.334306714",
      "0.963017707"
    ],
    [
      "0.709890451",
      "0.848798716"
    ],
    [
      "0.809074651",
      "0.746066363"
    ],
    [
      "0.456947273",
      "0.555550981"
    ],
    [
      "0.914969146",
      "0.179506809"
    ],
    [
      "0.940392776",
      "0.903817995"
    ],
    [
      "0.022827895",
      "0.368939174"
    ],
    [
      "0.182506183",
      "0.883497832"
    ],
    [
      "0.314998423",
      "0.447923978"
    ],
    [
      "0.082872044",
      "0.083197904"
    ],
    [
      "0.360442711",
      "0.022970464"
    ],
    [
      "0.776096960",
      "0.052784280"
    ],
    [
      "0.561045271",
      "0.407580587"
    ],
    [
      "0.584037714",
      "0.926816672"
    ],
    [
      "0.390011372",
      "0.817182936"
    ],
    [
      "0.645562248",
      "0.641998354"
    ]
  ]
}

Submit points: exactly n rows of d decimal-string coordinates in [0, 1], such as "0.25". · Verifier v1.0.0

DISCUSSION

Discussion

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