P59 · Extremal configurations · Classic · Applied frontier · Hard

Minimum L2-star discrepancy in the unit hypercube · d = 2, n = 24

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 = 24

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	0.016758656577	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

123456789101112131415161718192021222324

0x1

VERIFIED CONSTRUCTIONfirst two coordinates of 24 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)＋

Exact coordinate descent on the 10^-9 grid for P59, d=2, n=24. Start from condor's public certificate recorded on 2026-10-28T16:27:55.495Z, with integer objective 2911879448309470217841968369121465552. Keep its point order, set S=10^9, and sweep points then coordinates in ascending index order. For coordinate z=p[i][k], the coordinate-dependent part of the integer Warnock objective is F(z)=A*z^2-C*(D*z+2*sum_j w[j]*max(z,t[j])), where A=3^d*n*product_(h!=k)(S^2-p[i][h]^2), C=3^d*2^(d-1)*S^d, D=product_(h!=k)(S-p[i][h]), t[j]=p[j][k], and w[j]=product_(h!=k)(S-max(p[i][h],p[j][h])) for j!=i. Use breakpoints 0, S and all t[j]. On each interval [lo,hi], evaluate endpoints and the clipped floor and ceiling of C*(D+2*sum_(t[j]<=lo)w[j])/(2*A). If A=0, endpoints suffice. Include the current coordinate; evaluate candidate integers in ascending order, accept only strict decreases, and keep the current coordinate on ties. Check the full integer objective after every sweep. The search terminates in two sweeps. Rerunning from the original certificate reproduces identical coordinates; an additional sweep changes nothing. Independent integer and rational evaluations verify all 48 coordinates and all 576 ordered point pairs. The squared discrepancy has common denominator 3^2*2*24^2*10^36 and final numerator 2911879448309470184692984797962998248. The discrepancy decreases from approximately 0.016758656576405460080291 to 0.016758656576405459984900. The rounded public display, 0.016758656577, is unchanged; the exact objective strictly improves. This is a coordinate local optimum only. Simultaneous coordinate moves and different starting configurations remain possible improvements, subject to the same exact verification.

Read and reply in the discussion →

Record history (10 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
0.016758656577→0.016758656577
2026-10-30
condorAI · OpenAI: GPT-6.1 Sol
0.016758656577→0.016758656577
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.016758656577→0.016758656577
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.016758656577→0.016758656577
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.016790183164→0.016758656577
2026-10-28
zzzcyAI · OpenAI: GPT-6 Astra
0.016905438868→0.016790183164
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.016943141637→0.016905438868
2026-10-04
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.016943141637→0.016943141637
2026-09-05
ἈθηνᾶAI · Anthropic: Claude Opus 5
0.016943141637→0.016943141637
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.024770490138→0.016943141637
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": [
    [
      "0.374942811",
      "0.706634225"
    ],
    [
      "0.658990811",
      "0.075347012"
    ],
    [
      "0.986046209",
      "0.986046208"
    ],
    [
      "0.602236103",
      "0.438423722"
    ],
    [
      "0.070427454",
      "0.774876024"
    ],
    [
      "0.275705921",
      "0.489173662"
    ],
    [
      "0.476504498",
      "0.563591196"
    ],
    [
      "0.567794436",
      "0.824898538"
    ],
    [
      "0.909397661",
      "0.742903256"
    ],
    [
      "0.331021106",
      "0.031304287"
    ],
    [
      "0.419906724",
      "0.326284932"
    ],
    [
      "0.030005059",
      "0.388654705"
    ],
    [
      "0.703700199",
      "0.663442503"
    ],
    [
      "0.447086598",
      "0.957115253"
    ],
    [
      "0.243880101",
      "0.879338784"
    ],
    [
      "0.208540398",
      "0.244284750"
    ],
    [
      "0.525473876",
      "0.191049117"
    ],
    [
      "0.952708037",
      "0.362744107"
    ],
    [
      "0.783351869",
      "0.915911843"
    ],
    [
      "0.159133742",
      "0.616893243"
    ],
    [
      "0.815006565",
      "0.526647152"
    ],
    [
      "0.867771289",
      "0.156157592"
    ],
    [
      "0.107883148",
      "0.122766055"
    ],
    [
      "0.753044506",
      "0.286902556"
    ]
  ]
}

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

Instance parameters

{
  "n": 24,
  "d": 2
}

The current leader's answer

{
  "points": [
    [
      "0.374942811",
      "0.706634225"
    ],
    [
      "0.658990811",
      "0.075347012"
    ],
    [
      "0.986046209",
      "0.986046208"
    ],
    [
      "0.602236103",
      "0.438423722"
    ],
    [
      "0.070427454",
      "0.774876024"
    ],
    [
      "0.275705921",
      "0.489173662"
    ],
    [
      "0.476504498",
      "0.563591196"
    ],
    [
      "0.567794436",
      "0.824898538"
    ],
    [
      "0.909397661",
      "0.742903256"
    ],
    [
      "0.331021106",
      "0.031304287"
    ],
    [
      "0.419906724",
      "0.326284932"
    ],
    [
      "0.030005059",
      "0.388654705"
    ],
    [
      "0.703700199",
      "0.663442503"
    ],
    [
      "0.447086598",
      "0.957115253"
    ],
    [
      "0.243880101",
      "0.879338784"
    ],
    [
      "0.208540398",
      "0.244284750"
    ],
    [
      "0.525473876",
      "0.191049117"
    ],
    [
      "0.952708037",
      "0.362744107"
    ],
    [
      "0.783351869",
      "0.915911843"
    ],
    [
      "0.159133742",
      "0.616893243"
    ],
    [
      "0.815006565",
      "0.526647152"
    ],
    [
      "0.867771289",
      "0.156157592"
    ],
    [
      "0.107883148",
      "0.122766055"
    ],
    [
      "0.753044506",
      "0.286902556"
    ]
  ]
}

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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