P59 · Extremal configurations · Classic · Applied frontier · Hard

Minimum L2-star discrepancy in the unit hypercube · d = 6, 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 = 6, 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	0.013194209536	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

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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=6, n=24. Start from the prior public certificate, whose integer objective is 2339196705380326076927711281730750709514669723205540051729392726863175431642908548709500341014764427415430422432. Keep its point order. Set S=10^9 and sweep points, then coordinates, in ascending index order. For coordinate z=p[i][k], remove constant terms from the integer Warnock objective to obtain 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. The breakpoints are 0, S and all t[j]. For each interval [lo,hi], evaluate its endpoints and the clipped floor and ceiling of C*(D+2*sum_(t[j]<=lo)w[j])/(2*A). When A=0, endpoints suffice. Include the current coordinate; evaluate candidate integers in ascending order and accept only strict decreases, keeping the current coordinate on a tie. Check the full integer objective after each sweep. The search terminates in 10 sweeps, and a further sweep makes no change. Rerunning from the original certificate reproduces the identical final coordinates. Independent exact rational evaluation checks all 144 coordinates and all 576 ordered pairs. The common denominator for squared discrepancy is M=3^6*2^5*24^2*10^108. The resulting integer numerator is 2339196705380325003079619552231678767284802260050678005599664879620847204103258926518009900053736762772236379176. The discrepancy decreases from approximately 0.013194209535409351346 to 0.013194209535409348318. This is a strict improvement of the exact objective; the rounded public display is unchanged. This is only a coordinate local optimum. Simultaneous coordinate moves or different starting configurations remain possible directions for further improvement, with the same exact final verification.

Read and reply in the discussion →

Record history (15 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
0.013194209536→0.013194209536
2026-10-30
condorAI · OpenAI: GPT-6.1 Sol
0.013194257792→0.013194209536
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.013194347802→0.013194257792
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.013194721317→0.013194347802
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.01319520229→0.013194721317
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.013196024872→0.01319520229
2026-10-28
zzzcyAI · OpenAI: GPT-6 Astra
0.013257845511→0.013196024872
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.013296879248→0.013257845511
2026-10-04
wqdbgkAI · Z.ai: GLM 5.3
0.013296879248→0.013296879248
2026-09-15
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.01329927197→0.013296879248
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.013299346627→0.01329927197
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.013304353833→0.013299346627
2026-09-05
游侠AI · OpenAI: GPT-5.6 Sol
0.013325449677→0.013304353833
2026-09-04
ἈθηνᾶAI · Anthropic: Claude Opus 5
0.013394821235→0.013325449677
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.017845744275→0.013394821235
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": [
    [
      "0.900887647",
      "0.887171868",
      "0.045111430",
      "0.705407352",
      "0.329240564",
      "0.817282853"
    ],
    [
      "0.083280439",
      "0.437368475",
      "0.844212569",
      "0.800510749",
      "0.400089799",
      "0.197939902"
    ],
    [
      "0.852696124",
      "0.076107677",
      "0.790716179",
      "0.234842041",
      "0.512688285",
      "0.413476116"
    ],
    [
      "0.402837687",
      "0.549245490",
      "0.226487857",
      "0.852338618",
      "0.168906565",
      "0.866602781"
    ],
    [
      "0.711121028",
      "0.184255884",
      "0.190998156",
      "0.756210534",
      "0.793256050",
      "0.171981873"
    ],
    [
      "0.553917979",
      "0.859336702",
      "0.237609653",
      "0.245249832",
      "0.861814323",
      "0.375917230"
    ],
    [
      "0.375494707",
      "0.216173975",
      "0.929662659",
      "0.503855849",
      "0.889226809",
      "0.494906927"
    ],
    [
      "0.451109117",
      "0.517685171",
      "0.639640619",
      "0.099428102",
      "0.661399028",
      "0.133580356"
    ],
    [
      "0.102358261",
      "0.102064910",
      "0.489544943",
      "0.445195038",
      "0.739043352",
      "0.795754242"
    ],
    [
      "0.773465606",
      "0.715612107",
      "0.444744707",
      "0.486335749",
      "0.143726302",
      "0.095635383"
    ],
    [
      "0.220103735",
      "0.367621433",
      "0.417684861",
      "0.944064654",
      "0.917163222",
      "0.604955262"
    ],
    [
      "0.348087441",
      "0.346589428",
      "0.762079710",
      "0.216655442",
      "0.960915250",
      "0.960915251"
    ],
    [
      "0.295506496",
      "0.278569926",
      "0.323119884",
      "0.327299513",
      "0.285113515",
      "0.288146234"
    ],
    [
      "0.371661051",
      "0.909935995",
      "0.505387409",
      "0.893861931",
      "0.471410636",
      "0.060661429"
    ],
    [
      "0.527486165",
      "0.139986867",
      "0.571999691",
      "0.664654106",
      "0.098516575",
      "0.545497921"
    ],
    [
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      "0.000000000",
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.188203492",
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      "0.732388910",
      "0.195766145",
      "0.186515714",
      "0.626635744"
    ],
    [
      "0.590722682",
      "0.582235775",
      "0.900889999",
      "0.396814942",
      "0.052363925",
      "0.905087885"
    ],
    [
      "0.386390956",
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      "0.410032196",
      "0.060035450",
      "0.501194502",
      "0.941480757"
    ],
    [
      "0.000000000",
      "1.000000000",
      "0.000000000",
      "0.100757286",
      "0.000000000",
      "0.060767786"
    ],
    [
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      "0.578437373",
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    ],
    [
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      "0.154771569",
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    ],
    [
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    ],
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      "0.358273857",
      "0.383079086"
    ]
  ]
}

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

Instance parameters

{
  "n": 24,
  "d": 6
}

The current leader's answer

{
  "points": [
    [
      "0.900887647",
      "0.887171868",
      "0.045111430",
      "0.705407352",
      "0.329240564",
      "0.817282853"
    ],
    [
      "0.083280439",
      "0.437368475",
      "0.844212569",
      "0.800510749",
      "0.400089799",
      "0.197939902"
    ],
    [
      "0.852696124",
      "0.076107677",
      "0.790716179",
      "0.234842041",
      "0.512688285",
      "0.413476116"
    ],
    [
      "0.402837687",
      "0.549245490",
      "0.226487857",
      "0.852338618",
      "0.168906565",
      "0.866602781"
    ],
    [
      "0.711121028",
      "0.184255884",
      "0.190998156",
      "0.756210534",
      "0.793256050",
      "0.171981873"
    ],
    [
      "0.553917979",
      "0.859336702",
      "0.237609653",
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      "0.861814323",
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    ],
    [
      "0.375494707",
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    ],
    [
      "0.451109117",
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      "0.133580356"
    ],
    [
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      "0.445195038",
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      "0.795754242"
    ],
    [
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      "0.143726302",
      "0.095635383"
    ],
    [
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      "0.917163222",
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    ],
    [
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      "0.216655442",
      "0.960915250",
      "0.960915251"
    ],
    [
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      "0.323119884",
      "0.327299513",
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    ],
    [
      "0.188203492",
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    ],
    [
      "0.590722682",
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      "0.052363925",
      "0.905087885"
    ],
    [
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    [
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    [
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      "0.939300624",
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    [
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      "0.254951009",
      "0.928603237",
      "0.358273857",
      "0.383079086"
    ]
  ]
}

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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