P74 · Extremal configurations · Classic · Applied frontier · Hard

Covering an equilateral triangle with n equal discs · n = 23

Place n centres in a unit equilateral triangle. Every centre has the same covering radius; every point of the triangle must lie in at least one disc, and that radius is minimised.

Instancen = 23

ObjectiveMinimize the covering radius

Best known, unproven0.0937742911094478264The construction by Kari J. Nurmela appears in Figure 3 of Nurmela (2000); the paper's continuous value is 0.0937742911094478264. Optimality remains open; the paper publishes no directly importable coordinates; the site's nine-decimal contact-graph reconstruction verifies at 0.093985748465025, and its gap from the paper is disclosed rather than presented as source data.

Proven floor√(A(K)/(π·23))no arrangement can go below 0.077412566076258 · the record stands 21.4% above itHere K is the verifier's fixed-grid equilateral triangle and A(K) is its area. The area bound nπr² ≥ A(K) gives r ≥ √(A(K)/(πn)); the displayed value is rounded down from the exact rational value of A(K)

Formal definition

ContainerThe closed convex polygon with vertices (0,0), (1,0), and (0.5,0.866025404), the site's unambiguous nine-decimal representative of an equilateral triangle
SubmissionExactly n distinct points inside the container
ScoreTake the distance to the nearest centre at every place in the container, then the largest of those distances; rational Voronoi polygons make the verifier exact $r (P) = \max_{x \in K} \min_{i} ‖ x - p_{i} ‖$
ObjectiveMake the covering radius as small as possible $\min_{P} r (P)$

Open the full editor ↗

VERIFIED CONSTRUCTION23 equal discs of shared radius 0.093986; the marked place is the hardest to reach, and it is what sets the radius

Getting a feel for it

Corners and interior compete for centres

The three sharp corners demand attention, but pushing every centre outward leaves a hole in the middle. The best configurations are usually not a simple equally spaced triangular grid.

Published records and verifiable constructions are included

Nurmela (2000) collected and extended high-precision constructions for n=2–36. The site reconstructs n=7–36 from the contact graphs in Figures 2–4, showing both the paper's continuous value and the verifier-exact nine-decimal certificate. The n=9 and n=10 rows are proved optimal; the other n≤36 rows remain challengeable known bests. The paper explicitly reports no searches beyond n=36, so n=37–40 ship only transparent site starts obtained by filling successive farthest holes of n=36, without invented external records.

Source

Current leader

0.093985748465025

the covering radius

Answer sourceKari J. Nurmela

Solution methodPublished reference construction

Challenge this record ↗

ANSWER FORMAT

How to write your answer

The base runs from (0,0) to (1,0), with apex (0.5, 0.866025404). This explicitly defined nine-decimal representative of the equilateral triangle is the boundary the verifier actually uses.

Submit points: exactly n centres, with each coordinate a decimal string of at most nine places. The verifier computes the shared radius; smaller is better.

The current leader's answer

{
  "points": [
    [
      "0.952998291",
      "0.081388944"
    ],
    [
      "0.66705074",
      "0.078895956"
    ],
    [
      "0.812047669",
      "0.160890177"
    ],
    [
      "0.545314764",
      "0.061971578"
    ],
    [
      "0.639121832",
      "0.208729938"
    ],
    [
      "0.397841768",
      "0.054158046"
    ],
    [
      "0.778938327",
      "0.287713735"
    ],
    [
      "0.240947868",
      "0.049197152"
    ],
    [
      "0.481314102",
      "0.201834421"
    ],
    [
      "0.323031116",
      "0.195076371"
    ],
    [
      "0.080433451",
      "0.048618728"
    ],
    [
      "0.685690675",
      "0.384739653"
    ],
    [
      "0.558682749",
      "0.350164003"
    ],
    [
      "0.163369223",
      "0.190173661"
    ],
    [
      "0.400703761",
      "0.342974464"
    ],
    [
      "0.642152931",
      "0.500959339"
    ],
    [
      "0.242368464",
      "0.336126112"
    ],
    [
      "0.485537406",
      "0.489534339"
    ],
    [
      "0.325920142",
      "0.482806726"
    ],
    [
      "0.57245166",
      "0.634345312"
    ],
    [
      "0.411236451",
      "0.628409835"
    ],
    [
      "0.496631665",
      "0.772100034"
    ],
    [
      "0.812061617",
      "0.003089237"
    ]
  ]
}

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

Instance parameters

{
  "n": 23
}

The current leader's answer

{
  "points": [
    [
      "0.952998291",
      "0.081388944"
    ],
    [
      "0.66705074",
      "0.078895956"
    ],
    [
      "0.812047669",
      "0.160890177"
    ],
    [
      "0.545314764",
      "0.061971578"
    ],
    [
      "0.639121832",
      "0.208729938"
    ],
    [
      "0.397841768",
      "0.054158046"
    ],
    [
      "0.778938327",
      "0.287713735"
    ],
    [
      "0.240947868",
      "0.049197152"
    ],
    [
      "0.481314102",
      "0.201834421"
    ],
    [
      "0.323031116",
      "0.195076371"
    ],
    [
      "0.080433451",
      "0.048618728"
    ],
    [
      "0.685690675",
      "0.384739653"
    ],
    [
      "0.558682749",
      "0.350164003"
    ],
    [
      "0.163369223",
      "0.190173661"
    ],
    [
      "0.400703761",
      "0.342974464"
    ],
    [
      "0.642152931",
      "0.500959339"
    ],
    [
      "0.242368464",
      "0.336126112"
    ],
    [
      "0.485537406",
      "0.489534339"
    ],
    [
      "0.325920142",
      "0.482806726"
    ],
    [
      "0.57245166",
      "0.634345312"
    ],
    [
      "0.411236451",
      "0.628409835"
    ],
    [
      "0.496631665",
      "0.772100034"
    ],
    [
      "0.812061617",
      "0.003089237"
    ]
  ]
}

Submit points: exactly n centres, with each coordinate a decimal string of at most nine places. The verifier computes the shared radius; smaller is better. · Verifier v1.0.0

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