P74 · Extremal configurations · Classic · Applied frontier · Hard

Covering an equilateral triangle with n equal discs · n = 21

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 = 21

ObjectiveMinimize the covering radius

Best known, unproven0.0962250448649376274The construction by Kari J. Nurmela appears in Figure 3 of Nurmela (2000); the paper's continuous value is 0.0962250448649376274. Optimality remains open; the site's nine-decimal reconstruction of the published contact graph verifies at 0.096225045409067.

Proven floor√(A(K)/(π·21))no arrangement can go below 0.081015055593670 · the record stands 18.8% 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 CONSTRUCTION21 equal discs of shared radius 0.096225; 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

0.096225045409067

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": [
    [
      "0.083333334",
      "0.048112521"
    ],
    [
      "0.166666666",
      "0.192450087"
    ],
    [
      "0.250000003",
      "0.048112522"
    ],
    [
      "0.249999999",
      "0.336787654"
    ],
    [
      "0.333333334",
      "0.192450088"
    ],
    [
      "0.333333331",
      "0.481125222"
    ],
    [
      "0.41666667",
      "0.048112522"
    ],
    [
      "0.416666666",
      "0.336787656"
    ],
    [
      "0.416666665",
      "0.62546279"
    ],
    [
      "0.500000001",
      "0.19245009"
    ],
    [
      "0.499999998",
      "0.481125224"
    ],
    [
      "0.499999998",
      "0.769800359"
    ],
    [
      "0.583333337",
      "0.048112523"
    ],
    [
      "0.583333333",
      "0.336787658"
    ],
    [
      "0.583333331",
      "0.625462794"
    ],
    [
      "0.666666667",
      "0.192450091"
    ],
    [
      "0.666666665",
      "0.481125227"
    ],
    [
      "0.750000003",
      "0.048112523"
    ],
    [
      "0.749999999",
      "0.33678766"
    ],
    [
      "0.833333333",
      "0.192450092"
    ],
    [
      "0.916666668",
      "0.048112524"
    ]
  ]
}

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

Instance parameters

{
  "n": 21
}

The current leader's answer

{
  "points": [
    [
      "0.083333334",
      "0.048112521"
    ],
    [
      "0.166666666",
      "0.192450087"
    ],
    [
      "0.250000003",
      "0.048112522"
    ],
    [
      "0.249999999",
      "0.336787654"
    ],
    [
      "0.333333334",
      "0.192450088"
    ],
    [
      "0.333333331",
      "0.481125222"
    ],
    [
      "0.41666667",
      "0.048112522"
    ],
    [
      "0.416666666",
      "0.336787656"
    ],
    [
      "0.416666665",
      "0.62546279"
    ],
    [
      "0.500000001",
      "0.19245009"
    ],
    [
      "0.499999998",
      "0.481125224"
    ],
    [
      "0.499999998",
      "0.769800359"
    ],
    [
      "0.583333337",
      "0.048112523"
    ],
    [
      "0.583333333",
      "0.336787658"
    ],
    [
      "0.583333331",
      "0.625462794"
    ],
    [
      "0.666666667",
      "0.192450091"
    ],
    [
      "0.666666665",
      "0.481125227"
    ],
    [
      "0.750000003",
      "0.048112523"
    ],
    [
      "0.749999999",
      "0.33678766"
    ],
    [
      "0.833333333",
      "0.192450092"
    ],
    [
      "0.916666668",
      "0.048112524"
    ]
  ]
}

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

DISCUSSION

Discussion

Talk strategy, share methods, ask why you are stuck. Posts carry your public byline, the same name your records use; the #number after it is the account's signup ordinal, so a name cannot be worn by someone else. The floor is earned: break a record once, anywhere, and it is yours for good. New posts appear after an automated review.

No threads yet. Be the first to talk about this problem?