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P87 · Packing and covering · Classic · Hard

Equal discs covering a disc · n = 23

Cover the largest possible disc with n unit-radius discs. Maximise the covered disc radius.

Instancen = 23
ObjectiveMaximize covered disc radius
Best known, unproven4.000Source-precision interval: 4–4.001 (values inside match)exhibited site certificate 4.000739688Compiled by Erich Friedman; n=23 Found by Jeremy Tan in 2018. Source expression: r = 4.000+.

Formal definition

  • ContainerThe editor and certificate normalize to the closed unit disc centred at (1,1). This is a coordinate convention only; scores use the literature's unit-piece scale. Boundary contact is allowed.
  • SubmissionSubmit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.
  • ConstraintsThe n distinct centres lie in the target disc. Covering discs may overlap and protrude, but must cover every target point.
  • PrecisionInput decimals are exact rationals; regular-polygon vertices are algebraic, not rounded templates. Feasibility has no floating-point tolerance. A finite-decimal certificate is not a continuous optimality proof.
  • ObjectiveCovered disc radius R = 1/r, larger is better.
Open the full editor ↗
VERIFIED CONSTRUCTIONVERIFIED CONSTRUCTION

Getting a feel for it

Source and certificate

External targets are converted with their printed precision; decimal coordinate certificates are separate. Reconstruction loss never lowers the literature target, and a source construction is not automatically an optimality proof.

Source
The record-holding arrangement for Equal discs covering a disc n = 23, 4.000739688
Current leader

4.000739688

covered disc radius

Matches the best known
Answer sourceJeremy Tan
Solution methodPublished reference construction
Challenge this record ↗Submit a proof / idea ↓Share a proof or idea in the discussion. Accepted contributions can earn proof points.
Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

ANSWER FORMAT

How to write your answer

the closed unit disc centred at (1,1)

Submit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "0.870780029",
      "y": "0.565617776"
    },
    {
      "turn": "0.000000000",
      "x": "0.889771564",
      "y": "1.439585500"
    },
    {
      "turn": "0.000000000",
      "x": "1.435910456",
      "y": "0.990527557"
    },
    {
      "turn": "0.000000000",
      "x": "1.253933057",
      "y": "0.632240514"
    },
    {
      "turn": "0.000000000",
      "x": "1.269668818",
      "y": "1.356381487"
    },
    {
      "turn": "0.000000000",
      "x": "0.579313144",
      "y": "0.817520463"
    },
    {
      "turn": "0.000000000",
      "x": "0.993879703",
      "y": "1.000132996"
    },
    {
      "turn": "0.000000000",
      "x": "0.587637167",
      "y": "1.200581911"
    },
    {
      "turn": "0.000000000",
      "x": "1.658281669",
      "y": "1.418079221"
    },
    {
      "turn": "0.000000000",
      "x": "1.639498926",
      "y": "0.553719685"
    },
    {
      "turn": "0.000000000",
      "x": "0.491030090",
      "y": "1.645818047"
    },
    {
      "turn": "0.000000000",
      "x": "1.530948112",
      "y": "1.732167732"
    },
    {
      "turn": "0.000000000",
      "x": "0.755818410",
      "y": "0.125151221"
    },
    {
      "turn": "0.000000000",
      "x": "1.498641562",
      "y": "0.245459026"
    },
    {
      "turn": "0.000000000",
      "x": "0.463456213",
      "y": "0.376901224"
    },
    {
      "turn": "0.000000000",
      "x": "1.161420713",
      "y": "1.796682240"
    },
    {
      "turn": "0.000000000",
      "x": "1.126660472",
      "y": "0.197057685"
    },
    {
      "turn": "0.000000000",
      "x": "0.794052323",
      "y": "1.884630203"
    },
    {
      "turn": "0.000000000",
      "x": "1.847009728",
      "y": "1.176120854"
    },
    {
      "turn": "0.000000000",
      "x": "1.838559493",
      "y": "0.787251316"
    },
    {
      "turn": "0.000000000",
      "x": "0.171989328",
      "y": "0.628803914"
    },
    {
      "turn": "0.000000000",
      "x": "0.188895693",
      "y": "1.406814461"
    },
    {
      "turn": "0.000000000",
      "x": "0.173070609",
      "y": "1.017969381"
    }
  ],
  "radius": "0.249953778"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 23,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "0.870780029",
      "y": "0.565617776"
    },
    {
      "turn": "0.000000000",
      "x": "0.889771564",
      "y": "1.439585500"
    },
    {
      "turn": "0.000000000",
      "x": "1.435910456",
      "y": "0.990527557"
    },
    {
      "turn": "0.000000000",
      "x": "1.253933057",
      "y": "0.632240514"
    },
    {
      "turn": "0.000000000",
      "x": "1.269668818",
      "y": "1.356381487"
    },
    {
      "turn": "0.000000000",
      "x": "0.579313144",
      "y": "0.817520463"
    },
    {
      "turn": "0.000000000",
      "x": "0.993879703",
      "y": "1.000132996"
    },
    {
      "turn": "0.000000000",
      "x": "0.587637167",
      "y": "1.200581911"
    },
    {
      "turn": "0.000000000",
      "x": "1.658281669",
      "y": "1.418079221"
    },
    {
      "turn": "0.000000000",
      "x": "1.639498926",
      "y": "0.553719685"
    },
    {
      "turn": "0.000000000",
      "x": "0.491030090",
      "y": "1.645818047"
    },
    {
      "turn": "0.000000000",
      "x": "1.530948112",
      "y": "1.732167732"
    },
    {
      "turn": "0.000000000",
      "x": "0.755818410",
      "y": "0.125151221"
    },
    {
      "turn": "0.000000000",
      "x": "1.498641562",
      "y": "0.245459026"
    },
    {
      "turn": "0.000000000",
      "x": "0.463456213",
      "y": "0.376901224"
    },
    {
      "turn": "0.000000000",
      "x": "1.161420713",
      "y": "1.796682240"
    },
    {
      "turn": "0.000000000",
      "x": "1.126660472",
      "y": "0.197057685"
    },
    {
      "turn": "0.000000000",
      "x": "0.794052323",
      "y": "1.884630203"
    },
    {
      "turn": "0.000000000",
      "x": "1.847009728",
      "y": "1.176120854"
    },
    {
      "turn": "0.000000000",
      "x": "1.838559493",
      "y": "0.787251316"
    },
    {
      "turn": "0.000000000",
      "x": "0.171989328",
      "y": "0.628803914"
    },
    {
      "turn": "0.000000000",
      "x": "0.188895693",
      "y": "1.406814461"
    },
    {
      "turn": "0.000000000",
      "x": "0.173070609",
      "y": "1.017969381"
    }
  ],
  "radius": "0.249953778"
}

Submit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4. · Verifier v1.0.0

DISCUSSION

Discussion

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