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P87 · Packing and covering · Classic · Hard

Equal discs covering a disc · n = 13

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

Instancen = 13
ObjectiveMaximize covered disc radius
Best known, unproven2.884Source-precision interval: 2.884–2.885 (values inside match)exhibited site certificate 2.884791872Compiled by Erich Friedman; n=13 Found by Jeremy Tan in 2018. Source expression: r = 2.884+.

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 = 13, 2.884791872
Current leader

2.884791872

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": "0.756487555",
      "y": "1.004250563"
    },
    {
      "turn": "0.000000000",
      "x": "1.175013270",
      "y": "0.757429842"
    },
    {
      "turn": "0.000000000",
      "x": "1.183372283",
      "y": "1.236314388"
    },
    {
      "turn": "0.000000000",
      "x": "0.766947234",
      "y": "1.673617960"
    },
    {
      "turn": "0.000000000",
      "x": "0.743580073",
      "y": "0.334925893"
    },
    {
      "turn": "0.000000000",
      "x": "0.353021900",
      "y": "0.510163547"
    },
    {
      "turn": "0.000000000",
      "x": "0.370511223",
      "y": "1.512117460"
    },
    {
      "turn": "0.000000000",
      "x": "1.706594104",
      "y": "1.517275788"
    },
    {
      "turn": "0.000000000",
      "x": "1.688110961",
      "y": "0.458379238"
    },
    {
      "turn": "0.000000000",
      "x": "1.776372488",
      "y": "0.986448263"
    },
    {
      "turn": "0.000000000",
      "x": "1.211238665",
      "y": "1.847292364"
    },
    {
      "turn": "0.000000000",
      "x": "1.181539618",
      "y": "0.145851667"
    },
    {
      "turn": "0.000000000",
      "x": "0.148492025",
      "y": "1.014863256"
    }
  ],
  "radius": "0.346645458"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 13,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "0.756487555",
      "y": "1.004250563"
    },
    {
      "turn": "0.000000000",
      "x": "1.175013270",
      "y": "0.757429842"
    },
    {
      "turn": "0.000000000",
      "x": "1.183372283",
      "y": "1.236314388"
    },
    {
      "turn": "0.000000000",
      "x": "0.766947234",
      "y": "1.673617960"
    },
    {
      "turn": "0.000000000",
      "x": "0.743580073",
      "y": "0.334925893"
    },
    {
      "turn": "0.000000000",
      "x": "0.353021900",
      "y": "0.510163547"
    },
    {
      "turn": "0.000000000",
      "x": "0.370511223",
      "y": "1.512117460"
    },
    {
      "turn": "0.000000000",
      "x": "1.706594104",
      "y": "1.517275788"
    },
    {
      "turn": "0.000000000",
      "x": "1.688110961",
      "y": "0.458379238"
    },
    {
      "turn": "0.000000000",
      "x": "1.776372488",
      "y": "0.986448263"
    },
    {
      "turn": "0.000000000",
      "x": "1.211238665",
      "y": "1.847292364"
    },
    {
      "turn": "0.000000000",
      "x": "1.181539618",
      "y": "0.145851667"
    },
    {
      "turn": "0.000000000",
      "x": "0.148492025",
      "y": "1.014863256"
    }
  ],
  "radius": "0.346645458"
}

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

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