MinMax Arena
中/EN
P87 · Packing and covering · Classic · Hard

Equal discs covering a disc · n = 22

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

Instancen = 22
ObjectiveMaximize covered disc radius
Best known, unproven3.948Source-precision interval: 3.948–3.949 (values inside match)exhibited site certificate 3.948644545Compiled by Erich Friedman; n=22 Found by Jeremy Tan in 2018. Source expression: r = 3.948+.

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 = 22, 3.948644545
Current leader

3.948644545

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": "1.411181812",
      "y": "0.802063730"
    },
    {
      "turn": "0.000000000",
      "x": "1.411120465",
      "y": "1.198063631"
    },
    {
      "turn": "0.000000000",
      "x": "1.101477042",
      "y": "1.444917585"
    },
    {
      "turn": "0.000000000",
      "x": "1.101614865",
      "y": "0.555113846"
    },
    {
      "turn": "0.000000000",
      "x": "0.999999967",
      "y": "0.999999982"
    },
    {
      "turn": "0.000000000",
      "x": "0.543656619",
      "y": "0.999929322"
    },
    {
      "turn": "0.000000000",
      "x": "0.715529817",
      "y": "0.643172307"
    },
    {
      "turn": "0.000000000",
      "x": "0.715419312",
      "y": "1.356739542"
    },
    {
      "turn": "0.000000000",
      "x": "1.512597535",
      "y": "1.642981238"
    },
    {
      "turn": "0.000000000",
      "x": "1.512796698",
      "y": "0.357177584"
    },
    {
      "turn": "0.000000000",
      "x": "0.259075958",
      "y": "1.356668870"
    },
    {
      "turn": "0.000000000",
      "x": "0.259186470",
      "y": "0.643101647"
    },
    {
      "turn": "0.000000000",
      "x": "1.822302310",
      "y": "1.000127371"
    },
    {
      "turn": "0.000000000",
      "x": "1.203229738",
      "y": "0.110227702"
    },
    {
      "turn": "0.000000000",
      "x": "0.817144703",
      "y": "0.198286174"
    },
    {
      "turn": "0.000000000",
      "x": "1.822240958",
      "y": "1.396127276"
    },
    {
      "turn": "0.000000000",
      "x": "1.822363644",
      "y": "0.604127460"
    },
    {
      "turn": "0.000000000",
      "x": "1.202954095",
      "y": "1.889835188"
    },
    {
      "turn": "0.000000000",
      "x": "0.816896371",
      "y": "1.801657141"
    },
    {
      "turn": "0.000000000",
      "x": "0.431059655",
      "y": "0.286344634"
    },
    {
      "turn": "0.000000000",
      "x": "0.430838636",
      "y": "1.713479085"
    },
    {
      "turn": "0.000000000",
      "x": "0.087313266",
      "y": "0.999858649"
    }
  ],
  "radius": "0.253251461"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 22,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "1.411181812",
      "y": "0.802063730"
    },
    {
      "turn": "0.000000000",
      "x": "1.411120465",
      "y": "1.198063631"
    },
    {
      "turn": "0.000000000",
      "x": "1.101477042",
      "y": "1.444917585"
    },
    {
      "turn": "0.000000000",
      "x": "1.101614865",
      "y": "0.555113846"
    },
    {
      "turn": "0.000000000",
      "x": "0.999999967",
      "y": "0.999999982"
    },
    {
      "turn": "0.000000000",
      "x": "0.543656619",
      "y": "0.999929322"
    },
    {
      "turn": "0.000000000",
      "x": "0.715529817",
      "y": "0.643172307"
    },
    {
      "turn": "0.000000000",
      "x": "0.715419312",
      "y": "1.356739542"
    },
    {
      "turn": "0.000000000",
      "x": "1.512597535",
      "y": "1.642981238"
    },
    {
      "turn": "0.000000000",
      "x": "1.512796698",
      "y": "0.357177584"
    },
    {
      "turn": "0.000000000",
      "x": "0.259075958",
      "y": "1.356668870"
    },
    {
      "turn": "0.000000000",
      "x": "0.259186470",
      "y": "0.643101647"
    },
    {
      "turn": "0.000000000",
      "x": "1.822302310",
      "y": "1.000127371"
    },
    {
      "turn": "0.000000000",
      "x": "1.203229738",
      "y": "0.110227702"
    },
    {
      "turn": "0.000000000",
      "x": "0.817144703",
      "y": "0.198286174"
    },
    {
      "turn": "0.000000000",
      "x": "1.822240958",
      "y": "1.396127276"
    },
    {
      "turn": "0.000000000",
      "x": "1.822363644",
      "y": "0.604127460"
    },
    {
      "turn": "0.000000000",
      "x": "1.202954095",
      "y": "1.889835188"
    },
    {
      "turn": "0.000000000",
      "x": "0.816896371",
      "y": "1.801657141"
    },
    {
      "turn": "0.000000000",
      "x": "0.431059655",
      "y": "0.286344634"
    },
    {
      "turn": "0.000000000",
      "x": "0.430838636",
      "y": "1.713479085"
    },
    {
      "turn": "0.000000000",
      "x": "0.087313266",
      "y": "0.999858649"
    }
  ],
  "radius": "0.253251461"
}

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

Talk strategy, share methods, ask why you are stuck. Every signed-in user can post. 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. New posts appear after an automated review.

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